Heteroclinic Loop Research Articles

Computing the heteroclinic bifurcation curves in predator–prey systems with ratio-dependent functional response

Predator-prey models with Michaelis-Menten-Holling type ratio- dependent functional response exhibit very rich and complex dynamical behavior, such as the existence of degenerate equilibria, appearance of limit cycles and heteroclinic loops, and the coexistence of two attractive equilibria. In this paper, we study heteroclinic bifurcations of such a predator-prey model. We first calculate the higher order Melnikov functions by transforming the model into a Hamiltonian system and then provide an algorithm for computing higher order approximations of the heteroclinic bifurcation curves.

Large amplitude oscillations for a class of symmetric polynomial differential systems in R³

In this paper we study a class of symmetric polynomial differential systems in R³, which has a set of parallel invariant straight lines, forming degenerate heteroclinic cycles, which have their two singular endpoints at infinity. The global study near infinity is performed using the Poincaré compactification. We prove that for all n <FONT FACE=Symbol>Î</FONT> N there is epsilonn > 0 such that for 0 < epsilon < epsilonn the system has at least n large amplitude periodic orbits bifurcating from the heteroclinic loop formed by the two invariant straight lines closest to the x-axis, one contained in the half-space y > 0 and the other in y < 0.

Bifurcation of rough heteroclinic loop with orbit and inclination flips

In this paper, we are concerned with the heteroclinic loop bifurcations accompanied by one orbit flip and one inclination flip under some roughness conditions. The existence of one 1-periodic orbit, one 1-homoclinic orbit, two 1-periodic orbits, one double 1-periodic orbit is obtained, respectively, approximate expressions of the corresponding bifurcation curves (or surfaces) and the corresponding bifurcation graphs under nontwisted or twisted conditions are also given.

GENERATION OF SYMMETRIC PERIODIC ORBITS BY A HETEROCLINIC LOOP FORMED BY TWO SINGULAR POINTS AND THEIR INVARIANT MANIFOLDS OF DIMENSIONS 1 AND 2 IN ℝ3

In this paper we will find continuous periodic orbits passing near infinity for a class of polynomial vector fields in ℝ3. We consider polynomial vector fields that are invariant under a symmetry with respect to a plane Σ and that possess a "generalized heteroclinic loop" formed by two singular points e+ and e- at infinity and their invariant manifolds Γ and Λ. Γ is an invariant manifold of dimension 1 formed by an orbit going from e- to e+, Γ is contained in ℝ3 and is transversal to Σ. Λ is an invariant manifold of dimension 2 at infinity. In fact, Λ is the two-dimensional sphere at infinity in the Poincaré compactification minus the singular points e+ and e-. The main tool for proving the existence of such periodic orbits is the construction of a Poincaré map along the generalized heteroclinic loop together with the symmetry with respect to Σ.

Bifurcations of heterodimensional cycles with two saddle points

The bifurcations of 2-point heterodimensional cycles are investigated in this paper. Under some generic conditions, we establish the existence of one homoclinic loop, one periodic orbit, two periodic orbits, one 2-fold periodic orbit, and the coexistence of one periodic orbit and heteroclinic loop. Some bifurcation patterns different to the case of non-heterodimensional heteroclinic cycles are revealed.

Degenerate bifurcations of nontwisted heterodimensional cycles with codimension 3

Bifurcations of heterodimensional cycles with highly degenerate conditions are studied by establishing a suitable local coordinate system in three-dimensional vector fields. The existence, coexistence and noncoexistence of the periodic orbit, homoclinic loop, heteroclinic loop and double periodic orbit are obtained under some generic hypotheses. The bifurcation surfaces and the existence regions are located; the number of the bifurcation surfaces exhibits variety and complexity of the bifurcation of degenerate heterodimensional cycles. The corresponding bifurcation graph is also drawn.

Periodic orbits for a class of reversible quadratic vector field on [formula omitted

For a class of reversible quadratic vector fields on R 3 we study the periodic orbits that bifurcate from a heteroclinic loop having two singular points at infinity connected by an invariant straight line in the finite part and another straight line at infinity in the local chart U 2 . More specifically, we prove that for all n ∈ N , there exists ε n > 0 such that the reversible quadratic polynomial differential system x ˙ = a 0 + a 1 y + a 3 y 2 + a 4 y 2 + ε ( a 2 x 2 + a 3 x z ) , y ˙ = b 1 z + b 3 y z + ε b 2 x y , z ˙ = c 1 y + c 4 z 2 + ε c 2 x z in R 3 , with a 0 < 0 , b 1 c 1 < 0 , a 2 < 0 , b 2 < a 2 , a 4 > 0 , c 2 < a 2 and b 3 ∉ { c 4 , 4 c 4 } , for ε ∈ ( 0 , ε n ) has at least n periodic orbits near the heteroclinic loop.

