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A history of the mathematical theory of probability : from by Todhunter, I. (Isaac)

By Todhunter, I. (Isaac)

The beneficial reception which has been granted to my historical past of the Calculus of diversifications in the course of the 19th Century has inspired me to adopt one other paintings of a similar style. the topic to which I now invite cognizance has excessive claims to attention as a result of the sophisticated difficulties which it consists of, the dear contributions to research which it has produced, its vital useful purposes, and the eminence of these who've cultivated it.

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Often, linearized systems may not provide any indication of the existence of periodic solutions or limit cycles. An important result about periodic orbits and limit cycles is given in the following famous theorem called the Poincaré–Bendixson theorem [9]. 20), then L(C+) is a periodic orbit. Moreover, if C+ and L(C+) have no common regular point, then L(C+) is a limit cycle. 21 Discuss the stability of the limit cycles in the following problems: x1′ = − x2 + 2x1 ( x12 + x22 − 4), x2′ = x1 + 2x2 ( x12 + x22 − 4).

The method assumes that the human body is processing a particular medicine with a rate of change which is linear. Consider the following example. 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 ).

6  The characteristic equation of A is λ3 + 6λ2 + 11λ + 6 = 0. The eigenvalues are λ = −1, −2, −3. The matrix of eigenvectors (modal matrix) is obtained as  1  P =  −1  1 1 −2 4 1  −1   −1 −3  and P AP = D =  0  0 9  The transformation x = Py gives y′ = Dy. 0 −2 0 0  0 . −3  32 Introduction to Mathematical Modeling and Chaotic Dynamics To construct a Lyapunov function, we find the matrix B such that DB + BD = −I. We obtain B = diag(1/2, 1/4, 1/6). The Lyapunov function is given by V ( y ) = ( y , By ) = 1 2 1 2 1 2 y1 + y 2 + y 3 .

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