Analytical Conditions for the Existence of a Homoclinic Loop in Chua Circuits

We analyze a third-order piecewise-linear system that describes the operation of an electronicoscillator. This system can be used to investigate various scenarios for the onset of chaos, in particular, through separatrix loop destruction. Necessary and sufficient conditions are obtained for the existence of an equilibrium separatrix loop. Calculation of critical parameters is illustrated.     

On the nature of dynamic chaos in a neighborhood of a separatrix of a conservative system

In the present paper, we give a new treatment of the mechanism of generation of chaotic dynamics in a perturbed conservative system in a neighborhood of the separatrix contour of a hyperbolic singular point of the unperturbed system. We theoretically prove and justify by three numerical examples of classical Hamiltonian systems with one and a half degrees of freedom and by an example of a simply conservative three-dimensional system that the complication of the dynamics in a conservative system as the perturbation increases is caused by a nonlocal effect of multiplication of hyperbolic and elliptic cycles (and the tori surrounding them), which has nothing in common with the mechanism of separatrix splitting in classical Hamiltonian mechanics.     

Connection across a Separatrix with Dissipation

Dissipative perturbations of strongly nonlinear oscillators that correspond to slowly varying double-well potentials are considered. The method of averaging, which describes the solution as nearly periodic, fails as the trajectory approaches the unperturbed separatrix, a homoclinic orbit of the saddle point, significantly before it is captured in either well. Nevertheless, perturbed initial conditions corresponding to the boundary of the basin of attraction for each well, which are the perturbed stable manifolds of the saddle point, are accurately determined using only the method of averaging modified by Melnikov energy ideas near the separatrix. To determine the amplitude and phase of the captured oscillations after crossing the separatrix, a transition region is constructed consisting of a large sequence of nearly solitary pulses along the separatrix. The amplitude and phases of the slowly varying nonlinear oscillations away from the separatrix, both before and after capture, are matched to this transition region. In this way, analytic connection formulas across the separatrix are obtained and are shown to depend on the perturbed initial conditions.     

二次系统存在一类四次曲线分界线环的充要条件（续）

本义给出了二次系统存在一类四次曲线分界线环的充要条件，此类分界线环的方程为（ｙ＋ｃｘ￣２）－ｘ￣２（ｘ－ａ）（ｘ－ｂ）＝０。     

无穷远分界线的稳定性和产生极限环的条件

本文给出了无穷远分界线稳定性的判据以及从无穷远分界线产生极限环的条件，其结果包括了［７］，［８］的主要结果。     

Stability islands in domains of separatrix crossings in slow-fast Hamiltonian systems

We consider a two-degrees-of-freedom Hamiltonian system with one degree of freedom corresponding to fast motion and the other corresponding to slow motion. The ratio of typical velocities of changes of the slow and fast variables is the small parameter ɛ of the problem. At frozen values of the slow variables, there is a separatrix on the phase plane of the fast variables, and there is a region in the phase space (the domain of separatrix crossings) where the projections of phase points onto the plane of the fast variables repeatedly cross the separatrix in the process of evolution of the slow variables. Under a certain symmetry condition, we prove the existence of many (of order 1/ɛ) stable periodic trajectories in the domain of separatrix crossings. Each of these trajectories is surrounded by a stability island whose measure is estimated from below by a value of order ɛ. So, the total measure of the stability islands is estimated from below by a value independent of ɛ. The proof is based on an analysis of asymptotic formulas for the corresponding Poincaré map.     

Multidimensional Symplectic Separatrix Maps

Summary. We consider an a priori unstable (initially hyperbolic) near-integrable Hamiltonian system in a neighborhood of stable and unstable asymptotic manifolds of a family of hyperbolic tori. Such a neighborhood contains the most chaotic part of the dynamics. The main result of the paper is the construction of the separatrix map as a convenient tool for the studying of such dynamics. We present evidence that the separatrix map combined with the method of anti-integrable limit can give a large class of chaotic trajectories as well as diffusion trajectories. Received March 26, 2001; accepted November 5, 2001     

Stability islands in domains of separatrix crossings in slow-fast Hamiltonian systems

We consider a two-degrees-of-freedom Hamiltonian system with one degree of freedom corresponding to fast motion and the other corresponding to slow motion. The ratio of typical velocities of changes of the slow and fast variables is the small parameter ɛ of the problem. At frozen values of the slow variables, there is a separatrix on the phase plane of the fast variables, and there is a region in the phase space (the domain of separatrix crossings) where the projections of phase points onto the plane of the fast variables repeatedly cross the separatrix in the process of evolution of the slow variables. Under a certain symmetry condition, we prove the existence of many (of order 1/ɛ) stable periodic trajectories in the domain of separatrix crossings. Each of these trajectories is surrounded by a stability island whose measure is estimated from below by a value of order ɛ. So, the total measure of the stability islands is estimated from below by a value independent of ɛ. The proof is based on an analysis of asymptotic formulas for the corresponding Poincaré map. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2007, Vol. 259, pp. 243–255.     

