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In this paper, an age-structured predator–prey system with Beddington–DeAngelis (B–D) type functional response, prey refuge and harvesting is investigated, where the predator fertility function f(a) and the maturation function β (a) are assumed to be piecewise functions related to their maturation period τ . Firstly, we rewrite the original system as a non-densely defined abstract Cauchy problem and show the existence of solutions. In particular, we discuss the existence and uniqueness of a positive equilibrium of the system. Secondly, we consider the maturation period τ as a bifurcation parameter and show the existence of Hopf bifurcation at the positive equilibrium by applying the integrated semigroup theory and Hopf bifurcation theorem. Moreover, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are studied by applying the center manifold theorem and normal form theory. Finally, some numerical simulations are given to illustrate of the theoretical results and a brief discussion is presented. [ABSTRACT FROM AUTHOR]

We review a recent application of the ideas of normal form theory to systems (Hamiltonian ones or general ODEs) where the perturbing term is not periodic in one coordinate variable. The main difference from the standard case consists in the non-uniqueness of the normal form and the total absence of the small divisors problem. The exposition is quite general, so as to allow extensions to the case of more non-periodic coordinates, and more functional settings. Here, for simplicity, we work in the real-analytic class. [ABSTRACT FROM AUTHOR]

This article discusses the bifurcation analysis and control of a valve-controlled hydraulic cylinder system. The dynamic system of the valve-controlled hydraulic cylinder is established. Normal form theory and Hopf bifurcation theory are used to analyse the bifurcation characteristic at equilibria of the system. Then, a dynamic-state feedback control method is proposed. A nonlinear controller is set for the system to control the bifurcation with the method. By adjusting the control parameters, the delay of model bifurcation and the stability of the system can be changed. Numerical analysis verifies the correctness of bifurcation control.

In this paper, a new four-dimensional (4D) hyperchaotic biplane system is designed and presented. The dynamical properties of this new system are studied by means of tools such as bifurcation diagrams, Lyapunov exponents and phase diagrams. The Hopf bifurcation and periodic solutions of this hyperchaotic system are solved analytically. In addition, a new hyperchaotic control strategy is applied, and a comparative analysis of the controlled system is performed. [ABSTRACT FROM AUTHOR]

In this paper, a new four-dimensional (4D) hyperchaotic biplane system is designed and presented. The dynamical properties of this new system are studied by means of tools such as bifurcation diagrams, Lyapunov exponents and phase diagrams. The Hopf bifurcation and periodic solutions of this hyperchaotic system are solved analytically. In addition, a new hyperchaotic control strategy is applied, and a comparative analysis of the controlled system is performed.

We consider a family of Schrödinger equations with unbounded Hamiltonian quadratic nonlinearities on a generic tori of dimension d ≥ 1. We study the behavior of high Sobolev norms H s , s ≫ 1 , of solutions with initial conditions in H s whose H ρ -Sobolev norm, 1 ≪ ρ ≪ s , is smaller than ε ≪ 1. We provide a control of the H s -norm over a time interval of order O (ε − 2). Due to the lack of conserved quantities controlling high Sobolev norms, the key ingredient of the proof is the construction of a modified energy equivalent to the "low norm" H ρ (when ρ is sufficiently high) over a nontrivial time interval O (ε − 2). This is achieved by means of normal form techniques for quasi-linear equations involving para-differential calculus. The main difficulty is to control the possible loss of derivatives due to the small divisors arising form three waves interactions. By performing "tame" energy estimates we obtain upper bounds for higher Sobolev norms H s. [ABSTRACT FROM AUTHOR]

We construct a family of completely integrable systems -close to a slow-fast Hamiltonian system with two degrees of freedom which is described in the framework of the averaging method over slow-fast phase spaces with -symmetry. Our approach is based on the free-coordinate normalization procedure for slow-fast Hamiltonian system with two degree of freedom. [ABSTRACT FROM AUTHOR]

This article is concerned with the diffusion of a sport in a region, and the innovation diffusion model comprising of population classes, viz. nonadopters class, information class and adopters class. A qualitative analysis is carried out to assess the global asymptotic stability of the interior equilibrium for null delay. It has also been proved that the parameter τ (age gaps among sportspersons) in the intra-specific competition between the new players and the senior players can even destabilize the otherwise globally stable interior equilibrium state and the coexistence of all the populations is possible through periodic solutions due to Hopf bifurcation. With the help of normal form theory and center manifold arguments, the stability of bifurcating periodic orbits is determined. Numerical simulations have been executed in support of the analytical findings. [ABSTRACT FROM AUTHOR]

