Degenerate Equilibrium Point Research Articles

Response solutions of 3-dimensional degenerate quasi-periodic systems with small parameter

This paper considers a class of 3-dimensional real analytic nonlinear quasi-periodic systems with a small perturbation parameter, whose unperturbed part has a degenerate equilibrium point. Based on the Leray-Schauder Continuation Theorem [18] , [20] and using technique of outer parameter, we prove that the system has a small response solution for many sufficiently small parameters.

Impacts of strong Allee effect and hunting cooperation for a Leslie-Gower predator-prey system

In this paper, we propose a Leslie-Gower predator-prey system with strong Allee effect in prey and hunting cooperation among predators. To discuss the impacts of Allee effect and hunting cooperation, we choose the severity of Allee effect and the cooperative hunting coefficient as the main control parameters. First, using the topology equivalent method, type of singular point (0,0) is obtained. Moreover, the basin of attraction of (0,0) is studied. Then, the stability of all nonnegative equilibrium points, including type of degenerate equilibrium point, is discussed. Based on Sotomayor’s theorem, the existence of saddle-node bifurcation is derived. To determine the stability of limit cycles arising from the Hopf bifurcation, the first Lyapunov number is calculated. Meanwhile, by the rigorous mathematical proofs, we obtain that the bifurcating limit cycle is stable if the hunting cooperation coefficient and the severity of Allee effect are sufficiently small. Finally, numerical simulations are presented to validate the theoretical results and further explore the influences of Allee effect and hunting cooperation. These results show that strong Allee effect and hunting cooperation have significant impacts on dynamical behaviors of the system.

Quasi-periodic solutions for the two-dimensional systems with an elliptic-type degenerate equilibrium point under small perturbations

In this paper, we consider the existence of quasi-periodic solutions for the two-dimensional systems with an elliptic-type degenerate equilibrium point under small perturbations. We prove that under appropriate hypotheses there exist quasi-periodic solutions for perturbed ODEs near the equilibrium point for most parameter values.

Quasi-Periodic Solutions for Differential Equations with an Elliptic-Type Degenerate Equilibrium Point Under Small Perturbations

This work focuses on the existence of quasi-periodic solutions for ordinary and delay differential equations (ODEs and DDEs for short) with an elliptic-type degenerate equilibrium point under quasi-periodic perturbations. We prove that under appropriate hypotheses there exist quasi-periodic solutions for perturbed ODEs and DDEs near the equilibrium point for most parameter values, then apply these results to the delayed van der Pol’s oscillator with zero-Hopf singularity.

On the reducibility of a nonlinear periodic system with degenerate equilibrium

ABSTRACTIn this paper, we prove the reducibility of a class of nonlinear periodic differential equation with degenerate equilibrium point under small perturbations, and obtain a periodic solution near the equilibrium point. The result is new and more general, including that of the paper [X. Lu and J. Xu, On small perturbations of a nonlinear periodic system with degenerate equilibrium, Nonlinear Differ. Equ. Appl. 21 (2014), pp. 361–370 ].

Construction of response solutions for two classes of quasi-periodically forced four-dimensional nonlinear systems with degenerate equilibrium point under small perturbations

By developing two KAM theorems, in this paper, we show that two classes of quasi-periodically forced four-dimensional nonlinear systems with degenerate equilibrium point, a reversible system and a non-conservative system, admit a response solution under small perturbations. For the degenerate reversible system, applying special structure of unperturbed nonlinear term and Herman method, we successfully control the shift of equilibrium point, which is difficult in view of the degenerate linear term. For the degenerate non-conservative system, KAM method is brought into force even in completely degenerate case because of the restrictions on the smallness and average of perturbation. Moreover, arithmetic condition on the frequency is assumed to satisfy the Brjuno–Rüssmann's non-resonant condition. By the Pöschel–Rüssmann KAM method, we prove that these two kinds of perturbed systems can be reduced to a suitable normal form with zero as equilibrium point by a quasi-periodic transformation.

On small perturbation of four-dimensional quasi-periodic system with degenerate equilibrium point

This paper focuses on quasi-periodic perturbation of four dimensional nonlinear quasi-periodic system. Using the KAM method, the perturbed system can be reduced to a suitable normal form with zero as equilibrium point by a quasi-periodic transformation. Hence, the perturbed system has a quasi-periodic solution near the equilibrium point.

On small perturbations of a nonlinear periodic system with degenerate equilibrium

This paper considers the reducibility and existence of periodic solutions for a class of nonlinear periodic system with a degenerate equilibrium point under small perturbations. By introducing some parameter, we consider an equivalent periodic system. Then we prove that by an affine linear periodic transformation the parameterized periodic system is reducible to one with zero as an equilibrium. Topological degree theorem ensures that for some parameter the result can go back to the original system. Then, we obtain a small periodic solution.

