Transcription of Nonlinear Differential Equations
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Nonlinear Differential Equations
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
Fourier series For a periodicfunction one may write The Fourier series is a “best fit” in the least square sense of data fitting y(t +T) =y(t) ()cos( ) sin( ), 2 ( ) 1 0 ∑ ∞ = = + + n a n t bn n t a y t ω ω A general function may contain infinite number of components. In practice a good approximation is possible with about 10 ...
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