By G.C. Layek
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Extra resources for An Introduction to Dynamical Systems and Chaos
6 Graphical representation of f ðxÞ ¼ ðx þ x3 Þ (iii) Here f ð xÞ ¼ Àx À x3 . Then f ð xÞ ¼ 0 ) Àx À x3 ¼ 0 ) x ¼ 0; as x 2 R: So x ¼ 0 is the only ﬁxed point of the system. We now see that x_ [ 0 when x\0 and x_ \0 when x [ 0: This shows that the ﬁxed point x ¼ 0 is stable. The graphical representation of the flow generated by the system is displayed in Fig. 8. 8 Determine the equilibrium points and sketch the phase diagram in the neighborhood of the equilibrium points for the system represented as x_ þ x sgnðxÞ ¼ 0: Solution Given system is x_ þ x sgn ðxÞ ¼ 0; that is, x_ ¼ Àx sgn ðxÞ; where the function sgnðxÞ is deﬁned as 8 > < 1; x [ 0 sgn ðxÞ ¼ 0; x ¼ 0 > : À1; x\0 For equilibrium points, we have x_ ¼ 0 ) x sgn x ¼ 0 ) x ¼ 0: 26 1 Continuous Dynamical Systems Fig.
The Lipschitz condition jf ðt; x1 Þ À f ðt; x2 Þj K jx1 À x2 j; 8ðt; x1 Þ; ðt; x2 Þ 2R; K being constant, is satisﬁed on the Lipschitz R. Since jf ðt; xÞj ¼ x2 þ cos2 t x2 þ cos2 t x2 þ 1; and M ¼ maxjf ðt; xÞj ¼ 1 þ b2 in R. Therefore, from Picard’s theorem (if f ðt; xÞ is a continuous function in a rectangle R ¼ fðt; xÞ : jt À t0 j a; jx À x0 j b; a [ 0; b [ 0g and satisﬁes Lipschitz condition therein, then the initial value problem x_ ¼ f ðt; xÞ; xðt0 Þ ¼ x0 has a unique solution in the rectangle R0 ¼ fðt; xÞ : jt À t0 j h; jx À x0 j bg; where h ¼ minfa; b=M g; M ¼ maxjf ðt; xÞj for all ðt; xÞ 2 R; seen the books o Coddington and È bÉ b Levinson , Arnold ).
But it is the ﬁxed point of the given system, because x_ ¼ 0 , x ¼ 0: Therefore, /t ð0Þ ¼ 0 for all t 2 R: So the evolution operator of the system is given as /t ðxÞ ¼ 1 þx xt for all x 2 R: The evolution operator /t is not deﬁned for all t 2 R: For example, if t ¼ À1=x; x 6¼ 0; then /t is undeﬁned. Thus we see that the interval in which /t is deﬁned is completely dependent on x. We shall now examine the group properties of the evolution operator /t below: (i) /r /s 2 f/t ðxÞ; t 2 R; x 2 Rg 8r; s 2 R (Closure property) Now, y x : Take y ¼ 1 þ yr 1 þ xs x=1 þ sx x x ¼ ¼ xr ¼ 1 þ 1 þ xs 1 þ xs þ xr 1 þ xðs þ rÞ /r ðyÞ ¼ ¼ /s þ r 2 f/t ðxÞ; t 2 R; x 2 Rg 16 1 Continuous Dynamical Systems (ii) /t ð/s /r Þ ¼ ð/t /s Þ/r (Associative property) L:H:S: ¼ /t ðð/s /r ÞðxÞÞ ¼ /t ðyÞ ¼ ðwhere y ¼ /s ðzÞ; z ¼ /r ðxÞ ¼ x 1 þ rx y z x ¼ ¼ ; y ¼ /s ð/r ðxÞÞ 1 þ yt 1 þ zs 1 þ xðr þ sÞ x ¼ /t þ r þ s ðxÞ 1 þ xðt þ r þ sÞ R:H:S: ¼ ðð/t /s Þ/r ðxÞÞ ) L:H:S ¼ Now, y x ; y ¼ /s ðxÞ ¼ 1 þ yt 1 þ sx x ¼ /t þ s ðxÞ ¼ 1 þ xðt þ sÞ z x ; z ¼ /r ðxÞ ¼ /t þ s ð/r ÞðxÞ ¼ /t þ s ðzÞ ¼ 1 þ zðt þ sÞ 1 þ rx x ¼ /t þ s þ r ðxÞ /t þ s ð/r ÞðxÞ ¼ 1 þ xðt þ s þ rÞ /t ðyÞ ¼ Hence, /t ð/s /r ÞðxÞ ¼ ð/s /r Þ/t ðxÞ; 8x 2 R.