Heteroclinic Loop Research Articles

Limit cycles appearing from a generalized heteroclinic loop with a cusp and a nilpotent saddle

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 3n−1 limit cycles.

Codimension-3 Bifurcation in the p53 Regulatory Network Model

In this paper, a p53-Mdm2 mathematical model is analyzed to understand the biological implications of feedback loops in a p53 system. Results show that the model can undergo four types of codimension-3 Bogdanov–Takens bifurcations, including cusp, saddle, focus and elliptic. Specifically, we find new phenomena including the coexistence of four positive equilibria, two limit cycles, the coexistence of three stable states (two stable equilibria and one stable limit cycle, or three stable equilibria), a heteroclinic loop enclosing a smaller stable limit cycle and a larger stable limit cycle. These findings extend the understanding of the complex dynamics of the p53 system, and can provide some potential biological applications.

Global dynamics of a quintic Liénard system with -symmetry I: saddle case

In the paper we deal with a quintic Liénard system of the form with a -symmetry, where and a 3 b 2 ≠ 0. A complete study of this system with b 2 > 0, called the saddle case, is finished, showing that the system exhibits at most two limit cycles, and the necessary and sufficient conditions are obtained on the existence of two limit cycles and a two-saddle heteroclinic loop. We also present a global bifurcation diagram and the corresponding phase portraits of this system, including Hopf bifurcation, Bautin bifurcation, two-saddle heteroclinic loop bifurcation and double limit cycle bifurcation.

Periodic sinks and periodic saddle orbits induced by heteroclinic bifurcation in three-dimensional piecewise linear systems with two zones

For general three-dimensional piecewise linear systems, some explicit sufficient conditions are achieved for the existence of a heteroclinic loop connecting a saddle-focus and a saddle with purely real eigenvalues. Furthermore, certain sufficient conditions are obtained for the existence and number of periodic orbits induced by the heteroclinic bifurcation, through close analysis of the fixed points of the parameterized Poincaré map constructed along the hereroclinic loop. It turns out that the number can be zero, one, finite number or countable infinity, as the case may be. Some sufficient conditions are also acquired that guarantee these periodic orbits to be periodic sinks or periodic saddle orbits, respectively, and the main results are illustrated lastly by some examples.

Global Analysis of an Asymmetric Continuous Piecewise Linear Differential System with Three Linear Zones

In [Chen et al., 2020], the third author and other coauthors studied global dynamics of the following system: [Formula: see text] in the parameter region [Formula: see text]. To study completely the piecewise linear system, we consider the parameter region [Formula: see text] in this paper. Firstly, we study the local dynamics, such as the bifurcations of equilibria (including the equilibrium at infinity). Secondly, the number and stability of limit cycles are studied completely. Then, we analyze the existence of upper and lower saddle connections and homoclinic loops. Moreover, we show that there are no heteroclinic loops in this parameter region. Finally, we give the bifurcation diagram and all global phase portraits on the Poincaré disc are given as well as some numerical examples.

Sharp Bound of the Number of Zeros for a Liénard System with a Heteroclinic Loop

In the presented paper, the Abelian integral I h of a Liénard system is investigated, with a heteroclinic loop passing through a nilpotent saddle. By using a new algebraic criterion, we try to find the least upper bound of the number of limit cycles bifurcating from periodic annulus.

Generating Shilnikov chaos in 3D piecewise linear systems

We propose a construction algorithm that takes advantage of the geometric features of linear eigenspaces with focus-saddle and center-node equilibrium points to construct piecewise linear systems with Shilnikov chaos. Unlike previously published algorithms to generate chaos, we construct homoclinic (heteroclinic) orbits combining hyperbolic and nonhyperbolic domains by appropriately choosing locations where the trajectories move between linear domains through the switching surfaces at their crossing sections. We present two variants of our construction algorithm: On the first, a homoclinic orbit is constructed locating the exit and return points on the crossing section of the switching surface are connected by a nonhyperbolic center-node domain. In the second variant, we extend the construction method to construct heteroclinic loops with two switching surfaces between them. For the resulting systems a wide range of parameter values results on Shilnikov chaos. These chaotic systems are illustrated with numerical simulations.

Bifurcation of limit cycles near heteroclinic loops in near-Hamiltonian systems

In this paper, we study the bifurcation of limit cycles near a heterocilinc loop with hyperbolic saddles in a perturbed planar Hamiltonian system. We present a method for computing the coefficients in the corresponding expansion of the first order Melnikov function. With more those coefficients, more limit cycles could be determined around the heteroclinic loop. An example of studying limit cycles produced from a heteroclinic loop with 2 saddles is investigated to illustrate our method.

The saddle case of a nonsmooth Rayleigh–Duffing oscillator

We consider a single degree freedom oscillator in order to accurately represent some modeling with ship roll damping. The proposed oscillator is a nonsmooth Rayleigh–Duffing equation ẍ+aẋ+bẋ|ẋ|+cx+dx3=0. The main goal of this paper is to study the global dynamics of the nonsmooth Rayleigh–Duffing oscillator in the case d<0, i.e., the saddle case. The nonsmooth Rayleigh–Duffing oscillator is only C1 so that many classical theory cannot be applied directly. In order to see the tendency of evolutions in a large range, we study not only its finite equilibria but also the equilibria at infinity. We find necessary and sufficient conditions for existence of limit cycles and heteroclinic loops respectively. Finally, we give the complete global bifurcation diagram and classify all global phase portraits in the Poincaré disc in global parameters.

