By G.C. Layek
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Extra info for An Introduction to Dynamical Systems and Chaos
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! a ekt þ ta ekt þ $ a 3 ekt ; $1 $2 2 .. tnÀ1 x ðtÞ ¼ ðnÀ1Þ! a ekt þ Á Á Á þ $1 $n t2 a nÀ2 ekt 2!
Non-wandring points give asymptotic behavior of the orbit. In the above deﬁnition, if /ðt; UÞ \ U ¼ u, then the point p is called a wandering point. The examples of non-wandering points are ﬁxed points and periodic orbits of a system. For the undamped oscillator ð€x þ x ¼ 0Þ; all points are non-wandering in x_x phase plane while for the damped oscillator ð€x þ a_x þ x ¼ 0Þ; origin is the only non-wandering point. Attracting set A closed invariant set D & Rn for a flow /t is said to be an attracting T set if there exists some neighborhood U in D such that 8t !
Write down the relation between trapping zones T and absorbing sets. Prove that for an absorbing set A; t ! 0 /ðt; AÞ forms an attracting set. 27. Give the deﬁnition of invariant set of a flow. Write its importance in dynamical evolution of a system. Prove that the x-limit set, Kðx$ Þ; is invariant and it is nonempty and compact if the positive orbit c þ ðx$ Þ of $x is bounded. 28. If two orbits cðxÞ and cðyÞ of autonomous systems satisfy cðxÞ \ cðyÞ 6¼ u, prove that both the orbits are coinciding.
An Introduction to Dynamical Systems and Chaos by G.C. Layek