Deming Zhu

Bifurcations of heteroclinic loops

By generalizing the Floquet method from periodic systems to systems with exponential dichotomy, a local coordinate system is established in a neighborhood of the heteroclinic loop Γ to study the bifurcation problems of homoclinic and periodic orbits. Asymptotic expressions of the bifurcation surfaces and their relative positions are given. The results obtained in literature concerned with the 1-hom bifurcation surfaces are improved and extended to the nontransversal case. Existence regions of the 1-per orbits bifurcated from Γ are described, and the uniqueness and incoexistence of the 1-hom and 1-per orbit and the inexistence of the 2-hom and 2-per orbit are also obtained.

Multiple positive periodic solutions of a delayed discrete predator–prey system with type IV functional responses☆

Abstract In this paper, a discrete periodic predator–prey system with type IV functional responses and time delay is investigated. Using Gaines and Mawhin’s continuation theorem from coincidence degree theory as well as some prior estimates, we get sufficient conditions for the existence of positive periodic solutions of the system. This is also the first time that the coincidence degree theory has been used to obtain multiple positive periodic solutions in discrete ecological systems.

DEGENERATED HOMOCLINIC BIFURCATIONS WITH HIGHER DIMENSIONS

The degenerated homoclinic bifurcation for high dimensional system is considered. The existence, uniqueness, and incoexistence of the 1-homclinic orbit and 1-periodic orbit near Γ are studied under the nonresonant condition. Complicated bifurcation pattern is described under the resonant condition.

CODIMENSION 3 HOMOCLINIC BIFURCATION OF ORBIT FLIP WITH RESONANT EIGENVALUES CORRESPONDING TO THE TANGENT DIRECTIONS

Bifurcations of homoclinic orbit with orbit-flip and resonant eigenvalues corresponding to the tangent directions are investigated in a four-dimensional system. The existence, number, coexistence and incoexistence of 1-homoclinic orbit, 1-periodic orbit, 2n-homoclinic orbit and 2n-periodic orbit are given, and the bifurcation surfaces and the existence regions are also located.

HETERODIMENSIONAL CYCLE BIFURCATION WITH ORBIT-FLIP

The local moving frame approach is employed to study the bifurcation of a degenerate heterodimensional cycle with orbit-flip in its nontransversal orbit. Under some generic hypotheses, we provide the conditions for the existence, uniqueness and noncoexistence of the homoclinic orbit, heteroclinic orbit and periodic orbit. And we also present the coexistence conditions for the homoclinic orbit and the periodic orbit. But it is impossible for the coexistence of the periodic orbit and the persistent heterodimensional cycle or the coexistence of the homoclinic loop and the persistent heterodimensional cycle. Moreover, the double and triple periodic orbit bifurcation surfaces are established as well. Based on the bifurcation analysis, the bifurcation surfaces and the existence regions are located. An example of application is also given to demonstrate our main results.

BIFURCATIONS OF GENERIC HETEROCLINIC LOOP ACCOMPANIED BY TRANSCRITICAL BIFURCATION

The bifurcations of generic heteroclinic loop with one nonhyperbolic equilibrium p1 and one hyperbolic saddle p2 are investigated, where p1 is 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 p1 splits into two hyperbolic saddles

BIFURCATIONS OF HOMOCLINIC ORBIT CONNECTING TWO NONLEADING EIGENDIRECTIONS

p_1^0

CODIMENSION 3 HETEROCLINIC BIFURCATIONS WITH ORBIT AND INCLINATION FLIPS IN REVERSIBLE SYSTEMS

and

BIFURCATION OF ROUGH HETEROCLINIC LOOP WITH ORBIT FLIPS

p_1^1

A new existence result for impulsive dynamic equations on timescales

, a heteroclinic loop connecting