The effect of retardation along second vector of angular velocity on a new attitude system of quad-rotor unmanned aerial vehicle (QUAV) is examined in this paper. Catastrophic and hovering conditions of rotor dynamics are verified through bifurcation analysis. It is analyzed that disturbed rotor dynamics during yaw maneuvering lead towards the existence of oscillations and hamper the efficiency of attitude system. We have categorically signified the situation where remote controller of attitude system cannot prevent flipping and consequently, shortens the life of QUAV. This type of situation arises in a strong wind friction zone where positive drag force of rotor dynamics is impeded by the magnitude of rotational moment of inertia. Moreover, subcritical and zero-Hopf bifurcations aroused due to retardational effect in attitude system are analyzed with the aid of normal form and averaging theory. Numerical simulations are also provided for validation of analytical results. [ABSTRACT FROM AUTHOR]

In this paper, the zero-Hopf bifurcations are studied for a generalized Lorenz system. Firstly, by using the averaging theory and normal form theory, we provide sufficient conditions for the existence of small amplitude periodic solutions that bifurcate from zero-Hopf equilibria under appropriate parameter perturbations. Secondly, based on the Poincar'e compactification, the dynamic behavior of the generalized Lorenz system at infinity is described, and the zero-Hopf bifurcation at infinity is investigated. Additionally, for the above theoretical results, some related illustrations are given by means of the numerical simulation. [ABSTRACT FROM AUTHOR]

In this paper, chaos and bifurcation are explored for the controlled chaotic system, which is put forward based on the hybrid strategy in an unusual chaotic system. Behavior of the controlled system with variable parameter is researched in detain. Moreover, the normal form theory is used to analyze the direction and stability of bifurcating periodic solution.

Abstract Using center manifold theory, we investigate complex dynamics in a segmented disc dynamo with mechanical friction by making a determination of equilibrium states and analyzing local stability. We find that the system undergoes Hopf bifurcation for a key parameter. Normal form theory produces the formulae for determining a super- or sub-critical Hopf bifurcation; the stability of the bifurcating limit cycle is also determined. In addition, integrability of the system is studied carefully.

A dynamical system for optimal path of energy and resources between two cities in China is considered in this paper. We have discussed dynamics of variables and parameters involved in mentioned system. Bifurcation analysis around non-hyperbolic equilibria is also explained for codimension 1 and 2 bifurcations. Furthermore, double-zero eigenvalue condition is calculated for the proposed model. We have adopted methodology of the generalized vectors for existence of Bogdanov–Takens bifurcation critical point and used analytical computations instead of center manifold theorem for Bogdanov–Takens bifurcation around zero equilibria. Further, with the aid of bifurcation diagram, phase portraits and time history, we discussed occurrence of period doubling, Hopf bifurcation and chaotic region of our proposed model. Based on Lyapunov function and robust control, optimal controllers are designed using Hamilton-Jacobi theorem for the stability of disturbance and aperiodic solution in optimal transportation system (3) due to energy imports from city A to city B. [ABSTRACT FROM AUTHOR]

A normal form theory for non-quasiperiodic systems is combined with the special properties of the partially averaged Newtonian potential pointed out in Pinzari (Celest Mech Dyn Astron 131(5):22, 2019) to prove, in the averaged, planar three-body problem, the existence of a plenty of motions where, periodically, the perihelion of the inner body affords librations about one equilibrium position and its ellipse squeezes to a segment before reversing its direction and again decreasing its eccentricity (perihelion librations). [ABSTRACT FROM AUTHOR]

A nonlinear mathematical model with Holling II functional response describing the dynamics of nonadopter and adopters population in a stage structured innovation diffusion model, which incorporates the evaluation stage (multiple delays), is proposed. Firstly, we study the stability and the existence of periodic solutions via Hopf bifurcation with respect to both delays at the positive equilibrium by analyzing the distribution of the roots of the corresponding exponential characteristic equation obtained through the variational matrix. The direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined with the help of normal form theory and center manifold theorem. Meanwhile, various cases are discussed to examine the effect of different delays on the stability of delayed innovation diffusion system and are also established numerically. It is also observed that the cumulative density of external influences has a significant role in developing maturity stage (adoption stage) in the system. Finally, numerical simulations are carried out to support and supplement the analytical findings. [ABSTRACT FROM AUTHOR]

The main purpose of this article is to find a criterion for the nonexistence of formal first integrals for nonlinear systems under general resonance. An algorithm illustrates an application to a class of generalized Lokta-Volterra systems. Our result generalize the classical Poincare's nonintegrability theorem and the existing results in the literature.

