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Approximate Analytical Methods
Published in Daniel Zwillinger, Vladimir Dobrushkin, Handbook of Differential Equations, 2021
Daniel Zwillinger, Vladimir Dobrushkin
A separatrix is a curve that separates the phase plane into regions of accessibility; there are many possibilities including solutions that are simple closed curves and those with a terminus at an unstable saddle point. Figure 139.4 shows a separatrix for the ODE y′=x2+y2 having the initial condition y(0)=0.7134, along with the associated tangent field.
Nonlinear Vibration
Published in Haym Benaroya, Mark Nagurka, Seon Han, Mechanical Vibration, 2017
Haym Benaroya, Mark Nagurka, Seon Han
The trajectories are shown in Figure 11.21 for three energy levels. For E=ωn2 $ E = \omega_{n}^{2} $ the trajectory is closed, indicating periodic motion. For small values of E where E<ωn2 $ E < \omega_{n}^{2} $ , the motion is approximately harmonic. As E increases such that ωn2<E<2ωn2 $ \omega_{n}^{2} < E < 2\omega_{n}^{2} $ , the motion ceases to be approximately harmonic but remains periodic. For E≥2ωn2 $ E \ge 2\omega_{n}^{2} $ , the trajectory is open and the motion is rotary, like a propeller, as the mass rotates about the contact point. E=2ωn2 $ E = 2\omega_{n}^{2} $ is a separatrix, signifying the boundary between two types of motion. The separatrix also describes rotary motion.
Adiabatic and nonadiabatic dynamics in classical mechanics for coupled fast and slow modes: sudden transition caused by the fast mode against the slaving principle
Published in Molecular Physics, 2018
As for the theories about sudden change of state, quite popular is the theory of catastrophe [18], which is about unfolding of the degeneracy of fixed points in the so-called bifurcation theory. In nonlinear mechanics, sudden transition of state is frequently observed when a system parameter varies in a manner that the system passes across a separatrix. The so-called crisis [19,20] gives another characteristic mechanism, in which, for instance, a stable manifold suddenly disappears by contacting an unstable manifold. These sudden dynamics are all interesting and important, yet do not necessarily have direct relationship to hierarchical dynamics. These theories are concerned with the geometry of the set of solutions of differential equations.