Symmetric periodic orbits near a heteroclinic loop formed by two singular points and their invariant manifolds of dimension 1 and 2

In this paper we consider vector fields in that are invariant under a suitable symmetry and that possess a 'generalized heteroclinic loop' formed by two singular points (e+ and e−) and their invariant manifolds: one of dimension 2 (a sphere minus the points e+ and e−) and one of dimension 1 (the open diameter of the sphere having endpoints e+ and e−). In particular, we analyse the dynamics of the vector field near the heteroclinic loop by means of a convenient Poincare map, and we prove the existence of infinitely many symmetric periodic orbits near . We also study two families of vector fields satisfying this dynamics. The first one is a class of quadratic polynomial vector fields in , and the second one is the charged rhomboidal four-body problem.

Codimension 3 Non-resonant Bifurcations of Rough Heteroclinic Loops with One Orbit Flip*

Heteroclinic bifurcations in four dimensional vector fields are investigated by setting up a local coordinates near a rough heteroclinic loop. This heteroclinic loop has a principal heteroclinic orbit and a non-principal heteroclinic orbit that takes orbit flip. The existence, nonexistence, coexistence and uniqueness of the 1-heteroclinic loop, 1-homoclinic orbit and 1-periodic orbit are studied. The existence of the two-fold or three-fold 1-periodic orbit is also obtained.

SYMMETRIC PERIODIC ORBITS NEAR HETEROCLINIC LOOPS AT INFINITY FOR A CLASS OF POLYNOMIAL VECTOR FIELDS

For polynomial vector fields in ℝ3, in general, it is very difficult to detect the existence of an open set of periodic orbits in their phase portraits. Here, we characterize a class of polynomial vector fields of arbitrary even degree having an open set of periodic orbits. The main two tools for proving this result are, first, the existence in the phase portrait of a symmetry with respect to a plane and, second, the existence of two symmetric heteroclinic loops.

On the Bifurcations of a Hamiltonian Having Three Homoclinic Loops under Z 3 Invariant Quintic Perturbations

A cubic system having three homoclinic loops perturbed by Z3 invariant quintic polynomials is considered. By applying the qualitative method of differential equations and the numeric computing method, the Hopf bifurcation, homoclinic loop bifurcation and heteroclinic loop bifurcation of the above perturbed system are studied. It is found that the above system has at least 12 limit cycles and the distributions of limit cycles are also given.

Homoclinic and Periodic Orbits Arising Near the Heteroclinic Cycle Connecting Saddle–focus and Saddle Under Reversible Condition

In this paper, we study the dynamical behavior for a 4–dimensional reversible system near its heteroclinic loop connecting a saddle–focus and a saddle. The existence of infinitely many reversible 1–homoclinic orbits to the saddle and 2–homoclinic orbits to the saddle–focus is shown. And it is also proved that, corresponding to each 1–homoclinic (resp. 2–homoclinic) orbit Γ, there is a spiral segment such that the associated orbits starting from the segment are all reversible 1–periodic (resp. 2–periodic) and accumulate onto Γ. Moreover, each 2–homoclinic orbit may be also accumulated by a sequence of reversible 4–homoclinic orbits.

Heteroclinic bifurcation in a ratio-dependent predator-prey system

In this paper we study the heteroclinic bifurcation in a general ratio-dependent predator-prey system. Based on the results of heteroclinic loop obtained in [J. Math. Biol. 43(2001): 221-246], we give parametric conditions of the existence of the heteroclinic loop analytically and describe the heteroclinic bifurcation surface in the parameter space, so as to answer further the open problem raised in [J. Math. Biol. 42(2001): 489-506].

