Heteroclinic Loop Research Articles

Bifurcation of limit cycles from a Liénard system with a heteroclinic loop connecting two nilpotent saddles

In this article, we study the limit cycles bifurcated from a Lienard system with a heteroclinic loop connecting two nilpotent saddles. We apply expansion theory of a first-order Melnikov function to investigate the number of limit cycles near the heteroclinic loop and the center, and by some perturbation theory we find 3 limit cycles with 7 different distributions. Last, the least upper bound of the number of limit cycles bifurcated from the annulus is given by an algebraic criterion developed in J. Differ. Equ. 251, 1656–1669 (2011).

ASYMPTOTIC EXPANSIONS OF MELNIKOV FUNCTIONS AND LIMIT CYCLE BIFURCATIONS

In the study of the perturbation of Hamiltonian systems, the first order Melnikov functions play an important role. By finding its zeros, we can find limit cycles. By analyzing its analytical property, we can find its zeros. The main purpose of this article is to summarize some methods to find its zeros near a Hamiltonian value corresponding to an elementary center, nilpotent center or a homoclinic or heteroclinic loop with hyperbolic saddles or nilpotent critical points through the asymptotic expansions of the Melnikov function at these values. We present a series of results on the limit cycle bifurcation by using the first coefficients of the asymptotic expansions.

Bifurcation of limit cycles from a hyper-elliptic Hamiltonian system with a double heteroclinic loops

In this article, we consider the Liénard system of the form x ˙ =y, y ˙ =x(x−1)(x+1) ( x 2 − 3 ) +ε ( α + β x 2 + γ x 4 ) y with 0<ε≪1, a, b and c are real bounded parameters. We prove that the least upper bound of the number of isolated zeros of the corresponding Abelian integral I(h)= ∮ Γ h ( α + β x 2 + γ x 4 ) ydx is four (counting the multiplicity). This implies that the number of limit cycles that bifurcated from periodic orbits of the unperturbed system for ε=0 is less than or equal to four.MSC:34C05, 34C07, 34C08.

BIFURCATION OF ROUGH HETEROCLINIC LOOP WITH ORBIT FLIPS

In this paper, heteroclinic loop bifurcations with double orbit flips are investigated in four-dimensional vector fields. We obtain the bifurcation equations by setting up a local coordinate system near the rough heteroclinic orbit and establishing the Poincaré map. By means of the bifurcation equations, we investigate the existence, coexistence and noncoexistence of periodic orbit, homoclinic loop and heteroclinic loop under some nongeneric conditions. The approximate expressions of corresponding bifurcation curves (or surfaces) are also given. An example of application is also given to demonstrate the existence of the heteroclinic loop with double orbit flips.

Heteroclinic tangles in time-periodic equations

In this paper we prove that, when a heteroclinic loop is periodically perturbed, three types of heteroclinic tangles are created and they compete in the space of μ where μ is a parameter representing the magnitude of the perturbations. The three types are (a) transient heteroclinic tangles containing no Gibbs measures; (b) heteroclinic tangles dominated by sinks representing stable dynamical behavior; and (c) heteroclinic tangles with strange attractors admitting SRB measures representing chaos. We also prove that, as μ→0, the organization of the three types of heteroclinic tangles depends sensitively on the ratio of the unstable eigenvalues of the saddle fixed points of the heteroclinic connections. The theory developed in this paper is explicitly applicable to the analysis of various specific differential equations and the results obtained are well beyond the capacity of the classical Birkhoff–Melnikov–Smale method.

Bifurcations of a nongeneric heteroclinic loop with nonhyperbolic equilibria

In this paper, using the local moving frame approach, we investigate bifurcations of nongeneric heteroclinic loop with a nonhyperbolic equilibrium $p_1$ and a hyperbolic saddle $p_2$, where $p_1$ is assumed to undergo a transcritical bifurcation. Firstly, we establish the persistence of a nongeneric heteroclinic loop, the existence of a homoclinic loop and a periodic orbit when the transcritical bifurcation does not occur. Secondly, bifurcations of a nongeneric heteroclinic loop accompanied with a transcritical bifurcation are discussed. We obtain the existence of heteroclinic orbits, a homoclinic loop, a heteroclinic loop and a periodic orbit. Some bifurcation patterns different from the case of the generic heteroclinic loop accompanied with transcritical bifurcation are revealed. The results achieved here can be extended to higher dimensional systems.

