Complex oscillations in systems subject to periodic perturbation

2013

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Two types of stochastic motion are described. These are bifurcational processes with nonlinear behavior as a limit cycle (synchronized and on a torus) with limited switching of unstable trajectories Keywords: limit cycle, bifurcation, LCE spectrum 1. Introduction. The analysis of complex motions [8] is still of interest in modern nonlinear dynamics. Along with presenting original results, the monographs [1, 2, 7] provide detailed reviews of the state of the art in the study of the determinate and stochastic behavior of nonlinear systems that leads to chaos. An important characteristic of the dynamic behavior of systems is Lyapunov characteristic exponents (LCEs) (see [1, 2, etc.]). The signature of the LCE spectrum will be found based on a qualitative analysis. We will address nonlinear systems with three-dimensional phase space subject to periodic perturbation.Typical models of mechanics addressed by many authors will also be considered. Some dynamic properties of systems will be discussed for the first time.2. Stochastic Self-Oscillations in Parametrically Excited Oscillator. The monograph [7] addresses the oscillations of a periodically excited oscillator:

Section: Synchronized Self-oscillations Of Ionic Transport Through a mentioning

confidence: 99%

Bifurcation Processes in Periodically Perturbed Systems

Martynyuk

2013

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“…Systems with periodic perturbation. This is the widest class of problems [7,8]. A qualitative analysis shows that orbital loss of stability may occur and cause chaos in some cases [3,5].…”

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confidence: 99%

“…Analyzing the radicand in (1.3), we can find the boundaries of the domains of periodic and aperiodic points [8].…”

mentioning

confidence: 99%

Compound motion of conservative systems under periodic perturbation

2012

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A model is described in which chaos occurs as a motion of an autonomous two-frequency system. The boundaries for the domain of stochastic motions caused by bifurcational and chaotic processes are estimated Keywords: chaos, autonomous two-frequency system, orbital loss of stability, bifurcation Introduction. Physically, the complex behavior of a finite-dimensional system is due to the instability of individual motions. The instability of trajectories located within a bounded domain of the phase space leads to "mixing" and, as a consequence, stochastic motion. The approach used in this case is to set up variational equations with the coefficients of the characteristic equation dependent on a partial solution. The following types of systems are best studied (qualitatively).1. Systems with periodic perturbation. This is the widest class of problems [7,8]. A qualitative analysis shows that orbital loss of stability may occur and cause chaos in some cases [3,5].2. Two-frequency systems described by a system of fourth-order equations. Such systems are also within the limits of applicability of the approach [9].3. Lorentz-type systems. These are systems whose dimension is no less than three and that generate saddle cycles. In many cases, a chain of bifurcations leads from determinacy to chaos. Each such transition is accompanied by loss of stability of a simple manifold and birth of a new, more complicated stable manifold. Internal parametric fluctuations may affect the mechanism of compound motions and cause unexpected qualitative changes in the behavior of the system. Orbital stability analysis is the key aspect in the understanding of the mechanism of complex nonlinear dynamic phenomena.Here we conduct a qualitative analysis of systems of nonlinear conservative oscillators with periodic forcing. An important characteristic of the dynamic behavior of systems is Lyapunov characteristic exponents (LCEs) (see [1,2] and the references therein). We will estimate the initial perturbations for certain types of motion, including chaos.

“…Modern methods of qualitative analysis based on studies of Andronov, Birkhoff, Lyapunov, and Poincaré [1,[3][4][5] are developed rather intensively owing to applications in mechanics [6][7][8][9][10]. In the present paper, we will analyze the stochastic motions caused by bifurcational and chaotic processes in terms of Lyapunov characteristic exponents (LCEs).…”

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confidence: 99%

Estimating the chaos boundaries of a double pendulum

2011

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Loss of the orbital stability of a double pendulum is considered in terms of Lyapunov exponents. The boundaries of the domain of stochastic motion caused by bifurcational and chaotic processes are estimated Keywords: double pendulum, bifurcation, chaos Introduction. Modern methods of qualitative analysis based on studies of Andronov, Birkhoff, Lyapunov, and Poincaré [1,[3][4][5] are developed rather intensively owing to applications in mechanics [6-10]. In the present paper, we will analyze the stochastic motions caused by bifurcational and chaotic processes in terms of Lyapunov characteristic exponents (LCEs). The general approach followed to analyze the signature of the LCE spectrum involves a system of variational equations in which the coefficients of the matrix of the right-hand side depend on a partial solution. This makes it possible to establish the existence of solutions of certain quality.In the present paper, we study the stochastic oscillations of a double pendulum. We will distinguish stochastic oscillations generated by a bifurcation process and stochastic motions generated by chaos. The boundaries of the domains will be estimated in both cases. The signature of the LCE spectrum for various cases of stochastic motions will be given after a qualitative analysis. The present study is a continuation of the paper [8] concerned with the quasiperiodic and compound motion of a double pendulum. Preliminaries.A double pendulum is two simple pendulums connected in series. Let these pendulums be of equal length. Consider a particle A 1 of mass m 1 moving in a vertical plane along a circle of radius l 1 with center O and a particle A 2 of mass m 2 moving in a vertical plane and connected to the particle A 1 by a rigid imponderable bar of length l 2 . We will use Lagrangian coordinates: the angle x 1 between the vertical and the segment OA 1 and the angle x 3 between the vertical and the segment A A 1 2 . The kinetic and potential energies of the system are given by