For some values of " ? " and " ? ", the system is chaotic, but for others it tends to a stable periodic orbit.
32.
A steady state, or discretizing trajectories of a flow or iterating a map, periodic orbits and heteroclinic orbits can also be posed as a solution of F = 0.
33.
A periodic orbit of a flow is said to be hyperbolic if none of the eigenvalues of the Poincar?return map at a point on the orbit has absolute value one.
34.
Recently with Manfred Einsiedler, Philippe Michel and Akshay Venkatesh, he studied distributions of torus periodic orbits in some arithmetic spaces, generalizing theorems by Hermann Minkowski and Yuri Linnik.
35.
The periodic orbit ? of the continuous dynamical system is asymptotically stable if and only if the fixed point " p " of the discrete dynamical system is asymptotically stable.
36.
It is a theorem of John Milnor that every formal orbit portrait is realized by the actual orbit portrait of a periodic orbit of some quadratic one-complex-dimensional map.
37.
For Rydberg atoms and molecules, every orbit which is closed at the nucleus is also a periodic orbit whose period is equal to either the closure time or twice the closure time.
38.
In later years, Biham was involved in the development of methods for the calculation of unstable periodic orbits in chaotic systems, models for efficient simulations of traffic flow and quantum computation.
39.
Perhaps his best-known work is his introduction of cycle expansions & mdash; that is, expansions based on using periodic orbit theory to approximate chaotic dynamics in a controlled perturbative way.
40.
Employing automatic differentiation, Guckenheimer has constructed a new family of algorithms that compute bifurcations of periodic orbits as well as " generate rigorous computer proofs of the qualitative properties of numerically computed dynamical systems ".