具有二重抛物线解的二次系统

本文给出了具有二重抛物线解的二次系统的一般形状，并与具有并重抛物线解的二次系统相比较，证明了具有二重抛物线解的二次系统也有存在极限环的可能的，而且也是唯一的，但是二重抛物线解却是不可能成为二次系统的分界线不的。     

Integrable Geodesic Flows on Surfaces

We propose a new condition à{{\aleph}} which enables us to get new results on integrable geodesic flows on closed surfaces. This paper has two parts. In the first, we strengthen Kozlov’s theorem on non-integrability on surfaces of higher genus. In the second, we study integrable geodesic flows on a 2-torus. Our main result for a 2-torus describes the phase portraits of integrable flows. We prove that they are essentially standard outside what we call separatrix chains. The complement to the union of the separatrix chains is C ⁰-foliated by invariant sections of the bundle.     

具有高阶分界线环的多项式微分系统的近似系统

利用Hamilton系统的挑动系统得到了构造具有高阶分界线环的多项式微分系统的近似系统的一种方法，并举出了具体例子．     

Dynamics and chaos control of gyrostat satellite

We consider the chaotic motion of the free gyrostat consisting of a platform with a triaxial inertia ellipsoid and a rotor with a small asymmetry with respect to the axis of rotation. Dimensionless equations of motion of the system with perturbations caused by small asymmetries of the rotor are written in Andoyer-Deprit variables. These perturbations lead to separatrix chaos. For gyrostats with different ratios of moments of inertia heteroclinic and homoclinic trajectories are written in closed-form. These trajectories are used for constructing modified Melnikov function, which is used for determine the control that eliminates separatrix chaos. Melnikov function and phase space trajectory are built to show the effectiveness of the control.     

Separatrices and singular points

A new definition of separatrix is introduced and the theory developed is applied to the study of the topological structure of singular points. Each component of the complement of the separatrix set in a neighborhood of an isolated singular point is shown to satisfy one of seven types of behavior.     

Chaotic motions in the resonant separatrix band of a parametrically excited pendulum

The energy increment spectrum method is developed for the numerical prediction of the appearance and disappearance of a primary (2M: 1)-librational or (M: 1)-rotational resonant separatrix band in a parametrically excited pendulum. The analytical conditions for the presence of such separatrix bands are also obtained. Illustrations of the analytical and numerical results for the appearance and destruction of the resonant bands are given for a comparison.     

Separatrix Map Analysis for Fractal Scatterings in Weak Interactions of Solitary Waves

Previous studies have shown that fractal scatterings in weak interactions of solitary waves in the generalized nonlinear Schrödinger (NLS) equations are described by a universal second-order separatrix map. In this paper, this separatrix map is analyzed in detail, and hence a complete characterization of fractal scatterings in these weak interactions is obtained. In particular, scaling laws of these fractals are derived analytically for different initial conditions, and these laws are confirmed by direct numerical simulations. In addition, an analytical criterion for the occurrence of fractal scatterings is given explicitly.     

Uniqueness of Limit Cycle for the Quadratic Systems with Weak Saddle and Focus

It is proved that the quadratic system with a weak saddle has at most one limit cycle,andthat if this system has a separatrix cycle passing through the weak saddle,then the stability of theseparatrix cycle is contrary to that of the singular point surrounded by it.     

Uniqueness of Limit Cycle for the Quadratic Systems with Weak Saddle and Focus

It is proved that the quadratic system with a weak saddle has at most one limit cycle, and that if this system has a separatrix cycle passing through the weak saddle, then the stability of the separatrix cycle is contrary to that of the singular point surrounded by it.     

一个群体防卫捕-食系统的定性分析

该文对一个群体防卫捕一食系统进行了较全面的定性分析．讨论了分界线的相对位置，得到了极限环的存在性、唯一性以及分界线环的存在性，首次证明了群体防卫捕一食系统可以至少存在两个或三个极限环．     

The non-integrability of a rotating elliptical billiard

The problem of a point moving inside a rotating ellipse is considered; collisions at the boundary are assumed to be absolutely elastic. It is shown that this discrete dynamical system does not admit of an analytic integral independent of the energy integral. The proof of non-integrability is based on the method of separatrix splitting.     

三次系统的一类三曲线分界线环

本文证明了三次系统不可能同时存在三个三曲线x^3-3xy^2-1=0分界线环,但是可以同时存在两个三曲线分界线环,给出了同时存在两个三曲线分界线环的充要条件。