On the Number of Limit Cycles Bifurcated from Some Hamiltonian Systems with a Double Homoclinic Loop and a Heteroclinic Loop

In this paper, we study the number of bifurcated limit cycles from near-Hamiltonian systems where the corresponding Hamiltonian system has a double homoclinic loop passing through a hyperbolic saddle surrounded by a heteroclinic loop with a hyperbolic saddle and a nilpotent saddle, and obtain some new results on the lower bound of the maximal number of limit cycles for these systems. In particular, we study the bifurcation of limit cycles of the following system [Formula: see text] as an application of our results, where [Formula: see text] is a polynomial of degree five.

A Theoretical Framework for Lagrangian Descriptors

This paper provides a theoretical background for Lagrangian Descriptors (LDs). The goal of achieving rigorous proofs that justify the ability of LDs to detect invariant manifolds is simplified by introducing an alternative definition for LDs. The definition is stated for [Formula: see text]-dimensional systems with general time dependence, however we rigorously prove that this method reveals the stable and unstable manifolds of hyperbolic points in four particular 2D cases: a hyperbolic saddle point for linear autonomous systems, a hyperbolic saddle point for nonlinear autonomous systems, a hyperbolic saddle point for linear nonautonomous systems and a hyperbolic saddle point for nonlinear nonautonomous systems. We also discuss further rigorous results which show the ability of LDs to highlight additional invariants sets, such as [Formula: see text]-tori. These results are just a simple extension of the ergodic partition theory which we illustrate by applying this methodology to well-known examples, such as the planar field of the harmonic oscillator and the 3D ABC flow. Finally, we provide a thorough discussion on the requirement of the objectivity (frame-invariance) property for tools designed to reveal phase space structures and their implications for Lagrangian descriptors.

Bifurcation of an Orbit Homoclinic to a Hyperbolic Saddle of a Vector Field inR4

We perform a bifurcation analysis of an orbit homoclinic to a hyperbolic saddle of a vector field inR4. We give an expression of the gap between returning points in a transverse section by renormalizing system, through which we find the existence of homoclinic-doubling bifurcation in the case1+α>β>ν. Meanwhile, after reparametrizing the parameter, a periodic-doubling bifurcation appears and may be close to a saddle-node bifurcation, if the parameter is varied. These scenarios correspond to the occurrence of chaos. Based on our analysis, bifurcation diagrams of these bifurcations are depicted.

Bifurcations of Orbit and Inclination Flips Heteroclinic Loop with Nonhyperbolic Equilibria

The bifurcations of heteroclinic loop with one nonhyperbolic equilibrium and one hyperbolic saddle are considered, where the nonhyperbolic equilibrium is supposed to undergo a transcritical bifurcation; moreover, the heteroclinic loop has an orbit flip and an inclination flip. When the nonhyperbolic equilibrium does not undergo a transcritical bifurcation, we establish the coexistence and noncoexistence of the periodic orbits and homoclinic orbits. While the nonhyperbolic equilibrium undergoes the transcritical bifurcation, we obtain the noncoexistence of the periodic orbits and homoclinic orbits and the existence of two or three heteroclinic orbits.

THE NUMBER OF ZEROS OF ABELIAN INTEGRALS FOR A PERTURBATION OF HYPERELLIPTIC HAMILTONIAN SYSTEM WITH DEGENERATED POLYCYCLE

In this paper, we provide a complete study of the zeros of Abelian integrals obtained by integrating the 1-form (α + βx + x2)ydx over the compact level curves of the hyperelliptic Hamiltonian [Formula: see text]. Such a family of compact level curves is bounded by a polycycle passing through a nilpotent cusp and a hyperbolic saddle of this hyperelliptic Hamiltonian system, which is not the exceptional family of ovals proposed by Gavrilov and Iliev. It is shown that the least upper bound for the number of zeros of the related hyperelliptic Abelian integral is two, and this least upper bound can be achieved for some values of parameters (α, β). This implies that the Abelian integral still has Chebyshev property for this nonexceptional family of ovals. Moreover, we derive the asymptotic expansion of Abelian integrals near a polycycle passing through a nilpotent cusp and a hyperbolic saddle in a general case.

LIMIT CYCLE BIFURCATIONS NEAR A DOUBLE HOMOCLINIC LOOP WITH A NILPOTENT SADDLE

Homoclinic bifurcation is a difficult and important topic of bifurcation theory. As we know, a general theory for a homoclinic loop passing through a hyperbolic saddle was established by [Roussarie, 1986]. Then the method of stability-changing to find limit cycles near a double homoclinic loop passing through a hyperbolic saddle was given in [Han & Chen, 2000], and further developed by [Han et al., 2003; Han & Zhu, 2007]. For a homoclinic loop passing through a nilpotent saddle there are essentially two different cases, which we distinguish by cuspidal type and smooth type, respectively. For the cuspidal type a general theory was recently established in [Zang et al., 2008]. In this paper, we consider limit cycle bifurcation near a double homoclinic loop passing through a nilpotent saddle by studying the analytical property of the first order Melnikov functions for general near-Hamiltonian systems and obtain the conditions for the perturbed system to have 8, 10 or 12 limit cycles in a neighborhood of the loop with seven different distributions. In particular, for the homoclinic loop of smooth type, a general theory is obtained as a consequence. We finally consider some polynomial systems and find a lower bound of the maximal number of limit cycles as an application of our main results.

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.