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The influences of Allee effect and density‐dependent mortality on population growth are of great significance in ecology. In this paper, we first consider a Gause‐type predator–prey model with simplified Holling type IV functional response, strong Allee effect on prey, and density‐dependent mortality of predator. It is shown that the system exhibits rich and complex dynamics like bistability, local and global bifurcations such as transcritical bifurcation, saddle‐node bifurcation, Hopf bifurcation, cusp bifurcation, Bogdanov–Takens bifurcation, Bautin bifurcation, homoclinic bifurcation, saddle‐node bifurcation of limit cycle, and heteroclinic bifurcation. Next, we separately explore the influences of Allee effect and density‐dependent mortality on the dynamics of the same model. The results show that the strong Allee effect induces the occurrence of heteroclinic bifurcation and the reduction of the number of limit cycles, while the density‐dependent mortality stabilizes the system. Since the analytical expressions of the interior equilibria are difficult to derive, the results are verified numerically. [ABSTRACT FROM AUTHOR]

The tippedisk is a mechanical–mathematical archetype for friction-induced instabilities caused by geometry and interaction with a frictional support. The instability leads to a counterintuitive rise of the centre of gravity when an unbalanced disc is spun rapidly about an in-plane axis. To understand the qualitative behaviour of the tippedisk, a nonlinear analysis is performed, revealing the singularly perturbed structure of the system equations. Application of singular perturbation theory shows that the long-term behaviour is dominated by a two-dimensional slow manifold, on which the asymptotic dynamics takes place. Moreover, Melnikov theory is used to derive a closed-form approximation of a heteroclinic bifurcation, which allows general statements to be made about the dynamic behaviour of the tippedisk. [ABSTRACT FROM AUTHOR]

Jump and pull-in instability are common nonlinear dynamic behaviors leading to the loss of the performance reliability and structural safety of electrostatic micro gyroscopes. To achieve a better understanding of these initial-sensitive phenomena, the dynamics of a micro gyroscope system considering the nonlinearities of the stiffness and electrostatic forces are explored from a global perspective. Static and dynamic analyses of the system are performed to estimate the threshold of the detecting voltage for static pull-in, and dynamic responses are analyzed in the driving and detecting modes for the case of primary resonance and 1:1 internal resonance. The results show that, when the driving voltage frequency is a bit higher than the natural frequency, a high amplitude of the driving AC voltage may induce the coexistence of bistable periodic responses due to saddle-node bifurcation of the periodic solution. Basins of attraction of bistable attractors provide evidence that disturbance of the initial conditions can trigger a jump between bistable attractors. Moreover, the Melnikov method is applied to discuss the condition for pull-in instability, which can be ascribed to heteroclinic bifurcation. The validity of the prediction is verified using the sequences of safe basins and unsafe zones for dynamic pull-in. It follows that pull-in instability can be caused and aggravated by the increase in the amplitude of the driving AC voltage. [ABSTRACT FROM AUTHOR]

In this paper, we study heteroclinic bifurcation of limit cycles in a planar cubic near-Hamiltonian system by higher-order Melnikov functions. We compute the asymptotic expansion of the third-order Melnikov function near the heteroclinic loop L s and prove that this system can have five limit cycles around L s with proper perturbations. [ABSTRACT FROM AUTHOR]

The existence of nonmonotone traveling wave solutions of the three-species Lotka–Volterra competition diffusion system under strong competition is established. A traveling wave solution can be considered as a heteroclinic orbit of a vector field in R 6 . Under suitable assumptions on parameters of the equations, we apply a bifurcation theory of heteroclinic orbits to show that a three-species traveling wave can bifurcate from two two-species waves which connect to a common equilibrium. The three components of the three-species wave obtained are positive and have the profiles that one is a front, one is a back, and the third component is a pulse between the previous two with a long middle part close to a constant. As applications of our result, we find several explicit regions of parameters of the equations where the bifurcation of three-species traveling waves occur. [ABSTRACT FROM AUTHOR]

