By Ranjit Kumar Upadhyay
""The presentation is so transparent that any one with even a uncomplicated mathematical historical past can examine it and get a transparent photograph. ... in contrast to many different related textbooks, a wealthy reference part is given on the finish of every bankruptcy. The wary collection of labored out examples and workouts through the ebook is excellent. For an individual with earlier adventure of getting run into books in mathematical modeling and chaotic dynamics that quickly movement into complicated mathematical content material, the ebook deals a delightful recourse at an introductory point and accordingly will be very inspirational.""-MAA studies, Decembe. Read more...
summary: ""The presentation is so transparent that anybody with even a easy mathematical historical past can research it and get a transparent photograph. ... not like many different comparable textbooks, a wealthy reference part is given on the finish of every bankruptcy. The wary number of labored out examples and routines during the publication is great. For a person with prior event of getting run into books in mathematical modeling and chaotic dynamics that speedily flow into complicated mathematical content material, the ebook deals a delightful recourse at an introductory point and hence could be very inspirational.""-MAA stories, Decembe
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Additional resources for Introduction to Mathematical Modeling and Chaotic Dynamics
In every neighborhood of the origin, V(x1, x2) takes both positive and negative values. We obtain V *( x) = x12 + x22 − x1x2 sin(2x1 ). In any neighborhood of the origin, |x1|, |x2| are small and sin(2x1) ≈ 2x1. The third term on the right-hand side is of order of one higher than the first two terms. It is seen that V*(x) > 0 and therefore positive definite. 3, the zero solution is unstable. Construction of the Lyapunov Function and Testing of Stability We have noted above that there is no simple procedure for constructing the Lyapunov function V(x).
Assume that a drug is provided continuously for a period of 24 h and stopped. We are interested in finding the amount of medicine in the body at any given time. We consider this problem as a compartment model, where the rate of change is inflow minus outflow. If M is the amount of administered medicine in milligrams, D(t) is the rate at which the medicine is administered and P(M) is the processing rate, then the model equation can be written as dM = D(t) − P( M ). 1 M. Let M(0) = 0, that is, there is no medicine in the body at t = 0.
Iv. Negative semidefinite on Γ if and only if −V(x) is positive semidefinite on Γ. Let V(t, x) be a real-valued time-varying scalar continuous function in the variables t, x1, x2, . , xn. Then, V(t, x) is said to be i. Positive definite if there exists a positive-definite function V(x) such that V(t, 0) ≡ 0 and V(t, x) ≥ V(x) at all points x ≠ 0. ii. Positive semidefinite if V(t, 0) ≡ 0, V(t, x) = 0 at a few points x ≠ 0, and V(t, x) > 0 at all the remaining points x ≠ 0. iii. Negative definite if and only if −V(t, x) is positive definite.
Introduction to Mathematical Modeling and Chaotic Dynamics by Ranjit Kumar Upadhyay