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Sim_NonlinearOscillation

 Nonlinear Oscillations & Waves (Numerical Simulation)

m\frac{d^2}{dt^2}x = -2\zeta\frac{d}{dt}x - k x + f \sin \omega t \;\;\; (m=1)

\frac{d^2 x}{dt^2} -\mu (1-x^2)\frac{dx}{dt} +x = 0

\frac{d^2 x}{dt^2} +\mu \frac{dx}{dt} +(x^3 -x) = \gamma \cos(\omega t)

\frac{d^2 x}{dt^2} +\mu \frac{dx}{dt} +x^3 = \gamma \cos(\omega t)

\tau \frac{d}{dt}u = u - uv

\frac{d}{dt}v = uv -v

\tau \frac{\partial}{\partial t}u = -u(u+1)(u-1) -v

\frac{\partial}{\partial t}v = u - \gamma v -I

  • Oregonator

\tau \frac{\partial}{\partial t}u = u(1-u) - \frac{av(u-b)}{u+b} - v

\frac{\partial}{\partial t}v = u - v

  • Brusselator

\frac{\partial}{\partial t}u = a - (b+1)u + u^{2}v

\frac{\partial}{\partial t}v = bu - u^{2}v

m_{1}\frac{d^2}{dt^2}x_{1} = -k_{1}x_{1} + k^{\prime} \left( x_{2} - x_{1} \right)

m_{2}\frac{d^2}{dt^2}x_{2} = -k_{2}x_{2} - k^{\prime} \left( x_{2} - x_{1} \right)

\frac{d}{dt}\theta_{j} = \omega_{j} + \frac{K}{N} \sum_{k=1}^{N} \sin(\theta_{k}-\theta_{j})


  • Coupled Oscillator Systems (Case of the Millennium Bridge)
    • ref : S. H. Strogatz, "Crowd synchrony on the Millennium Bridge", Nature vol.438 p. 43

  • KdV equation

\frac{\partial}{\partial t}u + 6u\frac{\partial}{\partial x}u +\frac{\partial^3}{\partial x^3}u = 0

  • Kuramoto-Shivashinsky equation

\frac{\partial}{\partial t}u + u\frac{\partial}{\partial x}u +\frac{\partial^2}{\partial x^2}u +\frac{\partial^4}{\partial x^4}u = 0



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