This paper is to study the global dynamics of a van der Pol-Duffing oscillator with indefinite degree x ˙ = y , y ˙ = a x + x 2 n + 1 - δ (b + x 2 m) y , where a , b , δ ∈ R , m , n ∈ N + and δ ≠ 0 . By qualitative and bifurcation analysis, the oscillator contains abundant nonlinear phenomena, including the heteroclinic bifurcation, degenerate Hopf bifurcation, bifurcation of equilibria at infinity and pitchfork bifurcation. [ABSTRACT FROM AUTHOR]

Jump and pull-in instability are common nonlinear dynamic behaviors leading to the loss of the performance reliability and structural safety of electrostatic micro gyroscopes. To achieve a better understanding of these initial-sensitive phenomena, the dynamics of a micro gyroscope system considering the nonlinearities of the stiffness and electrostatic forces are explored from a global perspective. Static and dynamic analyses of the system are performed to estimate the threshold of the detecting voltage for static pull-in, and dynamic responses are analyzed in the driving and detecting modes for the case of primary resonance and 1:1 internal resonance. The results show that, when the driving voltage frequency is a bit higher than the natural frequency, a high amplitude of the driving AC voltage may induce the coexistence of bistable periodic responses due to saddle-node bifurcation of the periodic solution. Basins of attraction of bistable attractors provide evidence that disturbance of the initial conditions can trigger a jump between bistable attractors. Moreover, the Melnikov method is applied to discuss the condition for pull-in instability, which can be ascribed to heteroclinic bifurcation. The validity of the prediction is verified using the sequences of safe basins and unsafe zones for dynamic pull-in. It follows that pull-in instability can be caused and aggravated by the increase in the amplitude of the driving AC voltage.

Jump and excessive motion are undesirable phenomena in relative rotation systems, causing a loss of global integrity and reliability of the systems. In this work, a typical relative rotation system is considered in which jump, excessive motion, and their suppression via delayed feedback are investigated. The Method of Multiple Scales and the Melnikov method are applied to analyze critical conditions for bi-stability and initial-sensitive excessive motion, respectively. By introducing the fractal of basins of attraction and the erosion of the safe basin to depict jump and initial-sensitive excessive motion, respectively, the point mapping approach is used to present numerical simulations which are in agreement with the theoretical prediction, showing the validity of the analysis. It is found that jump between bistable attractors can be due to saddle–node bifurcation, while initial-sensitive excessive motion can be due to heteroclinic bifurcation. Under a positive coefficient of the gain, the types of delayed feedback can both be effective in reducing jump and initial-sensitive excessive motion. The results may provide some reference for the performance improvement of rotors and main bearings. [ABSTRACT FROM AUTHOR]

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In this paper, we study heteroclinic bifurcation and the appearance of chaos in time-perturbed piecewise smooth hybrid systems with discontinuities on finitely many switching manifolds. The unperturbed system has a heteroclinic orbit connecting hyperbolic saddles of the unperturbed system that crosses every switching manifold transversally, possibly multiple times. By applying a functional analytical method, we obtain a set of Melnikov functions whose zeros correspond to the occurrence of chaos of the system. As an application, we present an example of quasiperiodically excited piecewise smooth system with impacts formed by two linked rocking blocks. [ABSTRACT FROM AUTHOR]

The Rosenzweig-MacArthur model is a set of ordinary differential equations (ODEs) that provides an aggregate description of the dynamics of a predator-prey system. When including an Allee effect on the prey, this model exhibits bistability and contains a pitchfork bifurcation, a Hopf bifurcation and a heteroclinic bifurcation. We develop an agent-based model (ABM) on a two-dimensional, square lattice that encompasses the key assumptions of the aggregate model. Although the two modelling approaches - ODE and ABM - differ, both models exhibit similar bifurcation patterns. The ABM model's behaviour is richer and it is analysed using advanced statistical methods. In particular, singular spectrum analysis is used to robustly locate the transition between apparently random, small-amplitude fluctuations around a fixed point and stable, large-amplitude oscillations. Critical slowing down of model trajectories anticipates the heteroclinic bifurcation. Systematic comparison between the ABM and the ODE models' behaviour helps one understand the predator-prey system better; it provides guidance in model exploration and allows one to draw more robust conclusions on the nature of predator-prey interactions.

