By Ranjit Kumar Upadhyay
""The presentation is so transparent that anybody with even a uncomplicated mathematical history can examine it and get a transparent photograph. ... in contrast to many different comparable textbooks, a wealthy reference part is given on the finish of every bankruptcy. The wary choice of labored out examples and routines during the booklet is excellent. For somebody with prior event of getting run into books in mathematical modeling and chaotic dynamics that speedily stream into complex mathematical content material, the ebook deals a delightful recourse at an introductory point and as a result could be very inspirational.""-MAA studies, Decembe. Read more...
summary: ""The presentation is so transparent that any one with even a uncomplicated mathematical historical past can learn it and get a transparent photo. ... not like many different related textbooks, a wealthy reference part is given on the finish of every bankruptcy. The wary number of labored out examples and workouts through the e-book is great. For someone with prior event of getting run into books in mathematical modeling and chaotic dynamics that speedily movement into complicated mathematical content material, the ebook bargains a delightful recourse at an introductory point and accordingly should be very inspirational.""-MAA stories, Decembe
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Extra info for Introduction to Mathematical Modeling and Chaotic Dynamics
The characteristic equation of A is λ2 + 4 = 0, and the eigenvalues are λ = ±2i. This is a critical case where the real parts are zero. The test based on linearization cannot be applied. However, we can study the stability by constructing a Lyapunov function. Consider the positive-definite function V ( x1 , x2 ) = 4 x12 + x22 . We obtain V *( x1 , x2 ) = 8 x1 (− x2 − x15 ) + 2x2 ( 4 x1 − x25 ) = −2( 4 x16 + x26 ). V * (x1, x2) is negative definite. Therefore, the zero solution is asymptotically stable.
However, if no such 31 Introduction to Mathematical Modeling function can be determined, we cannot provide any conclusion about the stability. From the above discussion, λ = 0 is not an eigenvalue of A. , when the real parts of the eigenvalues of A are nonpositive and the real part of at least one eigenvalue is zero). 7) tend (converge) to zero as t → ∞ if and only if the real eigenvalues of A are negative and the complex eigenvalues have negative real parts. In such a case, the origin of the system is called an attractor.
13). 13) be a closed curve. We define the following: Stable limit cycle: If the periodic solution C is the positive limit set of the solutions contained in the interior and also of the solutions contained in the exterior of C, then the limit cycle is said to be stable. 5a). Unstable limit cycle: If the periodic solution C is the negative limit set of the solutions contained in the interior and also the solutions contained in the exterior of C, then the limit cycle is said to be unstable. 5b). 5 (a) Stable limit cycle.
Introduction to Mathematical Modeling and Chaotic Dynamics by Ranjit Kumar Upadhyay