Bifurcations of heteroclinic loop accompanied by pitchfork bifurcation

In this paper, using the local coordinate moving frame approach, we investigate bifurcations of generic heteroclinic loop with a hyperbolic equilibrium and a nonhyperbolic equilibrium which undergoes a pitchfork bifurcation. Under some generic hypotheses, the existence of homoclinic loop, heteroclinic loop, periodic orbit and three or four heteroclinic orbits is obtained. In addition, the non-coexistence conditions for homoclinic loop and periodic orbit are also given. Note that the results achieved here can be extended to higher dimensional systems.

BIFURCATION OF LIMIT CYCLES IN SMALL PERTURBATIONS OF A HYPER-ELLIPTIC HAMILTONIAN SYSTEM WITH TWO NILPOTENT SADDLES

In this paper we study the first-order Melnikov function for a planar near-Hamiltonian system near a heteroclinic loop connecting two nilpotent saddles. The asymptotic expansion of this Melnikov function and formulas for the first seven coefficients are given. Next, we consider the bifurcation of limit cycles in a class of hyper-elliptic Hamiltonian systems which has a heteroclinic loop connecting two nilpotent saddles. It is shown that this system can undergo a degenerate Hopf bifurcation and Poincare bifurcation, which emerges at most four limit cycles in the plane for sufficiently small positive e. The number of limit cycles which appear near the heteroclinic loop is discussed by using the asymptotic expansion of the first-order Melnikov function. Further more we give all possible distribution of limit cycles bifurcated from the period annulus.

Time-periodic perturbation of a Liénard equation with an unbounded homoclinic loop

We consider a quadratic Liénard equation with an unbounded homoclinic loop, which is a solution tending in forward and backward time to a non-hyperbolic equilibrium point located at infinity. Under small time-periodic perturbation, this equilibrium becomes a normally hyperbolic line of singularities at infinity. We show that the perturbed system may present homoclinic bifurcations, leading to the existence of transverse intersections between the stable and unstable manifolds of such a normally hyperbolic line of singularities. The global study concerning the infinity is performed using the Poincaré compactification in polar coordinates, from which we obtain a system defined on a set equivalent to a solid torus in R 3 , whose boundary plays the role of the infinity. The transversality of the manifolds is proved using the Melnikov method and implies, via the Birkhoff–Smale Theorem, a complex dynamical behaviour of the perturbed system solutions in the finite part of the phase space. Numerical simulations are performed in order to illustrate this behaviour, which could be called “ the chaos arising from infinity”, since it depends on the global structure of the Liénard equation, including the points at infinity. Although applied to a particular case, the analysis presented provides a geometrical approach to study periodic perturbations of homoclinic (or heteroclinic) loops to infinity of any planar polynomial vector field.

Generating Grid Multiwing Chaotic Attractors by Constructing Heteroclinic Loops Into Switching Systems

Over the last two decades, multiwing chaos generation has seen promising advances and becomes an active research field today. It is well known that there is a gap between theoretical design and engineering applications in multiwing chaos generation. That is, most theoretical designs of multiwing chaotic attractors with mathematical proofs or numerical verification have rather complex expressions; however, most engineering applications of multiwing chaotic attractors without theoretical supports have simple expressions. To bridge the gap between theoretical design and engineering applications in multiwing chaos generation, this paper introduces a novel practical approach for generating grid multiwing butterfly chaotic attractors from the multipiecewise Lu system by constructing heteroclinic loops. It should be particularly pointed out that the designed multiwing chaotic attractors exhibit typical heteroclinic chaos from the heteroclinic Shil'nikov theorem and also have clear potential engineering applications. The proposed method can be easily extended to the generalized Lorenz system family.

Bifurcation of limit cycles from a heteroclinic loop with a cusp

In this article, we study the expansion of the first Melnikov function of a near-Hamiltonian system near a heteroclinic loop with a cusp and a saddle or two cusps, obtaining formulas to compute the first coefficients of the expansion. Then we use the results to study the problem of limit cycle bifurcation for two polynomial systems.