The transition between strong and weak Allee effects in prey provides a simple regime shift in ecology. In this paper, we study the interplay between the functional response of Holling type IV and both strong and weak Allee effects. The model investigated here presents complex dynamics and high codimension bifurcations. In particular, nilpotent cusp singularity of order 3 and degenerate Hopf bifurcation of codimension 3 are completely analyzed. Remarkably it is the first time that three limit cycles are discovered in predator-prey models with multiplicative Allee effects. Moreover, a new unfolding of nilpotent saddle of codimension 3 with a fixed invariant line is discovered and fully developed, and the existence of codimension 2 heteroclinic bifurcation is proven. Our work extends the existing results of predator-prey systems with Allee effects. The bifurcation analysis and diagram allow us to give biological interpretations of predator-prey interactions. [ABSTRACT FROM AUTHOR]

The noise-induced transition of the augmented Lotka-Volterra system is investigated under vanishingly small noise. Populations will ultimately go extinct because of intrinsic noise, and different extinction routes may occur due to the Freidlin-Wentzell large deviation theory. The relation between the most probable extinction route (MPER) and heteroclinic bifurcation is studied in this paper. The MPERs and the quasi-potentials in different regimes of parameters are analyzed in detail. Before the bifurcation, the predator goes extinct, and the prey will survive for a long time. Then, the heteroclinic bifurcation changes the MPER wherein both species go extinct. The heteroclinic cycle plays a role in transferring the most probable extinction state. Moreover, the analyses of the weak noise limit can contribute to predicting the stochastic behavior under finite small noise. Both the heteroclinic bifurcation and the rotational deterministic vector field can reduce the action necessary for the MPER. [ABSTRACT FROM AUTHOR]

This paper studies the limit cycle bifurcation problem of a class of piecewise smooth differential polynomial systems of degree n by perturbing a piecewise cubic polynomial system having a generalized heteroclinic loop with a cusp and a nilpotent saddle. First, we provide all possible phase portraits of the unperturbed system on the plane with crossing periodic orbits and obtain a condition for the appearance of a generalized heteroclinic loop with a cusp and a nilpotent saddle by qualitative theoretical knowledge. Then, we investigate the algebraic structure of the first order Melnikov function and give its asymptotic expansion near the generalized heteroclinic loop with the help of analytical skills. Finally, we employ the expansion together with its coefficients to obtain the existence of at least 3 n − 1 limit cycles. [ABSTRACT FROM AUTHOR]

Jump and excessive motion are undesirable phenomena in relative rotation systems, causing a loss of global integrity and reliability of the systems. In this work, a typical relative rotation system is considered in which jump, excessive motion, and their suppression via delayed feedback are investigated. The Method of Multiple Scales and the Melnikov method are applied to analyze critical conditions for bi-stability and initial-sensitive excessive motion, respectively. By introducing the fractal of basins of attraction and the erosion of the safe basin to depict jump and initial-sensitive excessive motion, respectively, the point mapping approach is used to present numerical simulations which are in agreement with the theoretical prediction, showing the validity of the analysis. It is found that jump between bistable attractors can be due to saddle–node bifurcation, while initial-sensitive excessive motion can be due to heteroclinic bifurcation. Under a positive coefficient of the gain, the types of delayed feedback can both be effective in reducing jump and initial-sensitive excessive motion. The results may provide some reference for the performance improvement of rotors and main bearings.

In the last few years, Battelli and Fečkan have developed a functional analytic method to rigorously prove the existence of chaotic behaviors in time-perturbed piecewise smooth systems whose unperturbed part has a piecewise continuous homoclinic solution. In this paper, by applying their method, we study the appearance of chaos in time-perturbed piecewise smooth systems with discontinuities on finitely many switching manifolds whose unperturbed part has a hyperbolic saddle in each subregion and a heteroclinic orbit connecting those saddles that crosses every switching manifold transversally exactly once. We obtain a set of Melnikov type functions whose zeros correspond to the occurrence of chaos of the system. Furthermore, the Melnikov functions for planar piecewise smooth systems are explicitly given. As an application, we present an example of quasiperiodically excited three-dimensional piecewise linear system with four zones. [ABSTRACT FROM AUTHOR]

