199 results on '"Homoclinic orbit"'
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2. Bifurcations and Exact Solutions of the Generalized Radhakrishnan–Kundu–Lakshmanan Equation with the Polynomial Law.
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Yu, Mengke, Chen, Cailiang, and Zhang, Qiuyan
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POLYNOMIALS , *EQUATIONS , *DYNAMICAL systems , *BACKLUND transformations , *ORBITS (Astronomy) - Abstract
In this paper, we investigate the generalized Radhakrishnan–Kundu–Lakshmanan equation with polynomial law using the method of dynamical systems. By using traveling-wave transformation, the model can be converted into a singular integrable traveling-wave system. Then, we discuss the dynamical behavior of the associated regular system and we obtain bifurcations of the phase portraits of the traveling-wave system under different parameter conditions. Finally, under different parameter conditions, we obtain the exact periodic solutions, and the peakon, homoclinic and heteroclinic solutions. [ABSTRACT FROM AUTHOR]
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- 2023
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3. Grazing, Homoclinic Orbits and Chaos in a Single-Loop Feedback System with a Discontinuous Function.
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Horikawa, Yo
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ORBITS (Astronomy) , *DISCONTINUOUS functions , *GRAZING , *HARMONIC oscillators , *RING networks - Abstract
Bifurcations and chaos of a three-dimensional single-loop feedback system with a discontinuous piecewise linear feedback function are examined. Chaotic attractors are generated at the same time of the destabilization of foci accompanied with grazing. Multiple periodic solutions are connected with homoclinic orbits based at a pseudo saddle-focus, which satisfies the condition of Shil'nikov chaos formally. The generation of chaotic oscillations is shown in a circuit experiment on a linear ring oscillator with a comparator. The homoclinic bifurcations and chaos are also shown in a ring neural network with a nonmonotonic neuron. [ABSTRACT FROM AUTHOR]
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- 2023
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4. Grazing Bifurcations, Homoclinic Orbits and Chaos in a Ring Network of Linear Neuron-Like Elements with a Single Nonmonotonic Piecewise Constant Output Function.
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Horikawa, Yo
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RING networks , *ORBITS (Astronomy) , *GRAZING - Abstract
The bifurcations and chaos of a ring of three unidirectionally coupled neuron-like elements are examined as a minimal chaotic neural network. The output function of one neuron is nonmonotonic and piecewise constant while those of the other two neurons are linear. Two kinds of nonmonotonic output functions are considered and it is shown that periodic solutions undergo grazing bifurcations owing to discontinuity in the nonmonotonic functions. Chaotic attractors are created directly through a grazing bifurcation and homoclinic orbits based at pseudo steady states are generated. It is shown that homoclinic/heteroclinic orbits satisfying the condition of Shil'nikov chaos are caused by overshoot in the nonmonotonic functions. [ABSTRACT FROM AUTHOR]
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- 2023
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5. On Shilnikov attractors of three-dimensional flows and maps.
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Bakhanova, Yu. V., Gonchenko, S. V., Gonchenko, A. S., Kazakov, A. O., and Samylina, E. A.
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THREE-dimensional flow , *POINCARE maps (Mathematics) , *ATTRACTORS (Mathematics) - Abstract
We describe scenarios for the emergence of Shilnikov attractors, i.e. strange attractors containing a saddle-focus with two-dimensional unstable manifold, in the case of three-dimensional flows and maps. The presented results are illustrated with various specific examples. [ABSTRACT FROM AUTHOR]
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- 2023
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6. The 0:1 resonance bifurcation associated with the supercritical Hamiltonian pitchfork bifurcation.
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Zhou, Xing
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RESONANCE , *DUFFING equations , *BIFURCATION diagrams , *DEGREES of freedom , *ORBITS (Astronomy) , *MATHEMATICS - Abstract
We consider the non-semisimple 0:1 resonance (i.e. the unperturbed equilibrium has two purely imaginary eigenvalues ± i α ( α ∈ R and α > 0) and a non-semisimple double-zero one) Hamiltonian bifurcation with one distinguished parameter, which corresponds to the supercritical Hamiltonian pitchfork bifurcation. Based on BCKV singularity theory established by [H.W. Broer, S. -N. Chow, Y. Kim, and G. Vegter, A normally elliptic Hamiltonian bifurcation, Z. Angew. Math. Phys. 44 (3) (1993), pp. 389–432], this bifurcation essentially triggered by the reversible universal unfolding M = 1 2 p 2 + 1 4 q 4 + (λ + I 1) q 2 with respect to BCKV-restricted morphisms of the planar non-semisimple singularity 1 2 p 2 + 1 4 q 4 (the I 1 is regarded as distinguished parameter with respect to the external parameter λ). We first give the plane bifurcation diagram of the integrable Hamiltonian on each level of integral in detail, which is related to the usual supercritical Hamiltonian pitchfork bifurcation. Then, we use the S 1 -symmetry generated by the additional pair of imaginary eigenvalues ± i α to reconstruct the above plane bifurcation phenomenon caused by the zero eigenvalue pair into the case with two degrees of freedom. Finally, we prove the persistence of typical bifurcation scenarios (e.g. 2-dimensional invariant tori and the symmetric homoclinic orbit) under the small Hamiltonian perturbations, as proposed by [H.W. Broer, S. -N. Chow, Y. Kim, and G. Vegter, A normally elliptic Hamiltonian bifurcation, Z. Angew. Math. Phys. 44 (3) (1993), pp. 389–432]. An example system (the coupled Duffing oscillator) with strong linear coupling and weak local nonlinearity is given for this bifurcation. [ABSTRACT FROM AUTHOR]
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- 2023
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7. Scenarios for the creation of hyperchaotic attractors in 3D maps.
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Shykhmamedov, Aikan, Karatetskaia, Efrosiniia, Kazakov, Alexey, and Stankevich, Nataliya
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INVARIANT manifolds , *ORBITS (Astronomy) , *BIFURCATION diagrams , *POINCARE maps (Mathematics) , *LYAPUNOV exponents , *DIFFEOMORPHISMS - Abstract
We study bifurcation mechanisms for the appearance of hyperchaotic attractors in three-dimensional diffeomorphisms, i.e. such attractors whose orbits have two positive Lyapunov exponents in numerical experiments. In particular, periodic orbits belonging to the attractor should have two-dimensional unstable invariant manifolds. We discuss several bifurcation scenarios which create such periodic orbits inside the attractor. This includes cascades of supercritical period-doubling bifurcations of saddle periodic orbits and supercritical Neimark–Sacker bifurcations of stable periodic orbits, as well as various combinations of these cascades. These scenarios are illustrated by an example of the three-dimensional Mirá map. [ABSTRACT FROM AUTHOR]
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- 2023
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8. Homoclinic Bifurcations in a Class of Three-Dimensional Symmetric Piecewise Affine Systems.
