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First of all we should note that the term “phase” is used in physics and nonlinear dynamics in different senses. For example, we have already encountered the notion of “phase space”. The coordinates in this space are often denoted “phase point”, and the time evolution of a system is described by the motion of this point. Here, as well as in the expression “phase transitions”, the meaning of the word “phase” is completely different from what we understand by saying “the phase of oscillation”. In this book we hope that the exact meaning will always be clear from the context.

For this purpose we consider the behavior of the trajectories in the vicinity of the cycle. In other words, we look at what happens if a phase point is pushed off the limit cycle. ) system this would mean that we somehow perturb its periodic motion. , the phase 3 The number of required variables depends on the particular dynamical system and is called its dimension. 1. Periodic oscillation is represented by a closed curve in the phase space of the system: equivalent states x(t) and x(t + T ) (denoted by a number on the time plot) correspond to the same point on that curve (denoted by the same number).

Independently of its form, the limit cycle is stable in the transversal direction, whereas in the tangential direction the phase point is neither stable nor unstable. , the unperturbed and the perturbed ones, then we see that in the radial direction they converge with time, while in the direction along the cycle they neither converge nor diverge. In terms of nonlinear dynamics, the convergence/divergence properties of nearby trajectories are characterized by the Lyapunov exponents. Convergence of trajectories along some direction in the phase space corresponds to a negative Lyapunov exponent (Fig.

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