By G.C. Layek
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Extra resources for An Introduction to Dynamical Systems and Chaos
C-. 0 when x 0 8. Prove that the solutions of the initial value problem x_ ¼ x1=n when x [ 0 with xð0Þ ¼ 0 are not unique for n ¼ 2; 3; 4; . : 9. What do you mean by ﬁxed point of a system? Determine the ﬁxed points of the system x_ ¼ x2 À x; x 2 R: Show that solutions exist for all time and become unbounded in ﬁnite time. 10. Give mathematical deﬁnitions of ‘flow evolution operator’ of a system. Write the basic properties of an evolution operator of a flow. 11. Show that the dynamical system (or evolution) forms a dynamical group.
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 ! p and t ! 1. Without using analytical solution for this particular initial condition the same result can be found by drawing the graph of x versus t.
Then we obtain another 0 0 1 0 eigenvector @ 1 A. Clearly, these two eigenvectors are linearly independent. Thus, 1 we have two linearly independent eigenvectors corresponding to the repeated eigenvalue −2. Hence, the general solution of the system is given by 0 1 0 1 0 1 1 1 0 x$ ðtÞ ¼ c1 @ 1 Ae4t þ c2 @ 1 AeÀ2t þ c3 @ 1 AeÀ2t 2 0 1 where c1 , c2 and c3 are arbitrary constants. 8 Solve the system $x_ ¼ Ax$ where 2 À1 6 1 A¼6 4 0 0 À1 À1 0 0 3 0 0 0 0 7 7 0 À2 5 1 2 50 2 Linear Systems Solution Here matrix A has two pair of complex conjugate eigenvalues k1 ¼ À1 Æ i and k2 ¼ 1 Æ i.
An Introduction to Dynamical Systems and Chaos by G.C. Layek