On the study of limit cycles of a cubic polynomials system under Z4-equivariant quintic perturbation

This paper is concerned with the number and distribution of limit cycles of a perturbed cubic Hamiltonian system which has 5 centers and 4 saddle points. The singular point and singular close orbits’ stability theory and perturbation skills of differential equations are applied to study the Hopf, homoclinic loop and heteroclinic loop bifurcation of such system under Z 4 -equivariant quintic perturbation. It is found that the perturbed system has at least 16 limit cycles bifurcated from the focus. Further, at least 14 limit cycles with three different distributions appear in the heteroclinic loops bifurcation.

Bifurcations of Fine 3–point–loop in Higher Dimensional Space

By using the linear independent fundamental solutions of the linear variational equation along the heteroclinic loop to establish a suitable local coordinate system in some small tubular neighborhood of the heteroclinic loop, the Poincare map is constructed to study the bifurcation problems of a fine 3–point loop in higher dimensional space. Under some transversal conditions and the non–twisted condition, the existence, coexistence and incoexistence of 2–point–loop, 1–homoclinic orbit, simple 1–periodic orbit and 2–fold 1–periodic orbit, and the number of 1–periodic orbits are studied. Moreover, the bifurcation surfaces and existence regions are given. Lastly, the above bifurcation results are applied to a planar system and an inside stability criterion is obtained.

Bifurcations of limit cycles from quadratic non-Hamiltonian systems with two centres and two unbounded heteroclinic loops

We investigate the bifurcations of limit cycles in a class of planar quadratic integrable (non-Hamiltonian) systems with two centres, both surrounded by unbounded heteroclinic loops, under small quadratic perturbations. By a careful study of the number of zeros of Abelian integrals based on the geometric properties of some planar curves, defined by ratios of such integrals, we obtain complete results about the number and the distribution of limit cycles bifurcating from the two period annuli.

ON THE STUDY OF LIMIT CYCLES OF THE GENERALIZED RAYLEIGH–LIENARD OSCILLATOR

The Hopf bifurcation, saddle connection loop bifurcation and Poincaré bifurcation of the generalized Rayleigh–Liénard oscillator Ẍ+aX+2bX3+ε(c3+c2X2+c1X4+c4Ẋ2)Ẋ=0 are studied. It is proved that for the case a&lt;0, b&gt;0 the system has at most six limit cycles bifurcated from Hopf bifurcation or has at least seven limit cycles bifurcated from the double homoclinic loop. For the case a&gt;0, b&lt;0 the system has at most three limit cycles bifurcated from Hopf bifurcation or has three limit cycles bifurcated from the heteroclinic loop.

The cyclicity of the elliptic segment loops of the reversible quadratic Hamiltonian systems under quadratic perturbations

Denote by Q H and Q R the Hamiltonian class and reversible class of quadratic integrable systems. There are several topological types for systems belong to Q H ∩ Q R . One of them is the case that the corresponding system has two heteroclinic loops, sharing one saddle-connection, which is a line segment, and the other part of the loops is an ellipse. In this paper we prove that the maximal number of limit cycles, which bifurcate from the loops with respect to quadratic perturbations in a conic neighborhood of the direction transversal to reversible systems (called in reversible direction), is two. We also give the corresponding bifurcation diagram.

The cyclicity of the elliptic segment loops of the reversible quadratic Hamiltonian systems under quadratic perturbations

Denote by Q H and Q R the Hamiltonian class and reversible class of quadratic integrable systems. There are several topological types for systems belong to Q H ∩ Q R . One of them is the case that the corresponding system has two heteroclinic loops, sharing one saddle-connection, which is a line segment, and the other part of the loops is an ellipse. In this paper we prove that the maximal number of limit cycles, which bifurcate from the loops with respect to quadratic perturbations in a conic neighborhood of the direction transversal to reversible systems (called in reversible direction), is two. We also give the corresponding bifurcation diagram.

Bifurcations of limit cycles from Quintic Hamiltonian systems with a double figure eight loop

This paper deals with Liénard equations of the form x ̇ =y, y ̇ =P(x)+yQ(x,y) , with P and Q polynomial of degree 5 and 4, respectively. Attention goes to perturbations of the Hamiltonian vector fields with an elliptic Hamiltonian of degree six, exhibiting a double figure eight-loop. It is proved that the Hopf cyclicity is two, and it is also given by the new configurations of the limit cycles bifurcated from the homoclinic loop or heteroclinic loop for quintic system with quintic perturbations by using the methods of bifurcation theory and qualitative analysis.