WHAT ARE THE SEPARATRIX VALUES NAMED BY LEONTOVICH ON HOMOCLINIC BIFURCATION

For a given system, by using the Tkachev method which concerned with the proof of the stability of a multiple limit cycle, the exact computation formula of the third separatrix values named by Leontovich for the multiple limit cycle bifurcation was given, which was one of the main criterions for the number of limit cycles bifurcated from a homoclinic orbit and the stability of the homoclinic loop, and a computation formula for higher separatrix values was conjectured.

Problems in Homoclinic Bifurcation with Higher Dimensions

In this paper, a suitable local coordinate system is constructed by using exponential dichotomies and generalizing the Floquet method from periodic systems to nonperiodic systems. Then the Poincare map is established to solve various problems in homoclinic bifurcations with codimension one or two. Bifurcation diagrams and bifurcation curves are given.

Stability and Uniqueness of Periodic Orbits Produced During Homoclinic Bifurcation

Under a generic assumption, the existence and the uniqueness of the periodic orbit generating from a homoclinic bifurcation are shown, and the dimensions of its stable and unstable manifolds are given. In the case of a 3-dimensional system, our result revises the stability criterion given in [4,5].

Degenerate Orbit Flip Homoclinic Bifurcations with Higher Dimensions

Bifurcations of a degenerate homoclinic orbit with orbit flip in high dimensional system are studied. By establishing a local coordinate system and a Poincare map near the homoclinic orbit, the existence and uniqueness of 1-homoclinic orbit and 1-periodic orbit are given. Also considered is the existence of 2-homoclinic orbit and 2-periodic orbit. In additon, the corresponding bifurcation surfaces are given.

Microbial insecticide model and homoclinic bifurcation of impulsive control system

A new microbial insecticide mathematical model with density dependent for pest is proposed in this paper.First,the system without impulsive state feedback control is considered.The existence and stability of equilibria are investigated and the properties of equilibria under different conditions are verified by using numerical simulation.Since the system without pulse has two positive equilibria under some additional assumptions,the system is not globally asymptotically stable.Based on the stability analysis of equilibria,limit cycle,outer boundary line and Sotomayor's theorem,the existence of saddle-node bifurcation and global dynamics of the system are obtained.Second,we consider homoclinic bifurcation of the system with impulsive state feedback control.The existence of order-1 homoclinic orbit of the system is studied.When the impulsive function is slightly disturbed,the homoclinic orbit breaks and bifurcates order-1 periodic solution.The existence and stability of order-1 periodic solution are obtained by means of theory of semi-continuous dynamic system.

Analysis of a Shil’nikov Type Homoclinic Bifurcation

The bifurcation associated with a homoclinic orbit to saddle-focus including a pair of pure imaginary eigenvalues is investigated by using related homoclinie bifurcation theory. It is proved that, in a neighborhood of the homoclinic bifurcation value, there are countably infinite saddle-node bifurcation values, period-doubling bifurcation values and double-pulse homoclinic bifurcation values. Also, accompanied by the Hopf bifurcation, the existence of certain homoclinie connections to the periodic orbit is proved.

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 PERTURBED GENERALIZED HAMILTONIAN SYSTEMS

It this paper we obtain existence and bifurcation theorems for homoclinic orbits in three-dimeensional,time dependent and independent,perturbations of generalized Hamiltonian differential equations defined on three-dimensional Poisson manifolds.Thed we apply them to a truncated spectral model of the quasi-geostrophic flow on a cyclic β-plane.

Existence of periodic orbits and shift-invariant curve sequences near multiple homoclinics

In this paper, the complicated dynamics is studied near a double homoclinic loops with bellows configuration for general systems. For the non-twisted multiple homoclinics, the existence of periodic orbit with the specified route and the existence of shift-invariant curve sequences defined on the cross sections of multiple homoclinics corresponding to any specified one-side infinite sequences are given. In addition, the existence regions are also located.

ON THE COEFFICIENTS APPEARING IN THE EXPANSION OF MELNIKOV FUNCTIONS IN HOMOCLINIC BIFURCATIONS

We give computing formulas for the first three coefficients appearing in the expansion of the Melnikov function in homoclinic bifurcations.

Analysis of chaos behaviors of a bistable piezoelectric cantilever power generation system by the second-order Melnikov function

By applying the second order Melnikov function, the chaos behaviors of a bistable piezoelectric cantilever power generation system are analyzed. Firstly, the conditions for emerging chaos of the system are derived by the second order Melnikov function. Secondly, the effects of each item in chaos threshold expression are analyzed. The excitation frequency and resistance values, which have the most influence on chaos threshold value, are found. The result from the second order Melnikov function is more accurate compared with that from the first order Melnikov function. Finally, the attraction basins of large amplitude motions under different exciting frequency, exciting amplitude, and resistance parameters are given.

Bifurcation and Chaotic Dynamics of Homoclinic Systems in R^3

We consider perturbations which may or may not depend explicitly on time for the three-dimensional homoclinic systems. We obtain the existence and bifurcation theorems for transversal homoclinic points and homoclinic orbits, and illustrate our results with two examples.

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.