nonlinear - Dierential Equations Massoud Malek Nonlinear...

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Differential Equations Massoud Malek Nonlinear Systems of Ordinary Differential Equations Dynamical System . A dynamical system has a state determined by a collection of real numbers, or more generally by a set of points in an appropriate state space. Small changes in the state of the system correspond to small changes in the numbers. The evolution rule of the dynamical system is a fixed rule that describes what future states follow from the current state. The rule is deterministic: for a given time interval only one future state follows from the current state. The mathematical models used to describe the swinging of a clock pendulum, the flow of water in a pipe, or the number of fish each spring in a lake are examples of dynamical systems. Autonomous System . An autonomous differential equation is a system of ordinary dif- ferential equations which does not depend on the independent variable. It is of the form d dt X ( t ) = F ( X ( t )) , where X takes values in n-dimensional Euclidean space and t is usually time. It is distinguished from systems of differential equations of the form d dt X ( t ) = G ( X ( t ) , t ) , in which the law governing the rate of motion of a particle depends not only on the particle’s location, but also on time; such systems are not autonomous. Autonomous systems are closely related to dynamical systems. Any autonomous sys- tem can be transformed into a dynamical system and, using very weak assumptions, a dynamical system can be transformed into an autonomous systems. Jacobian Matrix . Consider the function F : IR n -→ IR m , where F ( x 1 , x 2 , . . . , x n ) = f 1 ( x 1 , x 2 , . . . , x n ) f 2 ( x 1 , x 2 , . . . , x n ) . . . . . . . . . . . . . . . . . . f m ( x 1 , x 2 , . . . , x n ) . The partial derivatives of f 1 ( x 1 , ..., x n ) , . . . , f m ( x 1 , ..., x n ) (if they exist) can be organized in an m × n matrix. The Jacobian matrix of F ( x 1 , x 2 , . . . , x n ) denoted by J F is as follows: J F ( x 1 , . . . , x n ) = ∂f 1 ∂x 1 ··· ∂f 1 ∂x n . . . . . . . . . ∂f m ∂x 1 ··· ∂f m ∂x n . Its importance lies in the fact that it represents the best linear approximation to a differentiable function near a given point. California State University, East Bay
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Massoud Malek Nonlinear Systems of Ordinary Differential Equations Page 2 Qualitative Analysis . Very often it is almost impossible to find explicitly or implicitly the solutions of a system (specially nonlinear ones). The qualitative approach as well as numerical one are important since they allow us to make conclusions regardless whether we know or not the solutions.
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This note was uploaded on 02/28/2012 for the course ENGINNEEER EML taught by Professor Zarda during the Spring '09 term at University of Central Florida.

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nonlinear - Dierential Equations Massoud Malek Nonlinear...

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