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

By G.C. Layek

ISBN-10: 8132225554

ISBN-13: 9788132225553

**Read Online or Download An Introduction to Dynamical Systems and Chaos PDF**

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**Additional resources for An Introduction to Dynamical Systems and Chaos**

**Example text**

The eigenvector for the eigenvalue k1 ¼ 4 1 1 is given as @ 1 A. The eigenvector corresponding to the repeated eigenvalue 2 k2 ¼ k3 ¼ À2 is ð e1 e2 e3 ÞT such that 0 3 @3 6 À3 À3 À6 10 1 0 1 0 3 e1 3 A@ e2 A ¼ @ 0 A e3 0 6 which is equivalent to 3e1 À 3e2 þ 3e3 ¼ 0; 3e1 À 3e2 þ 3e3 ¼ 0; 6e1 À 6e2 þ 6e3 ¼ 0; that is, e1 À e2 þ e3 ¼ 0. We 1 can choose e1 ¼ 1, e2 ¼ 1 and e3 ¼ 0, and so we can take one eigenvector 0 1 as @ 1 A. Again, we can choose e1 ¼ 0, e2 ¼ 1 and e3 ¼ 1. Then we obtain another 0 0 1 0 eigenvector @ 1 A.

Then each $x j ðtÞ ¼ $ a j ekj t , j = 1, 2, …, n is a solution of $x_ ¼ Ax$ . The general solution is a linear combination of the solutions x ðtÞ $j and is given by x ðtÞ ¼ $ n X j¼1 cj $x j ðtÞ where c1 ; c2 ; . ; cn are arbitrary constants. 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 (

According to the deﬁnition of ﬁxed point, the equilibrium points of this system are obtained as sin x ¼ 0 ) x ¼ npðn ¼ 0; Æ1; Æ2; . Þ: This simple looking autonomous system has inﬁnite numbers of equilibrium points in R: We can see that there are two kinds of equilibrium points. The equilibrium point where the flow is toward the point is called sink or attractor (neighboring trajectories approach asymptotically to the point as t ! 1 ). On the other hand, when the flow is away from the point, the point is called source or repellor (neighboring trajectories move away from the point as t !

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

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