Three-Dimensional Manifolds of Constant Energy and Invariants of Integrable Hamiltonian Systems

2018 ◽  
pp. 13-30
Author(s):  
Anatoly T. Fomenko ◽  
Kirill I. Solodskih
Keyword(s):  
Hamiltonian Systems ◽  
Three Dimensional ◽  
Integrable Hamiltonian Systems ◽  
Constant Energy

STOCHASTIC VERSIONS OF ANOSOV'S AND NEISTADT'S THEOREMS ON AVERAGING

2001 ◽  
Vol 01 (01) ◽  
pp. 1-21 ◽  
Author(s):  
YURI KIFER
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Hamiltonian Systems ◽  
Initial Conditions ◽  
Slow Motion ◽  
Averaging Principle ◽  
Integrable Hamiltonian Systems ◽  
Constant Energy ◽  
Slow Motions ◽  
Fast And Slow Motions

In systems which combine slow and fast motions the averaging principle says that a good approximation of the slow motion can be obtained by averaging its parameters in fast variables. This setup arises, for instance, in perturbations of Hamiltonian systems where motions on constant energy manifolds are fast and across them are slow. When these perturbations are deterministic Anosov's theorem says that the averaging principle works except for a small in measure set of initial conditions while Neistadt's theorem gives error estimates in the case of perturbations of integrable Hamiltonian systems. These results are extended here to the case of fast and slow motions given by stochastic differential equations.

Billiard books model all three-dimensional bifurcations of integrable Hamiltonian systems

2018 ◽  
Vol 209 (12) ◽  
pp. 1690-1727 ◽  
Author(s):  
V. V. Vedyushkina ◽  
I. S. Kharcheva
Keyword(s):  
Hamiltonian Systems ◽  
Three Dimensional ◽  
Integrable Hamiltonian Systems

Development of mixing and isotropy in inviscid homogeneous turbulence

1982 ◽  
Vol 120 ◽  
pp. 155-183 ◽  
Author(s):  
Jon Lee
Keyword(s):  
Dynamical Systems ◽  
Lower Order ◽  
Stokes System ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Fourier Space ◽  
Kolmogorov Entropy ◽  
Navier Stokes Equations ◽  
Constant Energy

We have investigated a sequence of dynamical systems corresponding to spherical truncations of the incompressible three-dimensional Navier-Stokes equations in Fourier space. For lower-order truncated systems up to the spherical truncation of wavenumber radius 4, it is concluded that the inviscid Navier-Stokes system will develop mixing (and a fortiori ergodicity) on the constant energy-helicity surface, and also isotropy of the covariance spectral tensor. This conclusion is, however, drawn not directly from the mixing definition but from the observation that one cannot evolve the trajectory numerically much beyond several characteristic corre- lation times of the smallest eddy owing to the accumulation of round-off errors. The limited evolution time is a manifestation of trajectory instability (exponential orbit separation) which underlies not only mixing, but also the stronger dynamical charac- terization of positive Kolmogorov entropy (K-system).

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