Homoclinic Loop Bifurcation Research Articles

Double homoclinic bifurcations by perturbing a class of cubic Z2-equivariant polynomial systems with nilpotent singular points

This paper focuses on the study of the limit cycle bifurcation of a general cubic Z2-equivariant Hamiltonian differential system with two nilpotent singular points. Initially, the necessary and sufficient conditions for the occurrence of two nilpotent singularities are provided in the unperturbed system, along with a complete description of all possible phase portraits on the plane. Subsequently, using a combination of the Melnikov function method and various analytical techniques, we derive the approximate expansion of the first order Melnikov function at the interval endpoint, as well as formula for its coefficients. These results can be utilized to analyze double homoclinic loop bifurcation of the perturbed system, finding the lower bound of limit cycles.

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.

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.

Loop Numbers for the Stability of Homoclinic Loops of Planar Vector Fields

This paper is devoted to the study of stability and bifurcations of homoclinic loops for planar vector fields. For a given homoclinic loop, a sequence of loop numbers can be defined such that the stability and bifurcations of the loop are determined by the first nonzero term of the sequence. Formulas for the first several loop numbers were established in the past. In this paper, we will introduce general formulas for the loop numbers for both the single and double homoclinic loops.

Bifurcations of resonant double homoclinic loops for higher dimensional systems

Bifurcations of resonant double homoclinic loops for higher dimensional systems

Scientific heritage of L.P. Shilnikov

This is the first part of a review of the scientific works of L.P. Shilnikov. We group his papers according to 7 major research topics: bifurcations of homoclinic loops; the loop of a saddle-focus and spiral chaos; Poincare homoclinics to periodic orbits and invariant tori, homoclinic in noautonous and infinite-dimensional systems; Homoclinic tangency; Saddlenode bifurcation — quasiperiodicity-to-chaos transition, blue-sky catastrophe; Lorenz attractor; Hamiltonian dynamics. The first two topics are covered in this part. The review will be continued in the further issues of the journal.

The Twisting Bifurcations of Double Homoclinic Loops with Resonant Eigenvalues

The twisting bifurcations of double homoclinic loops with resonant eigenvalues are investigated in four-dimensional systems. The coexistence or noncoexistence of large 1-homoclinic orbit and large 1-periodic orbit near double homoclinic loops is given. The existence or nonexistence of saddle-node bifurcation surfaces is obtained. Finally, the complete bifurcation diagrams and bifurcation curves are also given under different cases. Moreover, the methods adopted in this paper can be extended to a higher dimensional system.

Local and global bifurcations in a small power system

This paper presents a number of local and one global (homoclinic loop) bifurcation observed in a small power system involving a load tap changer transformer and an induction motor. Besides the well-known generic local bifurcations (saddle-node and generic Hopf), the paper presents examples of higher codimension bifurcations, such as fold with double zero eigenvalues, degenerate Hopf, and swallowtail bifurcation.

The inverse integrating factor and the Poincaré map

This work is concerned with planar real analytic differential systems with an analytic inverse integrating factor defined in a neighborhood of a regular orbit. We show that the inverse integrating factor defines an ordinary differential equation for the transition map along the orbit. When the regular orbit is a limit cycle, we can determine its associated Poincaré return map in terms of the inverse integrating factor. In particular, we show that the multiplicity of a limit cycle coincides with the vanishing multiplicity of an inverse integrating factor over it. We also apply this result to study the homoclinic loop bifurcation. We only consider homoclinic loops whose critical point is a hyperbolic saddle and whose Poincaré return map is not the identity. A local analysis of the inverse integrating factor in a neighborhood of the saddle allows us to determine the cyclicity of this polycycle in terms of the vanishing multiplicity of an inverse integrating factor over it. Our result also applies in the particular case in which the saddle of the homoclinic loop is linearizable, that is, the case in which a bound for the cyclicity of this graphic cannot be determined through an algebraic method.

Homoclinic Loop and Degenerate Hopf Bifurcations Introduced by Load Dynamics

This letter presents the occurrence and properties of a homoclinic loop bifurcation in a simple power system with induction motor and load tap changer (LTC). This global bifurcation is accompanied by the emergence of an unstable limit cycle, or the disappearance of a stable one. It is also shown that it may coincide with a Hopf bifurcation condition yielding a codimension-2 degenerate Hopf bifurcation.

Coexistence of Limit Cycles and Homoclinic Loops in a SIRS Model with a Nonlinear Incidence Rate

Recently, Ruan and Wang [J. Differential Equations, 188 (2003), pp. 135–163] studied the global dynamics of a SIRS epidemic model with vital dynamics and a nonlinear saturated incidence rate. Under certain conditions they showed that the model undergoes a Bogdanov–Takens bifurcation; i.e., it exhibits saddle-node, Hopf, and homoclinic bifurcations. They also considered the existence of none, one, or two limit cycles. In this paper, we investigate the coexistence of a limit cycle and a homoclinic loop in this model. One of the difficulties is to determine the multiplicity of the weak focus. We first prove that the maximal multiplicity of the weak focus is 2. Then feasible conditions are given for the uniqueness of limit cycles. The coexistence of a limit cycle and a homoclinic loop is obtained by reducing the model to a universal unfolding for a cusp of codimension 3 and studying degenerate Hopf bifurcations and degenerate Bogdanov–Takens bifurcations of limit cycles and homoclinic loops of order 2.