How to Generate Chaos from Switching System: A Saddle Focus of Index 1 and Heteroclinic Loop‐Based Approach

There exist two different types of equilibrium points in 3‐D autonomous systems, named as saddle foci of index 1 and index 2, which are crucial for chaos generation. Although saddle foci of index 2 have been usually applied for creating double‐scroll or double‐wing chaotic attractors, saddle foci of index 1 are further considered for chaos generation in this paper. A novel approach for constructing chaotic systems is investigated by applying the switching control strategy and yielding a heteroclinic loop which connects two saddle foci of index 1. A basic 3‐D linear system with an arbitrary normal direction of the eigenplane, possessing a saddle focus of index 1 whose corresponding eigenvalues satisfy the Shil′nikov inequality, is first introduced. Then a heteroclinic loop connecting two saddle foci of index 1 will be formed by applying the switching control strategy to the basic 3‐D linear system. The heteroclinic loop consists of an unstable manifold, a stable manifold, and a heteroclinic point. Under the necessary conditions for forming the heteroclinic loop, the intended two‐segmented piecewise linear system which exhibits the chaotic behavior in the sense of the Smale horseshoe can be finally constructed. An illustrative example is given, confirming the effectiveness of the proposed method.

NONRESONANT BIFURCATIONS OF HETEROCLINIC LOOPS WITH ONE INCLINATION FLIP

Heteroclinic bifurcations in four-dimensional vector fields are investigated by setting up local coordinates near a heteroclinic loop. This heteroclinic loop consists of two principal heteroclinic orbits, but there is one stable foliation that involves an inclination flip. The existence, nonexistence, coexistence and uniqueness of the 1-heteroclinic loop, 1-homoclinic orbit, and 1-periodic orbit are studied. Also, the nonexistence, existence of the 2-homoclinic and 2-periodic orbit are demonstrated.

Bifurcation analysis of a generalized Gause model with prey harvesting and a generalized Holling response function of type III

In this paper we study a generalized Gause model with prey harvesting and a generalized Holling response function of type III: p ( x ) = m x 2 a x 2 + b x + 1 . The goal of our study is to give the bifurcation diagram of the model. For this we need to study saddle-node bifurcations, Hopf bifurcation of codimension 1 and 2, heteroclinic bifurcation, and nilpotent saddle bifurcation of codimension 2 and 3. The nilpotent saddle of codimension 3 is the organizing center for the bifurcation diagram. The Hopf bifurcation is studied by means of a generalized Liénard system, and for b = 0 we discuss the potential integrability of the system. The nilpotent point of multiplicity 3 occurs with an invariant line and can have a codimension up to 4. But because it occurs with an invariant line, the effective highest codimension is 3. We develop normal forms (in which the invariant line is preserved) for studying of the nilpotent saddle bifurcation. For b = 0 , the reversibility of the nilpotent saddle is discussed. We study the type of the heteroclinic loop and its cyclicity. The phase portraits of the bifurcations diagram (partially conjectured via the results obtained) allow us to give a biological interpretation of the behavior of the two species.

Versal unfoldings of predator–prey systems with ratio-dependent functional response

In this paper we study the versal unfolding of a predator–prey system with ratio-dependent functional response near a degenerate equilibrium in order to obtain all possible phase portraits for its perturbations. We first construct the unfolding and prove its versality and degeneracy of codimension 2. Then we discuss all its possible bifurcations, including transcritical bifurcation, Hopf bifurcation, and heteroclinic bifurcation, give conditions of parameters for the appearance of closed orbits and heteroclinic loops, and describe the bifurcation curves. Phase portraits for all possible cases are presented.

Predator–prey system with strong Allee effect in prey

Global bifurcation analysis of a class of general predator-prey models with a strong Allee effect in prey population is given in details. We show the existence of a point-to-point heteroclinic orbit loop, consider the Hopf bifurcation, and prove the existence/uniqueness and the nonexistence of limit cycle for appropriate range of parameters. For a unique parameter value, a threshold curve separates the overexploitation and coexistence (successful invasion of predator) regions of initial conditions. Our rigorous results justify some recent ecological observations, and practical ecological examples are used to demonstrate our theoretical work.