Reducibility for a Class of Almost-Periodic Differential Equations with Degenerate Equilibrium Point under Small Almost-Periodic Perturbations

This paper focuses on almost-periodic time-dependent perturbations of an almost-periodic differential equation near the degenerate equilibrium point. Using the KAM method, the perturbed equation can be reduced to a suitable normal form with zero as equilibrium point by an affine almost-periodic transformation. Hence, for the equation we can obtain a small almost-periodic solution.

On quasi-periodic perturbations of hyperbolic-type degenerate equilibrium point of a class of planar systems

In this paper we consider two-dimensional nonlinear quasi-periodic system with small perturbations. Assume that the unperturbed system has a hyperbolic-type degenerate equilibrium point and the frequency satisfies the Diophantine conditions. Using the KAM iteration we prove that for sufficiently small perturbations, the system can be reduced by a nonlinear quasi-periodic transformation to a suitable normal form with an equilibrium point at the origin. Hence, for the system we can obtain a small quasi-periodic solution.

On small perturbation of two-dimensional quasi-periodic systems with hyperbolic-type degenerate equilibrium point

In this paper we consider small quasi-periodic perturbation of two-dimensional nonlinear quasi-periodic system with hyperbolic-type degenerate equilibrium point. By KAM method we prove that it can be reduced to a suitable normal form with zero as equilibrium point by a quasi-periodic transformation. Hence, the perturbed system has a quasi-periodic solution near the equilibrium point.

Reducibility for a class of nonlinear quasi-periodic differential equations with degenerate equilibrium point under small perturbation

AbstractIn this paper, using the Kolmogorov–Arnold–Moser method we prove reducibility of a class of nonlinear quasi-periodic differential equation with degenerate equilibrium point under small perturbation and obtain a quasi-periodic solution near the equilibrium point.

A note on the relation between period and energy of periodic orbits near equilibrium points

We investigate the growth of the periods of periodic solutions of ordinary differential equations which are arbitrarily close to an isolated equilibrium point. More precisely, we estimate the period of periodic orbits near a completely degenerate equilibrium point (the derivative of the vector field at the equilibrium point is identically zero) and show that it tends to infinity as the amplitude of the orbits tends to zero.Based on this, we give estimates on the relation of the period and the energy of periodic orbits near singular points in Hamiltonian systems.

Travelling Wavefronts in Reaction-Diffusion Equations with Convection Effects and Non-Regular Terms

This paper deals with the appearance of monotone bounded travelling wave solutions for a parabolic reaction-diffusion equation which frequently meets both in chemical and biological systems. In particular, we prove the existence of monotone front type solutions for any wave speed c ≥ c* and give an estimate for the threshold value c*. Our model takes into account both of a density dependent diffusion term and of a non-linear convection effect. Moreover, we do not require the main non-linearity g to be a regular C1 function; in particular we are able to treat both the case when g′(0) = 0, giving rise to a degenerate equilibrium point in the phase plane, and the singular case when g′(0) = +∞. Our results generalize previous ones due to Aronson and Weinberger [Adv. Math. 30 (1978), pp. 33–76 ], Gibbs and Murray (see Murray [Mathematical Biology, Springer-Verlag, Berlin, 1993 ]) and McCabe, Leach and Needham [SIAM J. Appl. Math. 59 (1998), pp. 870–899 ]. Finally, we obtain our conclusions by means of a comparison-type technique which was introduced and developed in this framework in a recent paper by the same authors.

Wheel torque proportioning, rear steering, and normal force control: a structural investigation

This paper is concerned with various analysis and design issues for a possible set of control channels which may be used to improve the maneuverability and handling of automobiles. Of the possible set of controls, the current paper is limited to variable wheel torque proportioning, rear steering, and normal force control. The analysis and design issues are studied using an extensive linear model representing the lateral, longitudinal, and vertical dynamics, including their interactions. For the investigation of the finite-time behavior of system dynamics, we first introduce the notion of a reachable set where the attainability of a particular state vector at a particular time instant is considered. Next, we examine the attainability of a particular output trajectory within a finite-time interval using the notion of output function controllability. Several structural properties of these concepts have been studied. Various degenerate operating points are identified, where the effectiveness of the control channels deteriorates. In the steady-state analysis, the reference tracking and regulation problems are considered. It is shown that for some input/output subsystems those problems are generically solvable due to the minimum phase structure of the subsystems. For some others, those problems are solvable provided that the equilibrium point is not a degenerate equilibrium point. Finally, we show that for some input/output subsystems those problems are generically unsolvable.