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Liu, Ruimin, Liu, Minghao, and Wu, Tiantian
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PIECEWISE affine systems , *POINCARE maps (Mathematics) , *LIMIT cycles , *DYNAMICAL systems , *INVARIANT sets , *ORBITS (Astronomy) - Abstract
Many physical and engineering systems have certain symmetric properties. Homoclinic orbits play an important role in studying the global dynamics of dynamical systems. This paper focuses on the existence and bifurcations of homoclinic orbits to a saddle in a class of three-dimensional one-parameter three-zone symmetric piecewise affine systems. Based on the analysis of the Poincaré maps, the systems have two types of limit cycles and do not have chaotic invariant sets near the homoclinic orbits. In addition, the paper provides a constant D to study the homoclinic bifurcations to limit cycles for the case | λ 1 | = λ 3 . Two examples with simulations of the homoclinic orbits and the limit cycles are given to illustrate the effectiveness of the results. [ABSTRACT FROM AUTHOR]
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- 2023
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9. Analytic and Algebraic Conditions for Bifurcations of Homoclinic Orbits II: Reversible Systems.
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Yagasaki, Kazuyuki
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ORBITS (Astronomy) , *GALOIS theory , *HAMILTONIAN systems , *SYSTEMS theory - Abstract
Following Part I, we consider a class of reversible systems and study bifurcations of homoclinic orbits to hyperbolic saddle equilibria. Here we concentrate on the case in which homoclinic orbits are symmetric, so that only one control parameter is enough to treat their bifurcations, as in Hamiltonian systems. First, we modify and extend arguments of Part I to show in a form applicable to general systems discussed there that if such bifurcations occur in four-dimensional systems, then variational equations around the homoclinic orbits are integrable in the meaning of differential Galois theory under some conditions. We next extend the Melnikov method of Part I to reversible systems and obtain some theorems on saddle-node, transcritical and pitchfork bifurcations of symmetric homoclinic orbits. We illustrate our theory for a four-dimensional system, and demonstrate the theoretical results by numerical ones. [ABSTRACT FROM AUTHOR]
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- 2023
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10. Minimal Chaotic Networks of Linear Neuron-Like Elements with Single Rectification: Three Prototypes.
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Horikawa, Yo
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JACOBIAN matrices , *VECTOR spaces , *PHASE space , *ORBITS (Astronomy) , *PROTOTYPES , *NEURAL circuitry - Abstract
Chaotic oscillations induced by single rectification in networks of linear neuron-like elements are examined on three prototype models: one nonautonomous system and two autonomous systems. The first is a system of coupled neurons with periodic input; the second is a system of three coupled neurons with six couplings; the third is a ring of four unidirectionally coupled neurons with one reverse coupling. In each system, the output function of one neuron is ramp and that of the others is linear. Each system is piecewise linear and the phase space is separated into two domains by a single border. Steady states, periodic solutions and homoclinic orbits are derived rigorously and their stability is evaluated with the eigenvalues of the Jacobian matrices. The bifurcation analysis of the three systems shows that chaotic attractors could be generated through cascades of period-doubling bifurcations of periodic solutions. [ABSTRACT FROM AUTHOR]
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- 2023
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11. Chaos analysis for a class of impulse Duffing-van der Pol system.
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Li, Shuqun and Zhou, Liangqiang
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BIFURCATION diagrams , *COMPUTER simulation , *TRANSVERSAL lines , *NONLINEAR oscillators , *ORBITS (Astronomy) , *MOTION - Abstract
Chaotic dynamics of an impulse Duffing-van der Pol system is studied in this paper. With the Melnikov method, the existence condition of transversal homoclinic point is obtained, and chaos threshold is presented. In addition, numerical simulations including phase portraits and time histories are carried out to verify the analytical results. Bifurcation diagrams are also given, from which it can be seen that the system may undergo chaotic motions through period doubling bifurcations. [ABSTRACT FROM AUTHOR]
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- 2023
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12. Existence of homoclinic orbit of Shilnikov type and the application in Rössler system.
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Ding, Yuting and Zheng, Liyuan
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ORBITS (Astronomy) , *MOUNTAIN pass theorem , *CHAOS theory , *CLINICS - Abstract
In this paper, we modify the methods of Zhou et al. (2004) and Shang and Han (2005) associated with proving the existence of a homoclinic orbit of Shilnikov type. We construct the series expressions of the solution based at a saddle-focus on stable and unstable manifolds, and give the sufficient conditions of the existence of homoclinic orbit and spiral chaos. Then, we consider the Rössler system with the typical parameters under which the system exhibits chaotic behavior. Using our modified method, we verify that there exists a homoclinic orbit of Shilnikov type in the Rössler system with a group of typical parameters, and prove the existence of spiral chaos by using the Shilnikov criterion, and we carry out numerical simulations to support the analytic results. [ABSTRACT FROM AUTHOR]
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- 2023
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13. Homoclinic Chaos in a Four-Dimensional Manifold Piecewise Linear System.
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Huang, Qiu, Liu, Yongjian, Li, Chunbiao, and Liu, Aimin
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LINEAR systems , *ORBITS (Astronomy) , *CHAOS synchronization , *COMPUTER simulation - Abstract
The existence of homoclinic orbits is discussed analytically for a class of four-dimensional manifold piecewise linear systems with one switching manifold. An interesting phenomenon is found, that is, under the same parameter setting, homoclinic orbits and chaos appear simultaneously in the system. In addition, homoclinic chaos can be suppressed to a periodic orbit by adding a nonlinear control switch with memory. These theoretical results are illustrated with numerical simulations. [ABSTRACT FROM AUTHOR]
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- 2022
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14. Odd and Even Functions in the Design Problem of New Chaotic Attractors.
- Author
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Belozyorov, Vasiliy Ye. and Volkova, Svetlana A.
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ORDINARY differential equations , *STATE feedback (Feedback control systems) , *AUTONOMOUS differential equations , *ORBITS (Astronomy) , *LORENZ equations , *LIMIT cycles , *QUADRATIC forms - Abstract
Let ⊂ ℝ n be a chaotic attractor generated by a quadratic system of ordinary differential equations ẋ = f (x). A method for constructing new chaotic attractors based on the attractor is proposed. The idea of the method is to replace the state vector x = (x 1 , ... , x n) T located on the right side of the original system with new vector u (x) ; where u (x) = K ⋅ (h 1 (x 1) , ... , h n (x n)) T , K ∈ ℝ n × n , and h i (x i) are odd power functions; i = 1 , ... , n. (In other words, a state feedback x → u (x) is introduced into the right side of the system under study: ẋ = f (x) → ẋ = f (u (x)).) As a result, the newly obtained system generates new chaotic attractors, which are topologically not equivalent (generally speaking) to the attractor . In addition, for an antisymmetric neural ODE system with a homoclinic orbit connected at a saddle point, the conditions for the occurrence of chaotic dynamics are found. [ABSTRACT FROM AUTHOR]
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- 2022
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15. Bifurcations and Chaos in Three-Coupled Ramp-Type Neurons.
