By G.C. Layek
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Additional info for An Introduction to Dynamical Systems and Chaos
M. Let $ $ eigenvectors. 2 Eigenvalue-Eigenvector Method 41 x ðtÞ ¼ $ where u $j m X j¼1 cj $ u j þ dj $v j ¼ expðaj tÞfa cosðbj tÞ À b j sinðbj tÞg, $j $ v $j ¼ expðaj tÞfa sinðbj tÞ þ b j $j $ cosðbj tÞg and cj ; dj ðj ¼ 1; 2; . ; mÞ are arbitrary constants. We discuss each of the above cases through speciﬁc examples below. 1 Find the general solution of the following linear homogeneous system using eigenvalue-eigenvector method: x_ ¼ 5x þ 4y y_ ¼ x þ 2y: Solution In matrix notation, the system can be written as $x_ ¼ Ax$ , where $x ¼ 5 4 x .
Then for k = 1, 2, …, m, any nonzero solution 0 is called a generalized eigenvector of A. For simof the equation ðA À kIÞk $v ¼ $ plicity consider a two dimensional system. Let the eigenvalues be repeated but only a 2 be a generalized eigenvector one eigenvector, say $ a 1 be linearly independent. Let $ of the 2 × 2 matrix A. Then $ a 2 can be obtained from the relation ¼$ a 1 ) Aa ¼ ka þ$ a 1 . So the general solution of the system is ðA À kIÞa $2 $2 $2 given by a 1 ekt þ c2 ðta ekt þ $ a ekt Þ: x ðtÞ ¼ c1 $ $1 $ 2 Similarly, P for an n × n matrix A, the general solution may be written as x$ ðtÞ ¼ ni¼1 ci $x i ðtÞ, where x ðtÞ $1 ¼$ a 1 ekt ; x 2 ðtÞ ¼ ta ekt þ $ a 2 ekt ; $1 $ x ðtÞ $3 t ¼ 2!
The system generates a flow /ðt; $x Þ: We give Liouville’s theorem which describes the time evolution of volume under the flow /ðt; $x Þ: Before this we now give the following lemma. 1 Consider an autonomous vector ﬁeld $x_ ¼ f ðx$ Þ; $x 2 Rn and generates a flow /t ðx$ Þ: Let D0 be a domain in Rn and /t ðD0 Þ be its evolution under the flow. If VðtÞ is the volume of Dt , then the time rate of change of volume is given as dV ¼ R r Á f d x . 1 (Liouville’s Theorem) Suppose r Á f ¼ 0 for a vector ﬁeld f.
An Introduction to Dynamical Systems and Chaos by G.C. Layek