Nine Limit Cycles in a 5-Degree Polynomials Liénard System

In this article, we study the limit cycles in a generalized 5-degree Liénard system. The undamped system has a polycycle composed of a homoclinic loop and a heteroclinic loop. It is proved that the system can have 9 limit cycles near the boundaries of the period annulus of the undamped system. The main methods are based on homoclinic bifurcation and heteroclinic bifurcation by asymptotic expansions of Melnikov function near the singular loops. The result gives a relative larger lower bound on the number of limit cycles by Poincaré bifurcation for the generalized Liénard systems of degree five.

Bifurcations in an economic model with fractional degree

A planar ODE system which models the industrialization of a small open economy is considered. Because fractional powers are involved, its interior equilibria are hardly found by solving a transcendental equation and the routine qualitative analysis is not applicable. We qualitatively discuss the transcendental equation, eliminating the transcendental term to polynomialize the expression of extreme value, so that we can compute polynomials to obtain the number of interior equilibria in all cases and complete their qualitative analysis. Orbits near the origin, at which the system cannot be extended differentiably, are investigated by using the GNS method. Then we display all bifurcations of equilibria such as saddle-node bifurcation, transcritical bifurcation and a codimension 2 bifurcation on a one-dimensional center manifold. Furthermore, we prove nonexistence of closed orbits, homoclinic loops and heteroclinic loops, exhibit global orbital structure of the system and analyze the tendency of the industrialization development.

Heteroclinic Chaotic Threshold in a Nonsmooth System with Jump Discontinuities

In smooth systems, the form of the heteroclinic Melnikov chaotic threshold is similar to that of the homoclinic Melnikov chaotic threshold. However, this conclusion may not be valid in nonsmooth systems with jump discontinuities. In this paper, based on a newly constructed nonsmooth pendulum, a kind of impulsive differential system is introduced, whose unperturbed part possesses a nonsmooth heteroclinic solution with multiple jump discontinuities. Using the recursive method and the perturbation principle, the effects of the nonsmooth factors on the behaviors of the nonsmooth dynamical system are converted to the integral items which can be easily calculated. Furthermore, the extended Melnikov function is employed to obtain the nonsmooth heteroclinic Melnikov chaotic threshold, which implies that the existence of the nonsmooth heteroclinic orbits may be due to the breaking of the nonsmooth heteroclinic loops under the perturbation of damping, external forcing and nonsmooth factors. It is worth pointing out that the form of the nonsmooth heteroclinic Melnikov function is different from the one of the nonsmooth homoclinic Melnikov function, which is quite different from the classical Melnikov theory.

Cyclicity of periodic annulus and Hopf cyclicity in perturbing a hyper-elliptic Hamiltonian system with a degenerate heteroclinic loop

In this paper, we study the cyclicity of periodic annulus and Hopf cyclicity in perturbing a quintic Hamiltonian system. The undamped system is hyper-elliptic, non-symmetric with a degenerate heteroclinic loop, which connects a hyperbolic saddle to a nilpotent saddle. We rigorously prove that the cyclicity is 3 for periodic annulus when the weak damping term has the same degree as that of the associated Hamiltonian system. This result provides a positive answer to the open question whether the annulus cyclicity is 3 or 4. When the smooth polynomial damping term has degree n, first, a transformation based on the involution of the Hamiltonian is introduced, and then we analyze the coefficients involved in the bifurcation function to show that the Hopf cyclicity is [2n+13]. Further, for piecewise smooth polynomial damping with a switching manifold at the y-axis, we consider the damping terms to have degrees l and n, respectively, and prove that the Hopf cyclicity of the origin is [3l+2n+43] ([3n+2l+43]) when l≥n (n≥l).

Global Dynamics of a Planar Filippov System with Symmetry

Chen [2016a, 2016b] studied global dynamics of the Filippov systems [Formula: see text], respectively. To study the global dynamics of [Formula: see text] completely, since the dynamics of [Formula: see text] is very simple, we are only interested in the global dynamics of [Formula: see text] in this paper. Firstly, we use Briot–Bouquet transformations and normal sector methods to discuss these degenerate equilibria at infinity. Secondly, we discuss the number of limit cycles completely. Then, the sufficient and necessary conditions of existence of the heteroclinic loop are found. To estimate the upper bound of the heteroclinic loop bifurcation function on parameter space, a result on the amplitude of a unique limit cycle of a discontinuous Liénard system is given. Finally, the complete bifurcation diagram and all global phase portraits are presented. The global dynamic property of system [Formula: see text] is totally different from systems [Formula: see text].

