Van der Pol’s differential equation is The equation describes a system with nonlinear damping, the degree of damping given by μ. If μ = 0 the system is linear and undamped, but for positive μ the system is nonlinear and damped. We will plot the phase portrait for the solution to Van der Pol’s equation in Python using SciPy’s new ODE solver ivp_solve . The function ivp_solve does not solve second-order systems of equations directly. It solves systems of first-order equations, but a second-order differential equation can be recast as a pair of first-order equations by introducing the first derivative as a new variable. Since y is the derivative of x , the phase portrait is just the plot of ( x , y ). If μ = 0, we have a simple harmonic oscillator and the phase portrait is simply a circle. For larger values of μ the solutions enter limiting cycles, but the cycles are more complicated than just circles. Here’s the Python code that made the plot. from scipy import linspace from ...
Very few of the 30,000 New Yorkers who are currently eligible for the city’s half-priced MetroCard program have signed up. Only 107 Manhattan residents have registered since the first phase of the initiative launched earlier this month, according to the Community Service Society, which only saw the Manhattan numbers. [ more › ] Gothamist http://bit.ly/2SeLebJ January 30, 2019 at 09:16PM
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