By G.C. Layek

ISBN-10: 8132225554

ISBN-13: 9788132225553

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23. Prove that the phase volume of a conservative system is constant. Is the converse true? Give reasons in support of your answer. 24. What can you say about time rate of change of phase volume element in a dissipative dynamical system? Explain it geometrically. Give an example of a dissipative system. 25. Prove that the a- and x-limit sets of a flow /t ðxÞ are contained in the non-wandering set of the flow /t ðxÞ: 26. Deﬁne absorbing set of a flow. Write down the relation between trapping zones T and absorbing sets.

Is the converse true? Give reasons in support of your answer. 24. What can you say about time rate of change of phase volume element in a dissipative dynamical system? Explain it geometrically. Give an example of a dissipative system. 25. Prove that the a- and x-limit sets of a flow /t ðxÞ are contained in the non-wandering set of the flow /t ðxÞ: 26. Deﬁne absorbing set of a flow. Write down the relation between trapping zones T and absorbing sets. Prove that for an absorbing set A; t ! 0 /ðt; AÞ forms an attracting set.

In R2 , the solution can be written as x ðtÞ ¼ $ 2 X cj $ a j ek j t ¼ c1 $ a 1 ek 1 t þ c2 $ a 2 ek 2 t : j¼1 Case II: Eigenvalues of A are real but repeated In this case matrix A may have either n linearly independent eigenvectors or only one or many (

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