Ordinary &amp; Partial Differential Equations - Reynolds (2000) - Chapter 3 - Nonlinear Systems; Loc

# Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 3 - Nonlinear Systems; Loc

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CHAPTER 3 Nonlinear Systems: Local Theory W e now turn our attention to nonlinear systems of ode s, which present a host of new challenges. Obtaining exact analytical solutions to such systems is usually impossible, so we must settle for qualitative descriptions of the dynamics. On the other hand, nonlinear systems can exhibit a wide variety of behaviors that linear systems cannot. Moreover, most dynamical phenomena in nature are inherently nonlinear. Consider a general system of the form x ± = f ( x , t ) , where x R n is a vector of unknowns and f ( x , t )= f 1 ( x 1 , x 2 , . . . x n , t ) f 2 ( x 1 , x 2 , . . . x n , t ) . . . f n ( x 1 , x 2 , . . . x n , t ) is a vector-valued function f : R n + 1 R n . In what follows, we shall work only with autonomous systems—those of the form x ± = f ( x ) where f : R n R n does not explicitly involve the independent variable t . This is actually not a severe restriction at all, because non-autonomous systems can be converted to autonomous ones by introducing an extra dependent variable. For example, the non-autonomous system ± x ± 1 x ± 2 ² = ± cos ( x 1 t )+ x 2 t 2 + x 2 1 ² 78

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nonlinear systems : local theory 79 becomes autonomous if we introduce a third dependent variable x 3 = t and corresponding ode x ± 3 = 1. The result is the autonomous system x ± 1 x ± 2 x ± 3 = cos ( x 1 x 3 )+ x 2 x 2 3 + x 2 1 1 , whose order is one larger than the original non-autonomous system. Hence, it will be sufFcient to consider autonomous systems only. ±or autonomous systems x ± = f ( x ) , it is straightforward to determine whether the system is linear or nonlinear. Defnition 3 . 0 . 5 . Consider the system x ± = f ( x ) , where f : R n R n . The system of ode s is linear if the function f satisFes f ( α x + y )= α f ( x f ( y ) for all vectors x , y R n and all scalars α R . Otherwise, the system of ode s is called nonlinear . Example 3 . 0 . 6 . The right-hand side of the ode d x d t = x 2 is f ( x x 2 , and f ( x + y ) = ( x + y ) 2 = x 2 + 2 xy + y 2 ² = x 2 + y 2 = f ( x f ( y ) . Therefore, the ode is nonlinear. Warning: Be careful when classifying ode s as autonomous/non-autonomous or linear/nonlinear. These concepts are independent of one another. Here are several one-dimensional examples to reinforce this point: ± d x d t = x is linear and autonomous. ± d x d t = x + cos t is linear and non-autonomous. ± d x d t = cos x is nonlinear and autonomous. ± d x d t = t + cos x is nonlinear and non-autonomous. There are very few analytical techniques for solving nonlinear ode s. Separation of variables is one such method, but it has very limited scope. Example 3 . 0 . 7 . Consider the nonlinear, autonomous ode d x d t = x - x 2 with initial condition x ( 0 1/2. The fact that the ode is autonomous makes separating the variables easy: 1 x - x 2 d x d t = 1.
80 Integrating both sides with respect to t yields # 1 x - x 2 d x = # d t , and a partial fractions decomposition simpliFes the left-hand side as # 1 x d x + # 1 1 - x d x = # d t .

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## This note was uploaded on 02/27/2012 for the course MATH 532 taught by Professor Reynolds during the Fall '11 term at VCU.

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Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 3 - Nonlinear Systems; Loc

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