By G.C. Layek

ISBN-10: 8132225554

ISBN-13: 9788132225553

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**Extra info for An Introduction to Dynamical Systems and Chaos**

**Sample text**

8 Conservative and Dissipative Dynamical Systems The dichotomy of dynamical systems in conservative versus dissipative is very important. They have some fundamental properties. Particularly, conservative systems are the integral part of Hamiltonian mechanics. We give here only the formal deﬁnitions of conservative and dissipative systems. 8 Conservative and Dissipative Dynamical Systems 27 Fig. 9 Graphical representation of the flow x_ ¼ Àx sgn x The conservative and dissipative systems are deﬁned with respect to the divergence of the corresponding vector ﬁeld, which in turn refers to the conservation of volume or area in their state space or phase plane, respectively as follows: A system is said to be conservative if the divergence of its vector ﬁeld is zero.

According to the above lemma, the change in phase area is given by AðtÞ ¼ cAð0ÞeÀat ; a [ 0 as t ! 1; c being a constant. 9 Find the phase volume element for the systems (i) x_ ¼ Àx; (ii) x_ ¼ ax À bxy; y_ ¼ bxy À cy where x; y ! 0 and a; b; c are positive constants. Solution (i) The flow of the system x_ ¼ Àx is attracted toward the point x ¼ 0: The time rate of change of volume element VðtÞ under the flow is given as Z dV ¼ À dt t¼0 dx ¼ ÀVð0Þ Dð0Þ or; VðtÞ ¼ Vð0ÞeÀt ! 0 as t ! 1: Hence the phase volume element VðtÞ shrinks exponentially.

4 Find the general solution of the linear system x_ ¼ 10x À y y_ ¼ 25x þ 2y Solution Given system can be written as x_ ¼ Ax$ ; where A ¼ $ 10 25 À1 2 x and $x ¼ : y The characteristic equation of matrix A is det(A À kIÞ ¼ 0 10 À k À1 ¼0 ) 25 2 À k ) k2 À 12k þ 45 ¼ 0 ) k ¼ 6 Æ 3i: Therefore, matrix A has a pair of complex conjugate eigenvalues 6 ± 3i. e1 be the eigenvector corresponding to the eigenvalue Let $e ¼ e2 λ = 6 + 3i. 2 Eigenvalue-Eigenvector Method 45 ðA À ð6 þ 3iÞI Þe$ ¼ $ 0 0 e1 10 À 6 À 3i À1 ¼ ) e2 0 25 2 À 6 À 3i 0 ð4 À 3iÞe1 À e2 ¼ ) 0 25e1 À ð4 þ 3iÞe2 ð4 À 3iÞe1 À e2 ¼ 0; 25e1 À ð4 þ 3iÞe2 ¼ 0: ) A nontrivial solution of this system is e1 ¼ 1; e2 ¼ 4 À 3i: 0 1 1 1 ¼$ a 1 þ ia þi ¼ , where $ a1 ¼ $2 À3 4 4 À 3i 4 Therefore $e ¼ 0 .

### An Introduction to Dynamical Systems and Chaos by G.C. Layek

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