Codimension 2 reversible heteroclinic bifurcations with inclination flips

In this paper, the heteroclinic bifurcation problem with real eigenvalues and two inclination-flips is investigated in a four-dimensional reversible system. We perform a detailed study of this case by using the method originally established in the papers “Problems in Homoclinic Bifurcation with Higher Dimensions” and “Bifurcation of Heteroclinic Loops,” and obtain fruitful results, such as the existence and coexistence of R-symmetric homoclinic orbit and R-symmetric heteroclinic loops, R-symmetric homoclinic orbit and R-symmetric periodic orbit. The double R-symmetric homoclinic bifurcation (i.e., two-fold R-symmetric homoclinic bifurcation) for reversible heteroclinic loops is found, and the existence of infinitely many R-symmetric periodic orbits accumulating onto a homoclinic orbit is demonstrated. The relevant bifurcation surfaces and the existence regions are also located.

Symmetric periodic orbits near a heteroclinic loop in [formula omitted] formed by two singular points, a semistable periodic orbit and their invariant manifolds

In this paper, we consider C 1 vector fields X in R 3 having a “generalized heteroclinic loop” L which is topologically homeomorphic to the union of a 2–dimensional sphere S 2 and a diameter Γ connecting the north with the south pole. The north pole is an attractor on S 2 and a repeller on Γ . The equator of the sphere is a periodic orbit unstable in the north hemisphere and stable in the south one. The full space is topologically homeomorphic to the closed ball having as boundary the sphere S 2 . We also assume that the flow of X is invariant under a topological straight line symmetry on the equator plane of the ball. For each n ∈ N , by means of a convenient Poincaré map, we prove the existence of infinitely many symmetric periodic orbits of X near L that gives n turns around L in a period. We also exhibit a class of polynomial vector fields of degree 4 in R 3 satisfying this dynamics.

Limit Cycles Near Homoclinic and Heteroclinic Loops

In the study of near-Hamiltonian systems, the first order Melnikov function plays an important role. It can be used to study Hopf, homoclinic and heteroclinic bifurcations, and the so-called weak Hilbert’s 16th problem as well. The form of expansion of the first order Melnikov function at the Hamiltonian value h 0 such that the curve defined by the equation H(x, y) = h 0 contains a homoclinic loop has been known together with the first three coefficients of the expansion. In this paper, our main purpose is to give an explicit formula to compute the first four coefficients appeared in the expansion of the first order Melnikov function at the Hamiltonian value h 0 such that the curve defined by the equation H(x, y) = h 0 contains a homoclinic or heteroclinic loop, where the formula for the fourth coefficient is new, and to give a way to find limit cycles near the loops by using these coefficients. As an application, we consider polynomial perturbations of degree 4 of quadratic Hamiltonian systems with a heteroclinic loop, and find 3 limit cycles near the loop.

BIFURCATIONS OF GENERIC HETEROCLINIC LOOP ACCOMPANIED BY TRANSCRITICAL BIFURCATION

The bifurcations of generic heteroclinic loop with one nonhyperbolic equilibrium p1and one hyperbolic saddle p2are investigated, where p1is assumed to undergo transcritical bifurcation. Firstly, we discuss bifurcations of heteroclinic loop when transcritical bifurcation does not happen, the persistence of heteroclinic loop, the existence of homoclinic loop connecting p1(resp. p2) and the coexistence of one homoclinic loop and one periodic orbit are established. Secondly, we analyze bifurcations of heteroclinic loop accompanied by transcritical bifurcation, namely, nonhyperbolic equilibrium p1splits into two hyperbolic saddles [Formula: see text] and [Formula: see text], a heteroclinic loop connecting [Formula: see text] and p2, homoclinic loop with [Formula: see text] (resp. p2) and heteroclinic orbit joining [Formula: see text] and [Formula: see text] (resp. [Formula: see text] and p2; p2and [Formula: see text]) are found. The results achieved here can be extended to higher dimensional systems.