A nonlinear modified form of Bass model involving the interactions of non-adopter and adopter populations has been proposed to describe the process of diffusion of a new technology in the presence of evaluation period (time delay). The basic aim is to model the diffusion of those technologies which require higher investments, and which require government subsidies for promotions in the various markets. We use government incentives and the costs in the form of external factors, as well as the internal word of mouth that considerably influence the non-adopters decisions. A qualitative analysis has been performed to determine the stability of the various equilibria. The Hopf bifurcation occurs near the positive equilibrium when the time delay passes some critical values. By applying the normal form theory and the center manifold reduction for functional differential equations, explicit formulae presenting stability properties of bifurcating periodic solutions have been computed. Moreover, the intra-specific competition has played an important role in establishing the maturity stage in the innovation diffusion model. Numerical analysis has been carried out to justify the correctness of our analytical findings. [ABSTRACT FROM AUTHOR]

We consider the spatial central force problem with a real analytic potential. We prove that for all analytic potentials, but for the Keplerian and the harmonic ones, the Hamiltonian fulfills a nondegeneracy property needed for the applicability of Nekhoroshev's theorem. We deduce stability of the actions over exponentially long times when the system is subject to an arbitrary analytic perturbation. The case where the central system is put in interaction with a slow system is also studied and stability over exponentially long time is proved. [ABSTRACT FROM AUTHOR]

The amplitude of limit cycles arising from Hopf bifurcation is estimated for nonlinear delay-differential equations by means of analytical formulas. An improved analytical estimation is introduced, which allows more accurate quantitative prediction of periodic solutions than the standard approach that formulates the amplitude as a square-root function of the bifurcation parameter. The improved estimation is based on special global properties of the system: the method can be applied if the limit cycle blows up and disappears at a certain value of the bifurcation parameter. As an illustrative example, the improved analytical formula is applied to the problem of stick balancing.

From the numerous field and roller rig tests, we observe one type of high speed railway vehicle has a small amplitude periodical oscillation and another one has a large amplitude periodical oscillation at loss of stability. The lateral dynamic models of the high speed bogies are established with the nonlinearity of wheel–rail contact and the non-smoothness of yaw damper taken into consideration. Following the bifurcation approach, simulation results of the two bogies are extensively computed with use of the path-following method. They demonstrate the supercritical and subcritical bifurcations exist in the different types of the considered vehicle systems. The application of Hopf normal theory makes a weak way to mathematically prove the bifurcation types for the two types of high speed railway vehicles Therefore the influence of parameters variation on the deduced function in normal form is further studied and the numerical method is also used for verification. The bifurcation analysis demonstrates that the yaw damper is one of the key parameters that can change the Hopf bifurcation type. [ABSTRACT FROM AUTHOR]

In this paper, a love-triangle model with nonlinear romantic sentimental interactions and four time delays is proposed. Regarding time delay as bifurcating parameter, the dynamical behaviors which include local asymptotical stability and Hopf bifurcation are studied in detail by analyzing the characteristic equation corresponding to the linearized system of a love-triangle model. When the delay passes through a sequence of critical values, we find that Hopf bifurcation occurs. Furthermore, stability and direction of bifurcating periodic solution are derived by applying the normal form theory and the center manifold theorem in our love-triangle model. Finally, an illustrative example is also used to support our theoretical results. [ABSTRACT FROM AUTHOR]

The amplitude of limit cycles arising from Hopf bifurcation is estimated for nonlinear delay-differential equations by means of analytical formulas. An improved analytical estimation is introduced, which allows more accurate quantitative prediction of periodic solutions than the standard approach that formulates the amplitude as a square-root function of the bifurcation parameter. The improved estimation is based on special global properties of the system: the method can be applied if the limit cycle blows up and disappears at a certain value of the bifurcation parameter. As an illustrative example, the improved analytical formula is applied to the problem of stick balancing. [ABSTRACT FROM AUTHOR]

This papers deals with the computationally costly simulation of the frequency response function of geometrically nonlinear structures exhibiting cyclic symmetry. Its objective is to propose a reduction method that extends the model of a single sector reduced via the direct normal form to a full cyclically symmetric system. After validation on simple structures, the methodology is applied on a 3D meshed blade, representative component of the new generation of high bypass ratio turbofans. Based on both the proposed reduction methodology and theoretical conditions that predict the possible appearance of internal resonances as a function of the number of sectors and the engine order of the excitation, the complex nonlinear dynamics of a finely meshed full fan cyclic system is then investigated. This could not have been achieved without an appropriate reduction method. Branch switching algorithm, stability analysis, and bifurcation tracking compose the numerical tools employed along the harmonic balance method to simulate the forced response of the structure. • The reduction strategy is based on normal form theory and cyclic symmetric properties. • It provides a substantial gain in computation time while maintaining a good accuracy. • It captures complex nonlinear phenomena as internal resonances. • It allows the analysis of complex finite elements design. [ABSTRACT FROM AUTHOR]

A mathematical model describing the dynamics of toxin producing phytoplankton-zooplankton interaction with instantaneous nutrient recycling is proposed. We have explored the dynamics of plankton ecosystem with multiple delays; one due to gestation period in the growth of phytoplankton population and second due to the delay in toxin liberated by TPP. It is established that a sequence of Hopf bifurcations occurs at the interior equilibrium as the delay increases through its critical value. The direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined using the theory of normal form and center manifold. Meanwhile, effect of toxin on the stability of delayed plankton system is also established numerically. Finally, numerical simulations are carried out to support and supplement the analytical findings. [ABSTRACT FROM AUTHOR]