Conformal Mapping - Methods and Applications (revised) by R. Schinzinger, P. Laura

By R. Schinzinger, P. Laura

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4. Stability of a point on the limit cycle illustrated by the dynamics of a light particle in a viscous fluid. (a) Transversely to the cycle, the point is in a state of stable equilibrium: perturbation of the amplitude rapidly decays to the stable value A. , that of the phase, neither grows nor decays. Steady growth of the phase can be represented by a light particle sliding in a viscous fluid along an inclined plane. Two such particles (corresponding to the unperturbed and perturbed phases, filled and open circles) slide with a constant lag.

Here we briefly summarize the properties of self-sustained oscillatory systems. This oscillator is an active system. It contains an internal source of energy that is transformed into oscillatory movement. Being isolated, the oscillator continues to generate the same rhythm until the source of energy expires. , without explicit time dependence) dynamical system. , on the transient to steady oscillation. The oscillation is stable to (at least rather small) perturbations: being disturbed, the oscillation soon returns to its original shape.

Moreover, sometimes we have no access to the parameters of the systems and coupling, but can only observe the oscillations. For example, the human organism contains a number of oscillators, such as the rhythmically contracting heart and respiratory system. Unlike the described “virtual” experiment with clocks, it is impossible (or, at least, very difficult) to tune these systems or to influence the coupling strength. The only way to detect the interaction is to analyze the oscillations registered under free-running conditions.

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