N-pulse homoclinic orbits in perturbations of resonant hamiltonian systems

In this paper we develop an analytical method to detect orbits doubly asymptotic to slow manifolds in perturbations of integrable, two-degree-of-freedom resonant Hamiltonian systems. Our energy-phase method applies to both Hamiltonian and dissipative perturbations and reveals families of multi-pulse solutions which are not amenable to Melnikov-type methods. As an example, we study a two-mode approximation of the nonlinear, nonplanar oscillations of a parametrically forced inextensional beam. In this problem we find unusually complicated mechanisms for chaotic motions and verify their existence numerically.     

Coexisting periodic orbits in vibro-impacting dynamical systems

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Detecting unstable periodic orbits and unstable quasiperiodic orbits in vibro-impact systems

In this paper, unstable dynamics is considered for the models of vibro-impact systems with linear differential equations coupled to an impact map. To provide a skeleton for the organization of chaotic attractors, we propose a method for detecting unstable periodic orbits embedded in chaotic attractors through a combination of unconstrained optimization technique and Poincaré map. Three numerical examples from different vibro-impact models demonstrate that the strategy can efficiently detect unstable periodic orbits in chaotic attractors. In order to explore the mechanism responsible for the creation of multi-dimensional tori attractors, we also present another method to detect unstable quasiperiodic orbits embedded multi-dimensional tori attractors by examining a specially transient time series. The upper bound and lower bound of the transient time series (in the Poincaré map) can be obtained by analyzing transient Lyapunov exponent and transient Lyapunov dimension. Some examples verify the effectiveness of the numerical algorithm.     

Stability analysis for multi-parameter linear periodic systems

Summary This paper is devoted to stability analysis of general linear periodic systems depending on real parameters. The Floquet method and perturbation technique are the basis of the development. We start out with the first and higher-order derivatives of the Floquet matrix with respect to problem parameters. Then the behaviour of simple and multiple multipliers of the system with a change of parameters is studied. Weak and strong interactions of multipliers in the complex plane are treated separately. The presented theory is exemplified and discussed. Received 27 April 1998; accepted for publication 5 November 1998     

Transversality between invariant manifolds of periodic orbits for a class of monotone dynamical systems

We consider a special class of monotone dynamical systems and show that in this special class the stable and unstable manifolds of two hyperbolic periodic orbits always intersect transversally. The proof is based on the existence of a family of positively invariant nested cones.This paper is dedicated to Jack Hale on the occasion of his 60th birthday.     

Stability of periodic motions of quasilinear systems

The stability of linear and quasilinear systems with periodic coefficients is analyzed. The properties of stability are established in terms of matrix-valued Lyapunov functions. An algorithm is developed to set up matrix-valued Lyapunov functions for linear quasiperiodic systems. A numerical example is given to illustrate the application of the algorithm Translated from Prikladnaya Mekhanika, Vol. 44, No. 10, pp. 101–113, October 2008.     

Optimization along families of periodic and quasiperiodic orbits in dynamical systems with delay

Nonlinear Dynamics - This paper generalizes a previously conceived, continuation-based optimization technique for scalar objective functions on constraint manifolds to cases of periodic and...     

Adaptive control designs for control-based continuation of periodic orbits in a class of uncertain linear systems

Nonlinear Dynamics - This paper proposes two novel adaptive control designs for the feedback signals used in the control-based continuation paradigm to track families of periodic orbits of...     

Analytical conditions for the existence of a two-parameter family of periodic orbits in nonautonomous dynamical systems

The two-parameter perturbation method, applied to the example of periodic oscillations in periodically driven nonlinear dynamical systems, is presented. The analytical conditions are given for the existence of a two-parameter family of periodic orbits in nonautonomous dynamical systems in both non-resonance and resonance cases.     

Double impact periodic orbits for an inverted pendulum

There exist many types of possible periodic orbits that impact at the walls for the inverted pendulum impacting between two rigid walls. Previous studies only focused on single impact periodic orbits and symmetric periodic orbits that bounce back and forth between the two walls. They respectively correspond to Types I and II orbits in the Chow, Shaw and Rand classification. In this paper we discuss two types of double impact periodic orbits that have not been studied before. The equations need to be solved for double impact orbits are transcendental and it is very hard to see the structure of the solutions. Consequently the analysis of double impact orbits is much more difficult than that of Types I and II orbits. A combination of analytical and numerical methods is employed to investigate the existence, stability and bifurcations of these orbits. Grazing bifurcations, which do not present for Types I and II orbits, are also observed.     

Relative periodic orbits in plane Poiseuille flow

A branch of relative periodic orbits is found in plane Poiseuille flow in a periodic domain at Reynolds numbers ranging from Re=3000$\mathit{Re}=3000$ to Re=5000$\mathit{Re}=5000$. These solutions consist in sinuous quasi-streamwise streaks periodically forced by quasi-streamwise vortices in a self-sustained process. The streaks and the vortices are located in the bulk of the flow. Only the amplitude, but not the shape, of the averaged velocity components does change as the Reynolds number is increased from 3000 to 5000. We conjecture that these solutions could therefore be related to large- and very large-scale structures observed in the bulk of fully developed turbulent channel flows.     

Utilizing true periodic orbits in chaos-based cryptography

Nonlinear Dynamics - Digital realizations of chaos-based cryptosystems suffer from lack of a reliable method for implementation. The common choice for implementation is to use fixed or...     

A semi-numerical method for periodic orbits in a bisymmetrical potential

We use a semi-numerical method to find the position and period of periodic orbits in a bisymmetrical potential, made up of a two dimensional harmonic oscillator, with an additional term of a Plummer potential, in a number of resonant cases. The results are compared with the outcomes obtained by the numerical integration of the equations of motion and the agreement is good. This indicates that the semi-numerical method gives general and reliable results. Comparison with other methods of locating periodic orbits is also made.     

Connecting orbits in perturbed systems

We apply Newton’s method and continuation techniques to determine heteroclinic connections in perturbed non-autonomous differential equations which do not exist for the underlying unperturbed system. This approach is particularly useful in a higher-dimensional context, where the numerical computation of invariant manifolds is very expensive. A detailed discussion of a four-dimensional model is presented, which describes a pendulum coupled to a harmonic oscillator.