By G.C. Layek
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Extra info for An Introduction to Dynamical Systems and Chaos
Consider a small perturbation quantity nðtÞ; away from the ﬁxed point xÃ ; such that xðtÞ ¼ xÃ þ nðtÞ: We see whether the perturbation grows or decays as time goes on. 6 Linear Stability Analysis 15 n_ ¼ x_ ¼ f ðxÞ ¼ f ðxÃ þ nÞ: Taylor series expansion of f ðxÃ þ nÞ gives n2 n_ ¼ f ðxÃ Þ þ nf 0 ðxÃ Þ þ f 00 ðxÃ Þ þ Á Á Á 2 According to linear stability analysis, we get n_ ¼ nf 0 ðxÃ Þ½* f ðxÃ Þ ¼ 0 Assuming f 0 ðxÃ Þ 6¼ 0; the perturbation nðtÞ grows exponentially if f 0 ðxÃ Þ [ 0 and decays exponentially if f 0 ðxÃ Þ\0: Linear theory fails if f 0 ðxÃ Þ ¼ 0 and then higher order derivatives must be considered in the neighborhood of ﬁxed point for stability analysis of the system.
From the geometric approach one can get local stability behavior of the equilibrium points of the system easily and is valid for all time. We shall now re-look the analytical solution of the system. The analytical solution can be expressed as À1 t ¼ logjtanðx=2Þj þ c ) xðtÞ ¼ 2 tan ðAet Þ where A is an integrating constant. Fig. 7 Analysis of One-Dimensional Flows 23 Let the initial condition be x0 ¼ xð0Þ ¼ p=4: Then from the above solution we obtain pﬃﬃﬃ pﬃﬃﬃ A ¼ tanðp=8Þ ¼ À1 þ 2 ¼ 1= 1 þ 2 : So the solution is expressed as xðtÞ ¼ 2 tan À1 et pﬃﬃﬃ : 1þ 2 We see that the solution xðtÞ !
Absorbing set A positive invariant compact subset B Rn is said to be an absorbing set if there exists a bounded subset C of Rn with C ' B such that tC [ 0 ) /ðt; CÞ & B 8t ! tC (see the book by Wiggins  for details). 32 1 Continuous Dynamical Systems Trapping zone An open set U in an invariant set D & Rn in an attracting set for a flow generated by a system is called a trapping zone. Let a set A be closed and invariant. The set A is said to be stable if and only if every neighborhood of A contains a neighborhood U of A which is trapping.
An Introduction to Dynamical Systems and Chaos by G.C. Layek