The study of differential equations is useful in to analyze the possible past or future with help of present information. In this paper, the behavior of solutions has been analyzed around the equilibrium points for Gause model. Finally, some results are worked out to exist the stable periodic orbit for mentioned predator-prey system.

By using the foundational solutions of the linear variational equation of the unperturbed system along the homoclinic orbit as the local current coordinates system of the system in the small neighborhood of the homoclinic orbit, we discuss the bifurcation problems of nondegenerated homoclinic loops. Under the nonresonant condition, existence, uniqueness, and incoexistence of 1-homoclinic loop a...

Homoclinic bifurcation with one orbit flip, two inclination flips and resonance in the tangent directions of homoclinic orbit is considered. By studying the associated successor functions constructed from a local active coordinate system, we prove the existence of double 1-periodic orbit, 1-homoclinic orbit, and also some coexistence conditions of 1-periodic orbit and 1-homoclinic orbit.

Motivated by the chaos suppression methods based on stabilizing an unstable periodic orbit, we study the stability of synchronized periodic orbits of coupled map systems when the period of the orbit is the same as the delay in the information transmission between coupled units. We show that the stability region of a synchronized periodic orbit is determined by the Floquet multiplier of the peri...

Harmonic inversion has already been proven to be a powerful tool for the analysis of quantum spectra and the periodic orbit orbit quantization of chaotic systems. The harmonic inversion technique circumvents the convergence problems of the periodic orbit sum and the uncertainty principle of the usual Fourier analysis, thus yielding results of high resolution and high precision. Based on the clo...

Niels Bohr successfully predicted in 1913 the energy levels for hydrogen atom by applying certain quantization rules to classically obtained periodic orbits. Many physicists tried apply similar methods other atoms. In his well-known 1921 paper, I. Langmuir established numerically existence of a orbit helium considered as classical three body problem. this we give an analytic proof Langmuir’s or...

This article concerns arbitrary finite heteroclinic networks in any phase space dimension whose vertices can be a random mixture of equilibria and periodic orbits. In addition, tangencies in the intersection of un/stable manifolds are allowed. The main result is a reduction to algebraic equations of the problem to find all solutions that are close to the heteroclinic network for all time, and t...

Given a dynamical system and a function f from the state space to the real numbers, an optimal orbit for f is an orbit over which the time average of f is maximal. In this paper we consider some basic mathematical properties of optimal orbits: existence, sensitivity to perturbations of f , and approximability by periodic orbits with low period. For hyperbolic systems, we conjecture that for (to...