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 ...
There are over 300 miles of scaffolding in this city. Those damp, dimly-lit plywood tunnels installed alongside construction sites are meant to keep construction materials from falling on pedestrians below. But sidewalk sheds can also be dangerous. Earlier this month, construction workers in Williamsburg were injured when a sidewalk shed gave way beneath them. Last summer, scaffolding in Brooklyn Heights fell outside a Starbucks, injuring three people below. And in 2017, a young woman suffered major spinal damage when scaffolding collapsed on her in SoHo. [ more › ] Gothamist http://bit.ly/2WtIwyy January 30, 2019 at 10:23PM
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