By G.C. Layek
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Extra resources for An Introduction to Dynamical Systems and Chaos
Nonlinear Dynamics: Integrability, Chaos and Patterns. Springer, 2003. 3. : Theory of Ordinary Differential Equations. McGraw Hill, New York (1955) 4. : Ordinary Differential Equations. MIT Press, Cambridge, MA (1973) 5. : Nonlinear Dynamics and Chaos with application to physics, biology, chemistry and engineering. C, Massachusetts (1994) References 35 6. : Stability, Instability and Chaos: An Introduction to the Theory of Nonlinear Differential Equations. Cambridge University Press. 1994 7. : Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edn.
From Fig. 5 we see that the ﬁxed point x ¼ 1 is stable whereas the ﬁxed point x ¼ 0 is unstable. 7 Find the ﬁxed points and analyze the local stability of the following systems (i) x_ ¼ x þ x3 (ii) x_ ¼ x À x3 (iii) x_ ¼ Àx À x3 Solution (i) Here f ð xÞ ¼ x þ x3 . Then for ﬁxed points f ð xÞ ¼ 0 ) x þ x3 ¼ 0 ) x ¼ 0; as x 2 R: So, 0 is the only ﬁxed point of the system. We now see that when x [ 0; x_ [ 0 and when x\0; x_ \0. Hence the ﬁxed point x ¼ 0 is unstable. The graphical representation of the flow generated by the system is displayed in Fig.
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 ! 1 as i ! 9 Some Deﬁnitions 31 /ðti ; xÞ ! p as i ! 1. If $x lies in the closed orbit C; then /ðti ; xÞ ¼ p for each i: Hence every point of C is a x-limit point of $x and so KðxÞ ¼ C for every x 6¼ 0: When j xj 1; the sequence fti g1 i¼1 with t\0 gives the a-limit set & f0g for jxj\1 .