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Nonlinear Differential Equations - Old Dominion University

11 Nonlinear Differential EquationsNonlinear Differential Equationsand The Beauty of Chaos2 Examples of Nonlinear Equations )()(2tkxdttxdm =Simple harmonic oscillator ( linear ODE)More complicated motion ( Nonlinear ODE)))(1)(()(2txtkxdttxdm =Other examples: weather patters, the turbulent motion of fluidsMost natural phenomena are essentially is special about Nonlinear ODE? For solving Nonlinear ODE we can use the same methods we use for solving linear Differential Equations What is the difference? Solutions of Nonlinear ODE may be simple, complicated, or chaotic Nonlinear ODE is a tool to study Nonlinear dynamic: chaos, fractals, solitons, attractors4A simple pendulumModel: 3 forces gravitational force frictional force is proportional to velocity periodic external force)cos(,.),sin(22tFdtdmgLdtdIextfgext fg = = =++=5 Equations22202022,,)cos()sin(mLFfmLLgImg Ltfdtddtd====+ = Computer simulation:there are very many web sites there are very many web sites with Java animation for the with Java animation for the simple pendulumsimple pendulum6 Case 1: A very simple pendulum)sin(2022 =dtdcode270 (0)= (t) )sin(2022 =dtd 2022 =dtdIs there any difference between the Nonlinear pendulumand the linear pendulum?

Nonlinear Differential Equations and The Beauty of Chaos 2 Examples of nonlinear equations 2 ( ) kx t dt d x t m =− Simple harmonic oscillator (linear ODE) More complicated motion (nonlinear ODE) ( )(1 ()) 2 ( ) kx t x t dt d x t m =− −α Other examples: weather patters, the turbulent motion of fluids Most natural phenomena are ...

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