This paper presents necessary and sufficient conditions for the existence of bright/dark solitary solutions in the Hodgkin–Huxley model. The second-order analytic solitary solutions are derived using the generalized differential operator technique. It is shown that the heteroclinic bifurcation in the Hodgkin–Huxley model yields a symmetry breaking effect. Trajectories of solitary solutions before the bifurcation lie on manifolds of one of the saddle points and the separatrix between periodic and non-periodic solutions. A new separatrix emerges after the heteroclinic bifurcation—but solitary solutions do not lie on this trajectory. This symmetry breaking effect is demonstrated using analytic and computational experiments. [ABSTRACT FROM AUTHOR]

We motivate and analyze a simple model for the formation of banded vegetation patterns. The model incorporates a minimal number of ingredients for vegetation growth in semiarid landscapes. It allows for comprehensive analysis and sheds new light onto phenomena such as the migration of vegetation bands and the interplay between their upper and lower edges. The key ingredient is the formulation as a closed reaction-diffusion system, thus introducing a conservation law that both allows for analysis and provides ready intuition and understanding through analogies with characteristic speeds of propagation and shock waves. [ABSTRACT FROM AUTHOR]

A generalized hyperbolic perturbation method for heteroclinic solutions is presented for strongly nonlinear self-excited oscillators in the more general form of x··+g(x)=ɛf(μ,x,x·). The advantage of this work is that heteroclinic solutions for more complicated and strong nonlinearities can be analytically derived, and the previous hyperbolic perturbation solutions for Duffing type oscillator can be just regarded as a special case of the present method. The applications to cases with quadratic-cubic nonlinearities and with quintic-septic nonlinearities are presented. Comparisons with other methods are performed to assess the effectiveness of the present method. [ABSTRACT FROM AUTHOR]

Heteroclinic bifurcation in a nonlinear rigid rocking block under external quasi-periodic excitation with two frequencies is investigated. By using the method of Melnikov type, we derive sufficient conditions under which the perturbed stable and unstable manifolds of the heteroclinic orbits intersect transversally. Then a complete description of the bifurcation sets and the chaotic zones in the parameter space are presented. Numerical simulations are performed for parameters chosen from the chaotic zones. The chaotic motions of the system are verified by computing the largest Lyapunov exponent of the system. • Heteroclinic bifurcation in a quasi-periodically excited rigid block investigated. • Complete description of the bifurcation sets and chaotic zones presented. • Numerical simulations performed for parameters chosen from the chaotic zones. • Chaotic motions verified by computing the largest Lyapunov exponent. [ABSTRACT FROM AUTHOR]

Analytical predictions of the triangle and clover heteroclinic bifurcations in the problem of self-oscillations stability loss near 1:3 resonance are provided using the method of nonlinear time transformation. The analysis was carried out considering the slow flow of a self-excited nonlinear Mathieu oscillator corresponding to the normal form near this 1:3 strong resonance. Using the Hamiltonian system of the corresponding slow flow near this resonance, the unperturbed zero-order approximation of the heteroclinic connections is established. Conditions of persistence of homoclinic connections in the perturbed first-order approximation of the heteroclinic connections provide close analytical approximations of the triangle and clover heteroclinic bifurcation curves, simultaneously. The analytical predictions are compared to the results obtained by numerical simulations for validation. [ABSTRACT FROM AUTHOR]

We discuss heteroclinic bifurcation in a class of periodically excited planar piecewise smooth systems with discontinuities on finitely many smooth curves intersecting at the origin. Assume that the unperturbed system has a hyperbolic saddle in each subregion, and those saddles are connected by a heteroclinic cycle that crosses every switching curve transversally exactly once. We present a method of Melnikov type to derive sufficient conditions under which the perturbed stable and unstable manifolds intersect transversally. Such transversal intersections imply that the corresponding Poincaré map has a transverse heteroclinic cycle. As applications, we present examples with 2 and 4 switching curves respectively. Our numerical simulations suggest that such transversal intersections result in the appearance of chaotic motions in those example systems. [ABSTRACT FROM AUTHOR]

