Since the time spent near the periodic orbit depends sensitively on how closely the system entered its vicinity ( in turn determined by what happened during the chaotic period ) the length of each phase is unpredictable.
42.
The "'CAPD library "'( "'Computer Assisted Proofs in Dynamics "') is a software chaotic dynamics, bifurcations, heteroclinic / homoclinic solutions and periodic orbits.
43.
Bifurcation theory considers a structure in phase space ( typically a fixed point, a periodic orbit, or an invariant torus ) and studies its behavior as a function of the parameter " ? ".
44.
For instance, Rabinowitz showed that on star-shaped level sets of a Hamiltonian function on a symplectic manifold, there were always periodic orbits ( Weinstein independently proved the special case of convex level sets ).
45.
It would take a hell of a lot of thrust to get from a near-escape trajectory to a periodic orbit around a galaxy .-- talk ) 18 : 45, 13 August 2011 ( UTC)
46.
Eventually the system gets far enough away from the periodic orbit to be affected by chaotic dynamics in the rest of the state space, until it gets close to the orbit again and returns to the nearly periodic behaviour.
47.
For example, by keeping " b " fixed at 0.3 the bifurcation diagram shows that for " a " = 1.25 the H�non map has a stable periodic orbit as an attractor.
48.
In mathematics, a dynamical system is chaotic if it is ( highly ) sensitive to initial conditions ( the so-called " butterfly effect " ), which requires the mathematical properties of topological mixing and dense periodic orbits.
49.
If this path arises from trivializing the symplectic vector bundle over a periodic orbit of a Hamiltonian vector field on a symplectic manifold or the Reeb vector field on a contact manifold, it is known as the Conley-Zehnder index.
50.
In 1935 at the Moscow University, Demidovich discussed his PhD thesis, " On the existence of the integral invariant on a system of periodic orbits " and the following year, he was awarded the degree of Ph . D.