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Horikawa, Yo
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POINCARE maps (Mathematics) , *NEURONS , *JACOBIAN matrices , *CHAOS theory , *ORBITS (Astronomy) , *NEURAL circuitry - Abstract
The bifurcations and chaos in autonomous systems of two- and three-coupled ramp-type neurons are considered. An asymmetric piecewise linear function is employed for the output function of neurons in order to examine changes in the bifurcations from a sigmoid output function to a ramp output function. Steady solutions in the systems are obtained exactly and they undergo discontinuous bifurcations because the systems are piecewise linear. Periodic solutions and homoclinic/heteroclinic orbits in the systems are obtained by connecting local solutions in linear domains at borders and solving transcendental equations. The bifurcations of the periodic solutions are calculated with the Poincaré maps and the Jacobian matrices, which are also derived rigorously. A stable periodic solution in a two-neuron oscillator of the Wilson–Cowan type with three couplings remains in the case of a ramp neuron. A chaotic attractor of Rössler type emerges in a network of three ramp neurons with six couplings, which is due to two saddle-focuses. The network consists of the two-neuron oscillator and one bypass neuron connected through three couplings. One-dimensional Poincaré maps show the generation of the chaotic attractor through a cascade of period-doubling bifurcations. Further, multiple homoclinic orbits based at a saddle are generated from the destabilization of two focuses when asymmetry in the output function is large. This homoclinicity causes qualitative change in the bifurcations of the periodic solutions as the output function of neurons changes from sigmoid to ramp. [ABSTRACT FROM AUTHOR]
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- 2022
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16. Global Bifurcation Behaviors and Control in a Class of Bilateral MEMS Resonators.
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Zhu, Yijun and Shang, Huilin
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MEMS resonators , *BIFURCATION theory , *POTENTIAL well , *BIFURCATION diagrams , *LIMIT cycles , *PERFORMANCE theory - Abstract
The investigation of global bifurcation behaviors the vibrating structures of micro-electromechanical systems (MEMS) has received substantial attention. This paper considers the vibrating system of a typical bilateral MEMS resonator containing fractional functions and multiple potential wells. By introducing new variations, the Melnikov method is applied to derive the critical conditions for global bifurcations. By engaging in the fractal erosion of safe basin to depict the phenomenon pull-in instability intuitively, the point-mapping approach is used to present numerical simulations which are in close agreement with the analytical prediction, showing the validity of the analysis. It is found that chaos and pull-in instability, two initial-sensitive phenomena of MEMS resonators, can be due to homoclinic bifurcation and heteroclinic bifurcation, respectively. On this basis, two types of delayed feedback are proposed to control the complex dynamics successively. Their control mechanisms and effect are then studied. It follows that under a positive gain coefficient, delayed position feedback and delayed velocity feedback can both reduce pull-in instability; nevertheless, to suppress chaos, only the former can be effective. The results may have some potential value in broadening the application fields of global bifurcation theory and improving the performance reliability of capacitive MEMS devices. [ABSTRACT FROM AUTHOR]
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- 2022
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17. A KdV–SIR Equation and Its Analytical Solutions for Solitary Epidemic Waves.
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Paxson, Wei and Shen, Bo-Wen
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ANALYTICAL solutions , *EPIDEMICS , *EQUATIONS , *ORBITS (Astronomy) , *LORENZ equations , *ECONOMIC impact - Abstract
Accurate predictions for the spread and evolution of epidemics have significant societal and economic impacts. The temporal evolution of infected (or dead) persons has been described as an epidemic wave with an isolated peak and tails. Epidemic waves have been simulated and studied using the classical SIR model that describes the evolution of susceptible (S), infected (I), and recovered (R) individuals. To illustrate the fundamental dynamics of an epidemic wave, the dependence of solutions on parameters, and the dependence of predictability horizons on various types of solutions, we propose a Korteweg–de Vries (KdV)–SIR equation and obtain its analytical solutions. Among classical and simplified SIR models, our KdV–SIR equation represents the simplest system that produces a solution with both exponential and oscillatory components. The KdV–SIR model is mathematically identical to the nondissipative Lorenz 1963 model and the KdV equation in a traveling-wave coordinate. As a result, the dynamics of an epidemic wave and its predictability can be understood by applying approaches used in nonlinear dynamics, and by comparing the aforementioned systems. For example, a typical solitary wave solution is a homoclinic orbit that connects a stable and an unstable manifold at the saddle point within the I – I ′ space. The KdV–SIR equation additionally produces two other types of solutions, including oscillatory and unbounded solutions. The analysis of two critical points makes it possible to reveal the features of solutions near a turning point. Using analytical solutions and hypothetical observed data, we derive a simple formula for determining predictability horizons, and propose a method for predicting timing for the peak of an epidemic wave. [ABSTRACT FROM AUTHOR]
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- 2022
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18. Homoclinic orbits and Jacobi stability on the orbits of Maxwell–Bloch system.
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Liu, Yongjian, Chen, Haimei, Lu, Xiaoting, Feng, Chunsheng, and Liu, Aimin
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ORBITS (Astronomy) , *CHAOS theory , *HAMILTON-Jacobi equations , *ORBIT method - Abstract
In this paper, we analytically and geometrically investigate the complexity of Maxwell–Bloch system by giving new insight. In the first place, the existence of homoclinic orbits is rigorously proved by means of the generalized Melnikov method. More precisely, for 6a−2b>c and d>0, it is certified analytically that Maxwell–Bloch system has two nontransverse homoclinic orbits. Secondly, Jacobi stability on the orbits of Maxwell–Bloch system is examined in view point of Kosambi–Cartan–Chern theory (KCC-theory). In other words, in the light of the deviation curvature tensor of the five corresponding invariant associated to the reformulated Maxwell–Bloch system, we further proved that Jacobi stability of all equilibria under appropriate parameters. Moreover, the deviation vector, as well as the curvature of the deviation vector near equilibrium points, is focused to interpret the chaotic behavior of Maxwell–Bloch system in Finsler geometry. [ABSTRACT FROM AUTHOR]
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- 2022
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19. Bifurcations and Exact Solutions of the Nonlinear Schrödinger Equation with Nonlinear Dispersion.
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Zhang, Qiuyan, Zhou, Yuqian, and Li, Jibin
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NONLINEAR Schrodinger equation , *DYNAMICAL systems , *DISPERSION (Chemistry) - Abstract
The nonlinear Schrödinger equation with nonlinear dispersion is investigated. By using the bifurcation-theoretic method of planar dynamical systems, bifurcations of phase portraits and exact periodic solutions, homoclinic and heteroclinic solutions, periodic peakons as well as compacton solutions for this planar dynamical system are obtained. Under different parameter conditions, solutions can be exactly obtained. Fifteen exact explicit solutions of the traveling wave system are derived. [ABSTRACT FROM AUTHOR]
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- 2022
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20. NEW GEOMETRIC VIEWPOINTS TO CHEN CHAOTIC SYSTEM.
