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 ...
Ever since Governor Andrew Cuomo descended into the depths of the L train tunnel and returned with an entirely new shutdown proposal, New Yorkers have been left to guess which, if any, of the sweeping mitigation measures would survive the last-minute intervention. Already, the MTA has discarded the 14th Street busway, the HOV lanes on the Williamsburg Bridge, and much of the plan for increased subway service — against the advice of transit advocates and their own internal recommendations. Plans for increased ferry service have also been scrapped, while the fate of the protected bike lane and other traffic-calming measures on Grand Street remains unclear. [ more › ] Gothamist https://ift.tt/2V3ePmH March 29, 2019 at 06:38PM
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