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Analysis of a nonsynchronized sinusoidally driven dynamical system. (English) Zbl 1090.34557

Summary: A nonautonomous system, i.e., a system driven by an external force, is usually considered as being phase synchronized with this force. In such a case, the dynamical behavior is conveniently studied in an extended phase space which is the product of the phase space \(\mathbb R^m\) of the undriven system by an extra dimension associated with the external force. The analysis is then performed by taking advantage of the known period of the external force to define a Poincaré section relying on a stroboscopic sampling. Nevertheless, it may so happen that the phase synchronization does not occur. It is then more convenient to consider the nonautonomous system as an autonomous system incorporating the subsystem generating the driving force. In the case of a sinusoidal driving force, the phase space is \(\mathbb R^{m+2}\) instead of the usual extended phase space \(\mathbb R^m\times S^1\). It is also demonstrated that a global model may then be obtained by using \(m+2\) dynamical variables with two variables associated with the driving force. The model obtained characterizes an autonomous system in contrast with a classical input/output model obtained when the driving force is considered as an input.

MSC:

34C28 Complex behavior and chaotic systems of ordinary differential equations
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