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XIAOTING LU, YONGJIAN LIU, AIMIN LIU, and CHUNSHENG FENG
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PARAMETERS (Statistics) , *MATHEMATICAL equivalence , *INTEGRALS , *MATHEMATICS theorems , *DIFFERENTIAL equations - Abstract
This paper presents new geometric viewpoints to Chen chaotic system. Firstly, the existence of two nontransverse homoclinic orbits in Chen system is rigorously proved beyond the classical parameters. Secondly, combined with the theory of tangent bundle, a new geometric viewpoint is given to explore chaos mechanism of Chen system. The fundamental geometric definition of tangent bundle and the essential role of nonlinear connection between the tangent space and the base space are described. By introducing the geometrical viewpoints of second order system governed by Lie-Poisson equation, some geometric invariants of Chen system can be obtained. Furthermore, the torsion tensor as one of the geometric invariants is obtained, and it gives the geometrical interpretation to the chaotic behaviour of Chen system. Finally, the torsion tensor of Chen system and Lorenz system are also compared. The obtaining results show that torsion tensor change will lead the Chen system from periodic to chaotic, which is not found in Lorenz system. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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21. Solitary Wave Solutions of Delayed Coupled Higgs Field Equation.
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Ji, Shu Guan and Li, Xiao Wan
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BOLTZMANN'S equation , *SINGULAR perturbations , *ORDINARY differential equations , *INVARIANT manifolds , *PERTURBATION theory , *HIGGS bosons , *EQUATIONS , *HAMILTONIAN systems - Abstract
This paper is devoted to the study of the solitary wave solutions for the delayed coupled Higgs field equation { u t t − u x x − α u + β f ∗ u | u | 2 − 2 u v − τ u ( | u | 2) x = 0 , v t t + v x x − β ( | u | 2) x x = 0. We first establish the existence of solitary wave solutions for the corresponding equation without delay and perturbation by using the Hamiltonian system method. Then we consider the persistence of solitary wave solutions of the delayed coupled Higgs field equation by using the method of dynamical system, especially the geometric singular perturbation theory, invariant manifold theory and Fredholm theory. According to the relationship between solitary wave and homoclinic orbit, the coupled Higgs field equation is transformed into the ordinary differential equations with fast variables by using the variable substitution. It is proved that the equations with perturbation also possess homoclinic orbit, and thus we obtain the existence of solitary wave solutions of the delayed coupled Higgs field equation. [ABSTRACT FROM AUTHOR]
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- 2022
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22. New solitary waves in a convecting fluid.
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Zhang, Lijun, Wang, Jundong, Shchepakina, Elena, and Sobolev, Vladimir
- Abstract
It is critical to examine the effects of small perturbations on solitary wave solutions to nonlinear wave equations because of the existence of the unavoidable perturbations in real applications. In this work, we aim to study the solitary wave solutions for the dissipative perturbed mKdV equation which is proposed to model the evolution of shallow wave in a convecting fluid. By using the geometric singular perturbation theorem, the three-dimensional traveling wave system is reduced to a planar dynamical system with regular perturbation. By examining the homoclinic bifurcations via Melnikov method, we show that not only some solitary waves with particularly chosen wave speed persist but also some new type of solitary waves appear under small perturbation. The theoretical analysis results are illustrated by numerical simulations. • The effect of Marangoni perturbation is considered. • Geometric singular perturbation theory and Melnikov's method are applied to prove the theorems. • Wave speed selection principle is determined. • A new type of solitary waves coexisting crest and trough is founded. • Wave profile of the new solitary waves is determined. [ABSTRACT FROM AUTHOR]
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- 2024
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23. Nonintegrability of time-periodic perturbations of single-degree-of-freedom Hamiltonian systems near homo- and heteroclinic orbits.
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Yagasaki, Kazuyuki
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SINGLE-degree-of-freedom systems , *HAMILTONIAN systems , *ORBITS (Astronomy) , *DUFFING equations , *TIME series analysis , *FOURIER series - Abstract
We consider time-periodic perturbations of single-degree-of-freedom Hamiltonian systems and study their real-meromorphic nonintegrability in the Bogoyavlenskij sense using a generalized version due to Ayoul and Zung of the Morales–Ramis theory. The perturbation terms are assumed to have finite Fourier series in time, and the perturbed systems are rewritten as higher-dimensional autonomous systems having the small parameter as a state variable. We show that if the Melnikov functions are not constant, then the autonomous systems are not real-meromorphically integrable near homo- and heteroclinic orbits. Our result is not just an extension of previous results for homoclinic orbits to heteroclinic orbits and provides a stronger conclusion than them for the case of homoclinic orbits. We illustrate the theory for two periodically forced Duffing oscillators and a periodically forced two-dimensional system. • Periodic perturbations of single-degree-of-freedom Hamiltonian systems are studied. • They are assumed to have homo- or heteroclinic orbits. • Their real-meromorphic nonintegrability near the orbits is discussed. • They are shown to be nonintegrable if the Melnikov functions are not constant. • The result is applied to three examples including the Duffing oscillators. [ABSTRACT FROM AUTHOR]
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- 2024
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24. Global analysis of energy-based swing-up control for soft robot.
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Xin, Xin and Yan, Yuhang
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ROBOT control systems , *GRAVITATIONAL potential , *GLOBAL analysis (Mathematics) , *MOBILE robots , *ORBITS (Astronomy) - Abstract
In this paper, we explore the energy-based swing-up control for a soft robot equipped with an actuated constant curvature soft pendulum and an unactuated rotational base joint. The aim is to swing the robot up towards its upright equilibrium point (UEP) with the pendulum at an upright position. We establish a necessary and sufficient condition to protect the control law from singularities and demonstrate the robot's potential motion towards a homoclinic orbit or a closed-loop equilibrium point. After examining the robot's closed-loop equilibrium points, we generate formulae to compute all such points and introduce two conditions concerning control parameters, eliminating all but the UEP and downward equilibrium point (DEP) with the robot in a downward position. We prove the instability of the equilibrium points with negative gravitational potential energy. This obviates the need for one of the two conditions concerning such equilibrium points. We also prove the robot's linear controllability at the UEP. Our obtained results reveal that, regardless of its initial state, the soft robot can be swung-up towards its UEP using the energy-based controller, provided that it meets the proposed control parameter conditions, and can subsequently be balanced around the UEP using a locally stabilizing controller. Our simulation investigations validate the presented theoretical results. [ABSTRACT FROM AUTHOR]
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- 2024
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25. Geometric singular perturbation analysis to the coupled Schrödinger equations.
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Li, Xiaowan and Ji, Shuguan
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SINGULAR perturbations , *INVARIANT manifolds , *PERTURBATION theory , *DIFFERENTIAL equations , *ORBITS (Astronomy) , *SCHRODINGER equation - Abstract
In this paper, we study the solitary wave solutions of the perturbed coupled Schrödinger equations i u t + u x x + b 1 f ∗ u | u | 2 + d | v | 2 u − τ u ( | u | 2) x = 0 , i v t + v x x + b 2 g ∗ v | v | 2 + d | u | 2 v − τ v ( | v | 2) x = 0. By combining the geometric singular perturbation theory, invariant manifold theory and Fredholm theory, we construct an invariant manifold for the associated differential equations and use this invariant manifold to obtain a homoclinic orbit. Furthermore, the solitary wave solutions of the perturbed coupled Schrödinger equations are obtained. [ABSTRACT FROM AUTHOR]
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- 2024
- Full Text
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26. Persistence of periodic and homoclinic orbits, first integrals and commutative vector fields in dynamical systems.