On the bifurcation of double homoclinic loops of a cubic system

A cubic symmetric system x ̇ = y + ε [ a 1 x + ( a 2 + a 3 + 1 ) x 3 + ( b 1 − 3 ) x 2 y + ( a 2 − 3 a 3 + 1 ) x y 2 + ( b 1 + 1 ) y 3 ] , y ̇ = x − x 3 + ε [ a 1 y + ( b 1 + 1 ) x 3 + ( a 2 + 3 a 3 − 1 ) x 2 y + ( b 1 − 3 ) x y 2 + ( a 2 − a 3 − 1 ) y 3 ] is considered. By computing the focus quantities and saddle quantities, we get the quantities which determine the stability of the double homoclinic loops appearing. Then combining Hopf bifurcation and double homoclinic loop bifurcation, we prove that seven limit cycles can bifurcated from the double homoclinic loops in the above cubic system. As far as we know, this result on the number of limit cycles bifurcated from double homoclinic loops of the cubic system is new.

The loop quantities and bifurcations of homoclinic loops

The stability and bifurcations of a homoclinic loop for planar vector fields are closely related to the limit cycles. For a homoclinic loop of a given planar vector field, a sequence of quantities, the homoclinic loop quantities were defined to study the stability and bifurcations of the loop. Among the sequence of the loop quantities, the first nonzero one determines the stability of the homoclinic loop. There are formulas for the first three and the fifth loop quantities. In this paper we will establish the formula for the fourth loop quantity for both the single and double homoclinic loops. As applications, we present examples of planar polynomial vector fields which can have five or twelve limit cycles respectively in the case of a single or double homoclinic loop by using the method of stability-switching.

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.

A new cubic system having eleven limit cycles

This paper concerns with the number and distribution of limit cycles of a perturbed cubic Hamiltonian system which has 5 centers and 4 saddle points. The stability analysis and bifurcation methods of differential equations are applied to study the homoclinic loop bifurcation under $Z_2$-equivariant cubic perturbation. It is proved that the perturbed system can have 11 limit cycles with two different distributions, one of which is already known, the other is new.

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 twisted homoclinic loops for degenerated cases

In this paper, the bifurcation problems of twisted and degenerated homoclinic loop for higher dimensional systems are studied. Under the nonresonant condition, the existence, uniqueness, and incoexistence of the 1-homoclinic loop and 1-periodic orbit near Γ are obtained, and the inexistence of the 2-homoclinic loop and the existence of 2-periodic orbit near Γ are also given.

Homoclinic orbits in a disease transmission model with nonlinear incidence and nonconstant population

Periodic oscillations are proved for an SIRS disease transmission model in which the size of the population varies and the incidence rate is a nonlinear function. For this particular incidence function, analytical techniques are used to show that, for some parameter values, periodic solutions can arise through a Hopf bifurcation and disappear through a homoclinic loop bifurcation. The existence of periodicity is important as it may indicate different strategies for controlling disease.

Finite cyclicity of finite codimension nondegerate homoclinic loops with real eigenvalues in ℝ3

In this paper we study homoclinic loops in ℝ3 which are nondegenerate in the sense of Sil’nikov ([21]) and with real principal eigenvalues in 1∶1 resonance, i.e. homoclinic loops which have thestrong inclination property and which are tangent to the principal eigenvectors. We are interested here in the higher codimensional cases. It is known that the dynamics of such systems is given by a 1-dimensional map. Using the ideas exposed in [5], we are able to show that, as for the “nontwisted” loops (cf. [17]), this 1-dimensional map admits a nice asymptotic expansion allowing to treat homoclinic loop bifurcations of arbitrarily high codimension and to exhibit an explicit bound for the number of isolated periodic solutions generated under small perturbations. The computations of the bound rely on derivation-division algorithms and Khovanskii’s fewnomials theory.

Complexity in the bifurcation structure of homoclinic loops to a saddle-focus

We report on the study of bifurcations of multi-circuit homoclinic loops in two-parameter families of vector fields in the neighbourhood of a main homoclinic tangency to a saddle-focus with characteristic exponents satisfying the Shil'nikov condition . We prove that one-parameter subfamilies of vector fields transverse to the main homoclinic tangency (1) may be tangent to subfamilies with a triple-circuit homoclinic loop; (2) may have a tangency of an arbitrarily high order to subfamilies with a multi-circuit homoclinic loop. These theorems show the high structural instability of one-parameter subfamilies of vector fields in the neighbourhood of a homoclinic tangency to a Shil'nikov-type saddle-focus. Implications for nonlinear partial differential equations modelling waves in spatially extended systems are briefly discussed.