By G.C. Layek
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Extra info for An Introduction to Dynamical Systems and Chaos
7 21 Analysis of One-Dimensional Flows As we know qualitative approach is the combination of analysis and geometry and is a powerful tool for analyzing solution behaviors of a system qualitatively. By drawing trajectories in phase line/plane/space, the behaviors of phase points may be found easily. ; (iii) Local and asymptotic solution behaviors of a system; (iv) Topological features of flows such as bifurcations, catastrophe, topological equivalence, transitiveness, etc. We shall now analyze a simple one-dimensional system as follows.
Iii) /0 ðxÞ ¼ 1 þxxÁ0 ¼ x; /0 is the identity operator. y x ; y ¼ /Àt ðxÞ ¼ 1 þ ty 1 À tx x ¼ x ¼ /0 ðxÞ ð/Àt is the inverse of /t Þ ¼ 1 À tx þ tx /t /Àt ðxÞ ¼ /t ðyÞ ¼ (iv) Hence the flow evolution operator forms a dynamical group. (v) /t /s ¼ /s /t Now, ð/t /s ÞðxÞ ¼ /t ðyÞ ¼ y x ; y ¼ /s ðxÞ ¼ 1 þ ty 1 þ xs x ¼ /t þ s ðxÞ 1 þ xðt þ sÞ z x ; z ¼ /t ðxÞ ¼ /s /t ðxÞ ¼ /s ðzÞ ¼ 1 þ sz 1 þ tx x x ¼ ¼ /s þ t ðxÞ ¼ 1 þ tx þ sx 1 þ ðs þ tÞx ¼ So, /t /s ¼ /s /t (Commutative property). 6 Linear Stability Analysis 17 Thus, the evolution operator /t forms a commutative group.
P as i ! 1. The x-limit set(cycle) is denoted by Kðx$ Þ and is deﬁned as & ' Kðx$ Þ ¼ $x 2 Rn j9fti g with ti ! 1 and /ðti ; $x Þ ! p as i ! 1 : Similarly, the a-limit set (cycle), lðx$ Þ; is deﬁned as & ' lðx$ Þ ¼ $x 2 Rn j9 fti gwith ti ! À1 and/ðti ; $x Þ ! p as i ! 1 : For example, consider a flow /ðt; xÞ on R2 generated by the system r_ ¼ crð1 À rÞ; h_ ¼ 1; c being a positive constant. For x 6¼ 0; let p be any point of the closed orbit C and take fti g1 i¼1 to be the sequence of t [ 0: The trajectory through x crosses the radial line through p: So, ti !