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# Simple Chaotic Oscillator: From Mathematical Model to Practical Experiment

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Simple Chaotic Oscillator: From Mathematical Model to Practical Experiment book

# Simple Chaotic Oscillator: From Mathematical Model to Practical Experiment

DOI link for Simple Chaotic Oscillator: From Mathematical Model to Practical Experiment

Simple Chaotic Oscillator: From Mathematical Model to Practical Experiment book

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## ABSTRACT

Introduction It is well known that many physical systems with some time evolution can be described by a set of generally nonlinear differential equations. It seems that an electronic circuit is an easy way how to construct a dynamical system which represents a given mathematical model. That is why we often use this approach for the purpose of studying dynamical motion, which can be simple (ﬁ xed points, limit cycles) as well as very complicated (chaos). We can also see some universality behind the construction of chaotic oscillators, because we are not interested in the concrete physical interpretation of the individual state variables. For example, chaos was recently reported in the scientiﬁ c ﬁ elds such as chemistry, mechanics, economy, biology, etc. For lumped electronic circuit, the necessary condition is a ﬁ nite number of the state variables and, of course, a complexicity of the synthesized network grows in accordance with growing dimension of the dynamical system. The

very ﬁ rst discovered chaotic oscillator was deeply described in [1], and is so far the only dynamical system where the presence of chaos was conﬁ rmed numerically, experimentally as well as mathematically. These equations, taking name after its discoverer as the Chua’s equations, belong to extensive group of dynamical systems covered by a following state space representation in compact matrix form

( )xwbxAx Th+= ,(1a) where 3ℜ∈X is a vector of the state variables, A is a square matrix and b, w are column vectors. The saturation-type nonlinearity of the form

and similarly for single inner region D0

3 =−+−= pspsps where E is the unity matrix. In the parameter space of some interest we can assume one ﬁ xed point per state space region, namely

bAx 1−±=outer ,( ) 0det ≠A (3b) The main property of the chaotic solution is in extreme sensitive to the changes of the initial conditions. It implies that we can not obtain closed-form analytic solution, so our analysis is restricted to the numerical integration.