29-nonlinear

29-nonlinear - NONLINEAR DYNAMICAL SYSTEMS Math 21b, O....

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Unformatted text preview: NONLINEAR DYNAMICAL SYSTEMS Math 21b, O. Knill Homework: 1,2,3,4,5* in handout SUMMARY. For linear ordinary differential equations (ODE) ˙ x = Ax , the eigenvalues and eigenvectors of A determine the behavior completely. For nonlinear systems explicit formulas for solutions are no more available in general. It even also happen that orbits go go off to infinity in finite time like in the case of ˙ x = x 2 where the solution is x ( t ) =- 1 / ( t- x (0)). With x (0) = 1, the solution ”reaches infinity” at time t = 1. Linearity is often too crude. The exponential growth ˙ x = ax of a bacteria colony for example is slowed down due to lack of food and the logistic model ˙ x = ax (1- x/M ) would be more accurate, where M is the population size for which bacteria starve so much that the growth has stopped: x ( t ) = M , then ˙ x ( t ) = 0. Nonlinear systems can be investigated with qualitative methods . In 2 dimensions ˙ x = f ( x, y ) , ˙ y = g ( x, y ), where chaos does not happen, the analysis of equilibrium points and linear approximation in general allows to understand the system quite well. Also in higher dimensions, where ODE’s can have ”chaotic” solutions, the analysis of equilibrium points and linear approximation at those points is a place, where linear algebra becomes useful. EQUILIBRIUM POINTS. A vector ~x is called an equilibrium point of d dt ~x = f ( ~x ) if f ( ~x ) = 0. If x (0) = x then x ( t ) = x for all times. The system ˙ x = x (6- 2 x- y ) , ˙ y = y (4- x- y ) for example has the four equilibrium points (0 , 0) , (3 , 0) , (0 , 4) , (2 , 2)....
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This note was uploaded on 04/06/2008 for the course MATH 21B taught by Professor Judson during the Spring '03 term at Harvard.

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