By combining the generalized Padé approximation method and the well-known Lindstedt-Poincaré method, a novel technique, referred to as the generalized Padé-Lindstedt-Poincaré method, is proposed for determining homo-/heteroclinic orbits of nonlinear autonomous oscillators. First, the classical Padé approximation method is generalized. According to this generalization, the numerator and denominator of the Padé approximant are extended from polynomial functions to a series composed of any kind of continuous function, which means that the generalized Padé approximant is not limited to some certain forms, but can be constructed variously in solving different matters. Next, the generalized Padé approximation method is introduced into the Lindstedt-Poincaré method's procedure for solving the perturbation equations. Via the proposed generalized Padé-Lindstedt-Poincaré method, the homo-/heteroclinic bifurcations of the generalized Helmholtz-Duffing-Van der Pol oscillator and $$\Phi ^{6}$$ -Van der Pol oscillator are predicted. Meanwhile, the analytical solutions to these oscillators are also calculated. To illustrate the accuracy of the present method, the solutions obtained in this paper are compared with those of the Runge-Kutta method, which shows the method proposed in this paper is both effective and feasible. Furthermore, the proposed method can be also utilized to solve many other oscillators. [ABSTRACT FROM AUTHOR]

In this paper, a novel construction of solutions of nonlinear oscillators are proposed which can be called as the quadratic generalized harmonic function. Based on this novel solution, a modified generalized harmonic function Lindstedt-Poincaré method is presented which call the quadratic generalized harmonic function perturbation method. Via this method, the homoclinic and heteroclinic bifurcations of Duffing-harmonic-van de Pol oscillator are investigated. The critical value of the homoclinic and heteroclinic bifurcation parameters are predicted. Meanwhile, the analytical solutions of homoclinic and heteroclinic orbits of this oscillator are also attained. To illustrate the accuracy of the present method, all the above-mentioned results are compared with those of Runge-Kutta method, which shows that the proposed method is effective and feasible. In addition, the present method can be utilized in study many other oscillators. [ABSTRACT FROM AUTHOR]

Recently, an evolutionary game dynamics model taking into account the environmental feedback has been proposed to describe the co-evolution of strategic actions of a population of individuals and the state of the surrounding environment; correspondingly a range of interesting dynamic behaviors have been reported. In this paper, we provide new theoretical insight into such behaviors and discuss control options. Instead of the standard replicator dynamics, we use a more realistic and comprehensive model of replicator–mutator dynamics , to describe the strategic evolution of the population. After integrating the environment feedback, we study the effect of mutations on the resulting closed-loop system dynamics. We prove the conditions for two types of bifurcations, Hopf bifurcation and Heteroclinic bifurcation , both of which result in stable limit cycles. These limit cycles have not been identified in existing works, and we further prove that such limit cycles are in fact persistent in a large parameter space and are almost globally stable. In the end, an intuitive control policy based on incentives is applied, and the effectiveness of this control policy is examined by analysis and simulations. [ABSTRACT FROM AUTHOR]

In this work we have considered a prey-predator model with strong Allee effect in the prey growth function, Holling type-II functional response and density dependent death rate for predators. It presents a comprehensive study of the complete global dynamics for the considered system. Especially to see the effect of the density dependent death rate of predator on the system behavior, we have presented the two parametric bifurcation diagrams taking it as one of the bifurcation parameters. In course of that we have explored all possible local and global bifurcations that the system could undergo, namely the existence of transcritical bifurcation, saddle node bifurcation, cusp bifurcation, Hopf-bifurcation, Bogdanov-Takens bifurcation and Bautin bifurcation respectively. [ABSTRACT FROM AUTHOR]

From the saddle-node-Hopf bifurcation point of view, this paper considers a modified Leslie–Gower predator-prey model with time delay and the Michaelis–Menten type prey harvesting. Firstly, we discuss the stability of the equilibria, obtain the critical conditions for the saddle-node-Hopf bifurcation, and give the completion bifurcation set by calculating the universal unfoldings near the saddle-node-Hopf bifurcation point by using the normal form theory and center manifold theorem. Then we derive the parameter conditions for the existence of monostable coexistence equilibrium and the parameter regions in which both the prey-extinction and the coexistence equilibrium (or coexistence periodic or quasi-periodic solutions) are simultaneously stabilized. We also investigate the heteroclinic bifurcation, and describe the phenomenon that the periodic behavior disappears as through the heteroclinic bifurcation. Finally, some numerical simulations are performed to support our analytic results. [ABSTRACT FROM AUTHOR]