- Author
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Motonaga, Shoya and Yagasaki, Kazuyuki
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DYNAMICAL systems , *INTEGRALS , *DUFFING equations , *SINGLE-degree-of-freedom systems , *HARMONIC oscillators , *HAMILTONIAN systems , *VECTOR fields - Abstract
We study persistence of periodic and homoclinic orbits, first integrals and commutative vector fields in dynamical systems depending on a small parameter É› > 0 and give several necessary conditions for their persistence. Here we treat homoclinic orbits not only to equilibria but also to periodic orbits. We also discuss some relationships of these results with the standard subharmonic and homoclinic Melnikov methods for time-periodic perturbations of single-degree-of-freedom Hamiltonian systems, and with another version of the homoclinic Melnikov method for autonomous perturbations of multi-degree-of-freedom Hamiltonian systems. In particular, we show that a first integral which converges to the Hamiltonian or another first integral as the perturbation tends to zero does not exist near the unperturbed periodic or homoclinic orbits in the perturbed systems if the subharmonic or homoclinic Melnikov functions are not identically zero on connected open sets. We illustrate our theory for four examples: the periodically forced Duffing oscillator, two identical pendula coupled with a harmonic oscillator, a periodically forced rigid body and a three-mode truncation of a buckled beam. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
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27. Microbial insecticide model and homoclinic bifurcation of impulsive control system.
- Author
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Wang, Tieying
- Subjects
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INSECTICIDES , *STATE feedback (Feedback control systems) , *FEEDBACK control systems , *LIMIT cycles , *GEOGRAPHIC boundaries , *DYNAMICAL systems - Abstract
A new microbial insecticide mathematical model with density dependent for pest is proposed in this paper. First, the system without impulsive state feedback control is considered. The existence and stability of equilibria are investigated and the properties of equilibria under different conditions are verified by using numerical simulation. Since the system without pulse has two positive equilibria under some additional assumptions, the system is not globally asymptotically stable. Based on the stability analysis of equilibria, limit cycle, outer boundary line and Sotomayor's theorem, the existence of saddle-node bifurcation and global dynamics of the system are obtained. Second, we consider homoclinic bifurcation of the system with impulsive state feedback control. The existence of order-1 homoclinic orbit of the system is studied. When the impulsive function is slightly disturbed, the homoclinic orbit breaks and bifurcates order-1 periodic solution. The existence and stability of order-1 periodic solution are obtained by means of theory of semi-continuous dynamic system. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
28. Existence conditions for bifurcations of homoclinic orbits in a railway wheelset model.
- Author
-
Wang, Xingang and Cao, Hongjun
- Subjects
- *
ORBITS (Astronomy) , *GALOIS theory , *DIFFERENTIAL equations , *POTENTIAL well , *BIFURCATION diagrams , *HOPF bifurcations - Abstract
This paper investigates the bifurcations of homoclinic orbits to hyperbolic saddle points in a simplified railway wheelset model with cubic and quintuple nonlinear terms. Using Melnikov's method, the sufficient conditions for the existence of the supercritical and the subcritical pitchfork bifurcations of homoclinic orbits are proven. To determine the integrability of the variational equations around homoclinic orbits in the meaning of differential Galois theory, the corresponding Fuchsian second-order differential equation for the normal variational equation and the Riemann P function are obtained. It is shown that the coefficients of the linear terms and the cubic coupling terms play a very significant role on influencing the existence of homoclinic orbits. While, the cubic coupling terms have little effect on the size of the left-hand and right-hand potential wells of homoclinic orbits. These results are beneficial to explore the key mechanism of hunting stability of a simplified railway wheelset model. • The sufficient conditions for pitchfork bifurcations of homoclinic orbits are proven. • The integrability of variational equations around homoclinic orbits is determined. • Linear and cubic coupling term coefficients affect the existence of homoclinic orbits. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
29. Phenomena of Bifurcation and Chaos in the Dynamically Loaded Hyperelastic Spherical Membrane Based on a Noninteger Power-Law Constitutive Model.
- Author
-
Zhao, Zhentao, Yuan, Xuegang, Niu, Datian, Zhang, Wenzheng, and Zhang, Hongwu
- Subjects
- *
ORDINARY differential equations , *BIFURCATION diagrams , *VARIATIONAL principles , *POLYNOMIAL chaos , *DYNAMICAL systems - Abstract
The phenomena of bifurcation and chaos are studied for a class of second order nonlinear nonautonomous ordinary differential equations, which may be formulated by the nonlinear radially symmetric motion of the dynamically loaded hyperelastic spherical membrane composed of the Rivlin–Saunders material model with a noninteger power-law exponent. Firstly, based on the variational principle, the governing equation describing the problem is obtained with the spherically symmetric deformation assumption. Then, the dynamic characteristics of the system are qualitatively analyzed in detail in terms of different values of material parameters. Particularly, for a given constant load, the parameter spaces describing the bifurcation behaviors of equilibrium curves are established and the characteristics of equilibrium points are presented; for a periodically perturbed load, the quasi-periodic and chaotic behaviors are discussed for the systems with two and three equilibrium points, respectively. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
30. Bogdanov–Takens bifurcation in a predator–prey model with age structure.
- Author
-
Liu, Zhihua and Magal, Pierre
- Subjects
- *
PREDATORY animals , *AGE , *CAUCHY problem , *HOPF bifurcations - Abstract
The results obtained in this article aim at analyzing Bogdanov–Takens bifurcation in a predator–prey model with an age structure for the predator. Firstly, we give the existence result of the Bogdanov–Takens singularity. Then we describe the bifurcation behavior of the parameterized predator–prey model with Bogdanov–Takens singularity. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
31. A nonzero solution for bounded selfadjoint operator equations and homoclinic orbits of Hamiltonian systems.
- Author
-
Mingliang Song and Runzhen Li
- Subjects
- *
SELFADJOINT operators , *OPERATOR equations , *HAMILTONIAN systems , *ORBIT method , *EXISTENCE theorems - Abstract
We obtain an existence theorem of nonzero solution for a class of bounded selfadjoint operator equations. The main result contains as a special case the existence result of a nontrivial homoclinic orbit of a class of Hamiltonian systems by Coti Zelati, Ekeland and Séré. We also investigate the existence of nontrivial homoclinic orbit of indefinite second order systems as another application of the theorem. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
32. Existence of a homoclinic orbit in a generalized Liénard type system.
- Author
-
Gharahasanlou, Tohid Kasbi, Roomi, Vahid, and Akbarfam, Aliasghar Jodayree
- Subjects
- *
DYNAMICAL systems , *CLINICS - Abstract
The object of this paper is to study the existence and nonexistence of an important orbit in a generalized Liénard type system. This trajectory is doubly asymptotic to an equilibrium solution, i.e., an orbit which lies in the intersection of the stable and unstable manifolds of a critical point. Such an orbit is called a homoclinic orbit. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
33. Melnikov analysis of chaos in a simple SIR model with periodically or stochastically modulated nonlinear incidence rate.