Global dynamics of the Josephson equation in TS1

The Josephson equation ϕ˙=y,y˙=−sin⁡ϕ+ϵ(a−(1+γcos⁡ϕ)y) was researched by Sanders and Cushman (1986) [12] for its phase portraits when ϵ>0 is small by applying the averaging method. The parameter ϵ can actually be large or even any real number in the practical application of this model. When |ϵ| is not small, we cannot apply the averaging method because the system is not near-Hamiltonian. For general ϵ∈R, we present complete dynamics and more complex bifurcations of the Josephson equation in TS1, including saddle-node bifurcation, Hopf bifurcation, Bogdanov-Takens bifurcation, homoclinic loop bifurcation, two-saddle heteroclinic loop bifurcation, upper saddle connection bifurcation and lower saddle connection bifurcation. Moreover, we prove the monotonicity of bifurcation functions with respect to parameters and the nonexistence of a two-saddle heteroclinic loop for all a≠0.

Emergence of a square chaotic attractor through the collision of heteroclinic orbits

In this work we introduce a square chaotic attractor based on the collision of two heteroclinic orbits. Before the collision, the system presents the coexistence of two double scroll attractors that are generated via piecewise linear (PWL) systems that deal with two saddle-foci equilibria of different classes, i.e. the dimensions of the unstable and stable manifolds of one of the equilibrium points are one and two, respectively, and vice versa for the unstable and stable manifolds associated with the second equilibrium point. The new class is constructed based on the existence of a heteroclinic loop by linear affine systems with two saddle-focus equilibrium points of different types. The chaotic behavior of the proposed system is tested by the maximum Lyapunov exponent and the 0–1 chaos test. This new class of PWL dynamical systems exhibits different behaviors and allows the generation of different basins of attraction for the coexistence of two or more attractors. Therefore, a mechanism to establish bistability is provided and how it can be controlled (i.e. bistability annihilation) via varying system’s parameters is demonstrated.

Perturbation of a Period Annulus Bounded by a Saddle–Saddle Cycle in a Hyperelliptic Hamiltonian Systems of Degree Seven

In this paper, we study the limit cycle bifurcation by perturbing the period annuluses of two perturbed hyper-elliptic Hamiltonian systems of degree seven. The period annuluses are bounded by heteroclinic loops, inside or outside of which there exist two nilpotent cusps. The bifurcation function is Abelian integral which is the first-order approximation of the Poincare map. The sharp bounds of the number of limit cycles bifurcated from the periodic annuluses are obtained by Chebyshev criterion and asymptotic analysis.

The Dynamics of a Kind of Liénard System with Sixth Degree and Its Limit Cycle Bifurcations Under Perturbations

In this paper, the different topological types of phase portrait of the unperturbed Lienard system $$ {\dot{x}}=y,\ \ {\dot{y}}=-g(x)$$ are given, where $$\deg {g(x)}=6$$. We find that the expansion of the Melnikov function near any of closed orbits appeared in the above phase portraits, except a heteroclinic loop with a hyperbolic saddle and a nilpotent saddle of order one, has been studied. In this paper, we give the expansion of the Melnikov function near this kind of heteroclinic loop. Further, we present the conditions to obtain limit cycles bifurcated from a compound loop with a hyperbolic saddle and a nilpotent saddle of order one, and apply it to study the number of limit cycles for a kind of Lienard system under perturbations.

Hopf Bifurcation and Hopf-Pitchfork Bifurcation in an Integro-Differential Reaction-Diffusion System

We study the bifurcations of small amplitude time-periodic solutions and chaotic solutions of a two-component integro-differential reaction-diffusion system in one spatial dimension. The system has doubly degenerate points and triply degenerate points. The following results are obtained. (I) Around the doubly degenerate points, a reduced two-dimensional dynamical system on the center manifold is obtained. We find that the small amplitude stable time-periodic solutions can bifurcate from the non-uniform stationary solutions through the Hopf bifurcations for all $n$. (II) Around the triply degenerate point, a three-dimensional dynamical system on the center manifold is obtained. The reduced system can be transformed into normal form for the Hopf-Pitchfork bifurcation. The truncated normal form can possess the invariant tori and the heteroclinic loop. Furthermore, the system under the non $S^{1}$-symmetric perturbation may possess the Shil'nikov type homoclinic orbit. Numerical results for the integro-differential reaction-diffusion system are presented and found to be convincing.

Versal unfolding of a nilpotent Liénard equilibrium within the odd Liénard family

Recently attentions were paid to versal unfolding of degenerate equilibria within a restricted family of vector fields for a practical sense. The family of Liénard systems is such a family of significant physical sense but only within the family of even Liénard systems there were found results for a nilpotent degenerate equilibrium. In this paper we discuss versal unfolding of a nilpotent Liénard equilibrium within the family of odd Liénard systems. As the well-known Bogdanov-Takens normal form is not available for the odevity, we prove that the degeneracy within the family is of codimension 2 and find its versal unfolding. We further discuss the unfolding system, displaying all its bifurcations such as pitchfork bifurcation, center-saddle bifurcation, center bifurcation and homoclinic (heteroclinic) loop bifurcation, which show how a single homoclinic loop, a twin homoclinic loop and a heteroclinic loop arise from a center or a (degenerate) saddle.