We prove that genuine nonuniformly hyperbolic dynamics emerge when flows in RN with homoclinic loops or heteroclinic cycles are subjected to certain time-periodic forcing. In particular, we establish the emergence of strange attractors and Sinai--Ruelle--Bowen (SRB) measures with strong statistical properties (central limit theorem, exponential decay of correlations, etc.). We identify and study the mechanism responsible for the nonuniform hyperbolicity: saddle point shear. Our results apply to concrete systems of interest in the biological and physical sciences, such as May--Leonard models of Lotka--Volterra dynamics. [ABSTRACT FROM AUTHOR]

In this paper, the method of nonlinear time transformation is applied to obtain analytical approximation of heteroclinic connections in the problem of stability loss of self-oscillations near 1:4 resonance. As example, we consider the case of parametric and self-excited oscillator near the 1:4 subharmonic resonance. The method uses the unperturbed heteroclinic connection in the slow flow to determine conditions under which the perturbed heteroclinic connection persists. The results show that for small values of damping, the nonlinear time transformation method can predict well both the square and clover heteroclinic connection near the 1:4 resonance. The analytical finding is confirmed by comparisons to the results obtained by numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we consider two coupled oscillators exhibiting both transient chaos and energy transfer from mechanical to electrical oscillators. Melnikov method is applied to these oscillators with linear damping and strongly nonlinear coupling terms in order to study the possibility of existence of chaos and transversal heteroclinic orbits and their control in a dynamical system. The energy transfer is studied using a qualitative measure of the system which can be obtained by computing the energy dissipated in it. At last, the numerical simulation is carried out for this system. [ABSTRACT FROM AUTHOR]

In this article, we investigate a prey- predator model with Allee effect and state-dependent impulsive harvesting. We obtain the sufficient conditions for the existence and uniqueness of order-1 periodic solution of system (1.2) by means of the geometry theory of semicontinuous dynamic system and the method of successor function. We also obtain that system (1.2) exhibits the phenomenon of heteroclinic bifurcation about parameter $$\alpha $$ . The methods used in this article are novel and prove the existence of order-1 periodic solution and heteroclinic bifurcation. [ABSTRACT FROM AUTHOR]

Abstract: Based on the stretch-twist-fold flow, a generalized stretch-twist-fold flow is introduced. By choosing an appropriate bifurcation parameter, Hopf bifurcations occur in this system when the bifurcation parameter exceeds a critical value. The formulae for determining the direction of the Hopf bifurcations and the stability of bifurcating periodic solutions are presented. In addition, the paper also investigates the bifurcations of the heteroclinic orbits for this system. The existence and its associated existing regions are given for two heteroclinic orbits, respectively. Finally, some numerical simulations for justifying the theoretical analysis are presented. [Copyright &y& Elsevier]

In this paper, we make a bifurcation analysis of a mathematical model for an electric circuit formed by the four fundamental electronic elements: one memristor, one capacitor, one inductor and one resistor. The considered model is given by a discontinuous piecewise linear system of ordinary differential equations, defined on three zones in ℝ3, determined by |z| < 1 (called the central zone) and |z| > 1 (the external zones). We show that the z-axis is filled by equilibrium points of the system, and analyze the linear stability of the equilibria in each zone. Due to the existence of this line of equilibria, the phase space ℝ3 is foliated by invariant planes transversal to the z-axis and parallel to each other, in each zone. In this way, each solution is contained in a three-piece invariant set formed by part of a plane contained in the central zone, which is extended by two half planes in the external zones. We also show that the system may present nonlinear oscillations, given by the existence of infinitely many periodic orbits, each one belonging to one such invariant set and passing by two of the three zones or passing by the three zones. These orbits arise due to homoclinic and heteroclinic bifurcations, obtained varying one parameter in the studied model, and may also exist for some fixed sets of parameter values. This intricate phase space may bring some light to the understanding of these memristor properties. The analytical and numerical results obtained extend the analysis presented in [Itoh & Chua, 2009; Messias et al., 2010]. [ABSTRACT FROM AUTHOR]