- Author
-
Shi, Yanxiang
- Subjects
- *
MOTION , *NOISE , *POLYNOMIAL chaos - Abstract
In this paper, Melnikov analysis of chaos in a simple SIR model with periodically or stochastically modulated nonlinear incidence rate and the effect of periodic and bounded noise on the chaotic motion of SIR model possessing homoclinic orbits are detailed investigated. Based on homoclinic bifurcation, necessary conditions for possible chaotic motion as well as sufficient condition are derived by the random Melnikov theorem, and to establish the threshold of bounded noise amplitude for the onset of chaos. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
34. Dynamical analysis of a two-species competitive system with state feedback impulsive control.
- Author
-
Xu, Jing, Huang, Mingzhan, and Song, Xinyu
- Subjects
- *
DIFFERENTIAL geometry , *DIFFERENTIAL equations , *PSYCHOLOGICAL feedback - Abstract
In this paper, three competitive systems with different kinds of state-dependent control are presented and investigated. The existence of the order-1 homoclinic orbit and order-1 periodic solution of the two systems that incorporate just one kind of state-dependent control is obtained by applying differential equation geometry theory, and the stability of the order-1 periodic solution of each system is also given. Besides, sufficient conditions for the existence and stability of the order-2 periodic solution of the system that incorporate two kinds of state-dependent control are gained by successor function method and analogue of Poincaré criterion, respectively. Finally, numerical simulations are carried out to verify the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
35. Homoclinic Bifurcations and Chaos in the Fishing Principle for the Lorenz-like Systems.
- Author
-
Leonov, G. A., Mokaev, R. N., Kuznetsov, N. V., and Mokaev, T. N.
- Subjects
- *
CLINICS , *NUMERICAL analysis - Abstract
In this article using an analytical method called Fishing principle we obtain the region of parameters, where the existence of a homoclinic orbit to a zero saddle equilibrium in the Lorenz-like system is proved. For a qualitative description of the different types of homoclinic bifurcations, a numerical analysis of the obtained region of parameters is organized, which leads to the discovery of new bifurcation scenarios. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
36. Resonant Chaotic Dynamics of a Symmetric Cross-Ply Composite Laminated Plate Under Transverse and In-Plane Excitations.
- Author
-
Ma, W. S. and Zhang, W.
- Subjects
- *
LAMINATED materials , *COMPOSITE plates , *ORTHOTROPIC plates , *PARTIAL differential equations , *VON Karman equations , *GALERKIN methods - Abstract
The resonant chaotic dynamics of a symmetric cross-ply composite laminated plate are studied using the exponential dichotomies and an averaging procedure for the first time. The partial differential governing equations of motion for the symmetric cross-ply composite laminated plate are derived by using Reddy's third-order shear deformation plate theory and von Karman type equation. The partial differential governing equations of motion are discretized into two-degree-of-freedom nonlinear systems including the quadratic and cubic nonlinear terms by using Galerkin method. There exists a fixed point of saddle-focus in the linear part for two-degree-of-freedom nonlinear system. The Melnikov method containing the terms of the nonhyperbolic mode is developed to investigate the resonant chaotic motions of the symmetric cross-ply composite laminated plate. The obtained results indicate that the nonhyperbolic mode of the symmetric cross-ply composite laminated plate does not affect the critical conditions in the occurrence of chaotic motions in the resonant case. When the resonant chaotic motion occurs, we can draw a conclusion that the resonant chaotic motions of the hyperbolic subsystem are shadowed for the full nonlinear system of the symmetric cross-ply composite laminated plate. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
37. Dissonant points and the region of influence of non-saddle sets.
- Author
-
Barge, Héctor and Sanjurjo, José M.R.
- Subjects
- *
TOPOLOGICAL property , *POINT set theory , *ATTRACTORS (Mathematics) , *MANIFOLDS (Mathematics) - Abstract
The aim of this paper is to study dynamical and topological properties of a flow in the region of influence of an isolated non-saddle set. We see, in particular, that some topological conditions are sufficient to guarantee that these sets are attractors or repellers. We study in detail the existence of dissonant points of the flow, which play a key role in the description of the region of influence of a non-saddle set. These points are responsible for much of the dynamical and topological complexity of the system. We also study non-saddle sets from the point of view of the Conley index theory and consider, among other things, the case of flows on manifolds with trivial first cohomology group. For flows on these manifolds, dynamical robustness is equivalent to topological robustness. We carry out a particular study of 2-dimensional flows and give a topological condition which detects the existence of dissonant points for flows on surfaces. We also prove that isolated invariant continua of planar flows with global region of influence are necessarily attractors or repellers. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
38. Modeling cooperative evolution in prey species using the snowdrift game with evolutionary impact on prey–predator dynamics.
- Author
-
Sahoo, Debgopal and Samanta, Guruprasad
- Subjects
- *
PREDATION , *TRANSIENTS (Dynamics) , *ENDANGERED species , *LIMIT cycles , *SPECIES , *ORBITS (Astronomy) - Abstract
Cooperative evolution refers to the collaborative process by which two or more individuals or species interact and adapt together some specific behavior, leading to mutual benefits for their survival and reproduction. In this article, we employ the snowdrift game to model cooperative evolution in prey species and examine its evolutionary impact on the dynamics of prey–predator interactions. The model system consists of a replicator equation that captures the evolution of cooperative behavior in prey species, along with two growth equations of prey and predator incorporating the factors that influence cooperative evolution on prey reproduction. Our analysis reveals that in long time run, the prey population becomes polymorphic, allowing cooperative and defective individuals to coexist and persist over time. However, a very low cooperative proportion poses a risk of extinction for both prey and predator. Again if majority of prey species exhibit cooperative behavior, the predator species still faces possible extinction due to the strong group defense mechanism employed by prey. At some intermediate level of cooperative proportions, the proposed model system exhibits more rich and complex dynamics such as oscillatory patterns, bi-stable phenomena, homoclinic orbit, and saddle–node bifurcations of limit cycles. Further, we explore the transient dynamics of the proposed system through which one can anticipate the potential shifts in population size and can identify critical thresholds in promoting sustainable coexistence. • A snowdrift game is introduced to model cooperative evolution in prey species. • A replicator equation is derived to capture evolutionary impact on prey–predator dynamics. • A very low cooperative proportion in prey poses a risk of extinction for both prey and predator. • At intermediate level of cooperative proportions, the model system exhibits some rich and complex dynamics. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
39. Dynamics of the Morse Oscillator: Analytical Expressions for Trajectories, Action-Angle Variables, and Chaotic Dynamics.
- Author
-
Krajňák, Vladimír and Wiggins, Stephen
- Subjects
- *
CHAOS theory , *HAMILTONIAN systems , *MORSE theory , *ENERGY function , *POTENTIAL functions , *POTENTIAL energy - Abstract
We consider the one degree-of-freedom Hamiltonian system defined by the Morse potential energy function (the "Morse oscillator"). We use the geometry of the level sets to construct explicit expressions for the trajectories as a function of time, their period for the bounded trajectories, and action-angle variables. We use these trajectories to prove sufficient conditions for chaotic dynamics, in the sense of Smale horseshoes, for the time-periodically perturbed Morse oscillator using a Melnikov type approach. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
40. Existence of Chaos in the Chen System with Linear Time-Delay Feedback.
- Author
-
Tian, Kun, Ren, Hai-Peng, and Grebogi, Celso
- Subjects
- *
CHAOS theory , *LINEAR systems , *CHAOS synchronization , *ATTRACTORS (Mathematics) , *EIGENVALUES - Abstract
It is mathematically challenging to analytically show that complex dynamical phenomena observed in simulations and experiments are truly chaotic. The Shil'nikov lemma provides a useful theoretical tool to prove the existence of chaos in three-dimensional smooth autonomous systems. It requires, however, the proof of existence of a homoclinic or heteroclinic orbit, which remains a very difficult technical problem if contigent on data. In this paper, for the Chen system with linear time-delay feedback, we demonstrate a homoclinic orbit by using a modified undetermined coefficient method and we propose a spiral involute projection method. In such a way, we identify experimentally the asymmetrical homoclinic orbit in order to apply the Shil'nikov-type lemma and to show that chaos is indeed generated in the Chen circuit with linear time-delay feedback. We also identify the presence of a single-scroll attractor in the Chen system with linear time-delay feedback in our experiments. We confirm that the Chen single-scroll attractor is hyperchaotic by numerically estimating the finite-time local Lyapunov exponent spectrum. By means of a linear scaling in the coordinates and the time, such a method can also be applied to the generalized Lorenz-like systems. The contribution of this work lies in: first, we treat the trajectories corresponding to the real eigenvalue and the image eigenvalues in different ways, which is compatible with the characteristics of the trajectory geometry; second, we propose a spiral involute projection method to exhibit the trajectory corresponding to the image eigenvalues; third, we verify the homoclinic orbit by experimental data. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
41. Dissipative periodic and chaotic patterns to the KdV–Burgers and Gardner equations.
- Author
-
Mancas, Stefan C. and Adams, Ronald
- Subjects
- *
DYNAMICAL systems , *EQUATIONS , *LIMIT cycles , *HOPF bifurcations , *ORBITS (Astronomy) - Abstract
We investigate the KdV-Burgers and Gardner equations with dissipation and external perturbation terms by the approach of dynamical systems and Shil'nikov's analysis. The stability of the equilibrium point is considered, and Hopf bifurcations are investigated after a certain scaling that reduces the parameter space of a three-mode dynamical system which now depends only on two parameters. The Hopf curve divides the two-dimensional space into two regions. On the left region the equilibrium point is stable leading to dissapative periodic orbits. While changing the bifurcation parameter given by the velocity of the traveling waves, the equilibrium point becomes unstable and a unique stable limit cycle bifurcates from the origin. This limit cycle is the result of a supercritical Hopf bifurcation which is proved using the Lyapunov coefficient together with the Routh-Hurwitz criterion. On the right side of the Hopf curve, in the case of the KdV-Burgers, we find homoclinic chaos by using Shil'nikov's theorem which requires the construction of a homoclinic orbit, while for the Gardner equation the supercritical Hopf bifurcation leads only to a stable periodic orbit. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
42. On Singular Orbits and Global Exponential Attractive Set of a Lorenz-Type System.
- Author
-
Wang, Haijun
- Subjects
- *
ORBITS (Astronomy) , *LYAPUNOV functions , *COMPUTER simulation - Abstract
This paper deals with some unsolved problems of the global dynamics of a three-dimensional (3D) Lorenz-type system: ẋ = a (y − x) , ẏ = c x − x z , ż = − b z + x y + e x 2 by constructing a series of Lyapunov functions. The main contribution of the present work is that one not only proves the existence of singularly degenerate heteroclinic cycles, existence and nonexistence of homoclinic orbits for a certain range of the parameters according to some known results and LaSalle theorem but also gives a family of mathematical expressions of global exponential attractive sets for that system with respect to its parameters, which is available only in very few papers as far as one knows. In addition, numerical simulations illustrate the consistence with the theoretical conclusions. The results together not only improve and complement the known ones, but also provide support in some future applications. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
43. Smooth Exact Traveling Wave Solutions Determined by Singular Nonlinear Traveling Wave Systems: Two Models.
- Author
-
Li, Jibin, Chen, Guanrong, and Deng, Shengfu
- Subjects
- *
NONLINEAR waves , *WAVE equation , *METAMATERIALS - Abstract
For a singular nonlinear traveling wave system of the first class, if there exist two node points of the associated regular system in the singular straight line, then the dynamics of the solutions of the singular system will be very complex. In this paper, two representative nonlinear traveling wave system models (namely, the traveling wave system of Green–Naghdi equations and the traveling wave system of the Raman soliton model for optical metamaterials) are investigated. It is shown that, if there exist two node points of the associated regular system in the singular straight line, then the singular system has no peakon, periodic peakon and compacton solutions, but rather, it has smooth periodic wave, solitary wave and kink wave solutions. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
44. New global bifurcation diagrams for piecewise smooth systems: Transversality of homoclinic points does not imply chaos.
- Author
-
Franca, M. and Pospíšil, M.
- Subjects
- *
BIFURCATION diagrams , *DIMENSIONAL analysis , *RATIONAL points (Geometry) , *PERTURBATION theory , *SURFACE analysis - Abstract
Abstract In this paper we consider some piecewise smooth 2-dimensional systems having a possibly non-smooth homoclinic γ → (t). We assume that the critical point 0 → lies on the discontinuity surface Ω 0. We consider 4 scenarios which differ for the presence or not of sliding close to 0 → and for the possible presence of a transversal crossing between γ → (t) and Ω 0. We assume that the systems are subject to a small non-autonomous perturbation, and we obtain 4 new bifurcation diagrams. In particular we show that, in one of these scenarios, the existence of a transversal homoclinic point guarantees the persistence of the homoclinic trajectory but chaos cannot occur. Further we illustrate the presence of new phenomena involving an uncountable number of sliding homoclinics. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
45. Complex dynamics of epidemic models on adaptive networks.
- Author
-
Zhang, Xiaoguang, Shan, Chunhua, Jin, Zhen, and Zhu, Huaiping
- Subjects
- *
OPTICAL bistability , *HYSTERESIS , *CENTER manifolds (Mathematics) , *HOPF bifurcations , *TOPOLOGY - Abstract
Abstract There has been a substantial amount of well mixing epidemic models devoted to characterizing the observed complex phenomena (such as bistability, hysteresis, oscillations, etc.) during the transmission of many infectious diseases. A comprehensive explanation of these phenomena by epidemic models on complex networks is still lacking. In this paper we study epidemic dynamics in an adaptive network proposed by Gross et al., where the susceptibles are able to avoid contact with the infectious by rewiring their network connections. Such rewiring of the local connections changes the topology of the network, and inevitably has a profound effect on the transmission of the disease, which in turn influences the rewiring process. We rigorously prove that the adaptive epidemic model investigated in this paper exhibits degenerate Hopf bifurcation, homoclinic bifurcation and Bogdanov–Takens bifurcation. Our study shows that adaptive behaviors during an epidemic may induce complex dynamics of disease transmission, including bistability, transient and sustained oscillations, which contrast sharply to the dynamics of classical network models. Our results yield deeper insights into the interplay between topology of networks and the dynamics of disease transmission on networks. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
46. The Smale horseshoe-type chaos in the general 3-D quadratic continuous-time system with a cubic term.
- Author
-
Samia, Rezzag
- Subjects
- *
TERMS & phrases , *ORBITS (Astronomy) - Abstract
This paper presents the existence of horseshoe-type chaos in the general 3-D quadratic continuous-time system with a cubic term. For the discussion of chaos, the bifurcate parameter value is limited to the requirement of the Shilnikov theorem. Also, The existence of the homoclinic orbit has been proven by using the undetermined coefficient method. Finally, an example is provided to illustrate the main result. [ABSTRACT FROM AUTHOR]
- Published
- 2019
47. Study on the Geometrical Properties and Existence of Orbit Homoclinic to a Saddle Point in n-Dimensional Autonomous Vector Field with Polynomials.
- Author
-
Xie, Lingli
- Subjects
- *
SADDLEPOINT approximations , *VECTOR fields , *POLYNOMIALS , *MANIFOLDS (Mathematics) , *BIFURCATION theory - Abstract
According to the theory of stable and unstable manifolds of an equilibrium point, we firstly find out some geometrical properties of orbits on the stable and unstable manifolds of a saddle point under some brief conditions of nonlinear terms composed of polynomials for n -dimensional time continuous system. These properties show that the orbits on stable and unstable manifolds of the saddle point will stay on the corresponding stable and unstable subspaces in the 𝜀 -neighborhood of the saddle point. Furthermore, the necessary conditions of existence for orbit homoclinic to a saddle point are exposed. Some examples including homoclinic bifurcation are given to indicate the application of the results. Finally, the conclusions are presented. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
48. On periodic solutions in the non-dissipative Lorenz model: the role of the nonlinear feedback loop.
- Author
-
Shen, B.-W.
- Abstract
In this study, we discuss the role of the linear heating term and nonlinear terms associated with a non-linear feedback loop in the energy cycle of the three-dimensional (X-Y-Z) non-dissipative Lorenz model (3D-NLM), where (X, Y, Z) represent the solutions in the phase space. Using trigonometric functions, we first present the closed-form solution of the nonlinear equation without the heating term (i.e. rX), (where τ is a non-dimensional time and r is the normalized Rayleigh number), a solution that has not been previously documented. Since the solution of the simplified 3D-NLM is oscillatory (wave-like) and since the nonlinear term (X3) is associated with the nonlinear feedback loop, here, we suggest that the nonlinear feedback loop may act as a restoring force. When the heating term is considered, the system yields three critical points. A linear analysis suggests that the origin (i.e. the trivial critical point) is a saddle point and that the other two non-trivial critical points (i.e. centers) are stable. Here, we provide an analysis for three types of solutions that are associated with these critical points. Two of the solutions represent closed curves that travel around one non-trivial critical point or all three critical points. The third type of solution, appearing to connect the stable and unstable manifolds of the saddle point, is called the homoclinic orbit. Using the solution that encloses one non-trivial critical point, here, we show that the competing impact of the nonlinear restoring force and the linear (heating) force determines the partition of the average available potential energy from the Y and Z modes. Based on the energy analysis, an energy cycle with four different regimes is identified. The cycle is only half of a 'large' cycle, displaying the wing pattern of a glass-winged butterfly. The other half cycle is anti-symmetric with respect to the origin. The two types of oscillatory solutions with either a small cycle or a large cycle are orbitally stable. As compared to the oscillatory solutions, the homoclinic orbit is not periodic because it "takes forever" to reach the origin. Two trajectories with starting points near the homoclinic orbit may be diverged because one moves with a small cycle and the other moves with a large cycle. Therefore, the homoclinic orbit is not orbitally stable. In a future study, dissipation and/or additional nonlinear terms will be further examined in order to determine how their interactions with the original nonlinear feedback loop and the heating term may change the periodic orbits (as well as homoclinic orbits) to become quasi-periodic orbits and chaotic solutions. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
49. ELLIPTIC SECTORS AND EULER DISCRETIZATION.
- Author
-
MOHDEB, NADIA, FRUCHARD, AUGUSTIN, and MEHIDI, NOUREDDINE
- Subjects
- *
EULER equations , *DISCRETIZATION methods , *EIGENANALYSIS , *EIGENFREQUENCIES , *EIGENVALUES - Abstract
In this work we are interested in the elliptic sector of autonomous differential systems with a degenerate equilibrium point at the origin, and in their Euler discretization. When the linear part of the vector field at the origin has two zero eigenvalues, then the differential system has an elliptic sector, under some conditions. We describe this elliptic sector and we show that the associated Euler discretized system has an elliptic sector converging to that of the continuous system when the step size of the discretization tends to zero. [ABSTRACT FROM AUTHOR]
- Published
- 2018
50. Elements of Contemporary Theory of Dynamical Chaos: A Tutorial. Part I. Pseudohyperbolic Attractors.
- Author
-
Gonchenko, A. S., Gonchenko, S. V., Kazakov, A. O., and Kozlov, A. D.
- Subjects
- *
CHAOS theory , *ATTRACTORS (Mathematics) , *PHENOMENOLOGICAL theory (Physics) , *ENERGY dissipation , *DIFFEOMORPHISMS - Abstract
The paper is devoted to topical issues of modern mathematical theory of dynamical chaos and its applications. At present, it is customary to assume that dynamical chaos in finite-dimensional smooth systems can exist in three different forms. This is dissipative chaos, the mathematical image of which is a strange attractor; conservative chaos, for which the entire phase space is a large "chaotic sea" with randomly spaced elliptical islands inside it; and mixed dynamics, characterized by the principal inseparability in the phase space of attractors, repellers and conservative elements of dynamics. In the present paper (which opens a series of three of our papers), elements of the theory of pseudohyperbolic attractors of multidimensional maps and flows are presented. Such attractors, as well as hyperbolic ones, are genuine strange attractors, but they allow the existence of homoclinic tangencies. We describe two principal phenomenological scenarios for the appearance of pseudohyperbolic attractors in one-parameter families of three-dimensional diffeomorphisms, and also consider some basic examples of concrete systems in which these scenarios occur. We propagandize new methods for studying pseudohyperbolic attractors (in particular, the method of saddle charts, the modified method of Lyapunov diagrams and the so-called LMP-method for verification of pseudohyperbolicity of attractors) and test them on the above examples. We show that Lorenz-like attractors in three-dimensional generalized Hénon maps and in a nonholonomic model of Celtic stone as well as figure-eight attractors in the model of Chaplygin top are genuine (pseudohyperbolic) ones. Besides, we show an example of four-dimensional Lorenz model with a wild spiral attractor of Shilnikov–Turaev type that was found recently in [Gonchenko et al., 2018]. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
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