scholarly journals Livschitz Theorem in Suspension Flows and Markov Systems: Approach in Cohomology of Systems

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 338
Author(s):  
Rosário D. Laureano
Keyword(s):  
Periodic Orbits ◽  
Systems Approach ◽  
Existence Of Solutions ◽  
Suspension Flows ◽  
Markov Systems ◽  
Fundamental Relationship ◽  
Natural Notion ◽  
Hyperbolic Flows ◽  
Cohomological Equations

It is presented and proved a version of Livschitz Theorem for hyperbolic flows pragmatically oriented to the cohomological context. Previously, it is introduced the concept of cocycle and a natural notion of symmetry for cocycles. It is discussed the fundamental relationship between the existence of solutions of cohomological equations and the behavior of the cocycles along periodic orbits. The generalization of this theorem to a class of suspension flows is also discussed and proved. This generalization allows giving a different proof of the Livschitz Theorem for flows based on the construction of Markov systems for hyperbolic flows.

Prevalence of rapid mixing in hyperbolic flows

1998 ◽  
Vol 18 (5) ◽  
pp. 1097-1114 ◽  
Author(s):  
DMITRY DOLGOPYAT
Keyword(s):  
Finite Type ◽  
Periodic Orbits ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Suspension Flow ◽  
Subshift Of Finite Type ◽  
Rapid Mixing ◽  
Hyperbolic Flows ◽  
Necessary And Sufficient

We provide necessary and sufficient conditions for a suspension flow, over a subshift of finite type, to mix faster than any power of time. Then we show that these conditions are satisfied if the flow has two periodic orbits such that the ratio of the periods cannot be well approximated by rationals.

scholarly journals Intermittent chaos driven by nonlinear Alfvén waves

2004 ◽  
Vol 11 (5/6) ◽  
pp. 691-700 ◽  
Author(s):  
E. L. Rempel ◽  
A. C.-L. Chian ◽  
A. J. Preto ◽  
S. Stephany
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits ◽  
Systems Approach ◽  
Alfven Waves ◽  
Space Plasmas ◽  
Alfvén Waves ◽  
Unstable Periodic Orbits ◽  
Phase Dynamics ◽  
Intermittent Chaos

Abstract. We investigate the relevance of chaotic saddles and unstable periodic orbits at the onset of intermittent chaos in the phase dynamics of nonlinear Alfvén waves by using the Kuramoto-Sivashinsky (KS) equation as a model for phase dynamics. We focus on the role of nonattracting chaotic solutions of the KS equation, known as chaotic saddles, in the transition from weak chaos to strong chaos via an interior crisis and show how two of these unstable chaotic saddles can interact to produce the plasma intermittency observed in the strongly chaotic regimes. The dynamical systems approach discussed in this work can lead to a better understanding of the mechanisms responsible for the phenomena of intermittency in space plasmas.

A Representation Theory-Based Model for the Rapid Pressure Strain Correlation of Turbulence

2018 ◽  
Vol 140 (8) ◽  
Author(s):  
J. P. Panda ◽  
H. V. Warrior
Keyword(s):  
Numerical Simulation ◽  
Dynamical Systems ◽  
Direct Numerical Simulation ◽  
Plane Strain ◽  
Present Model ◽  
Systems Approach ◽  
Satisfactory Performance ◽  
Hyperbolic Flows ◽  
Mean Strain ◽  
Rapid Pressure

In the presence of mean strain or rotation, the anisotropy of turbulence increases due to the rapid pressure strain term. In this paper, we consider the modeling of the rapid pressure strain correlation of turbulence. The anisotropy of turbulence in the presence of mean strain is studied and a new model is formulated by calibrating the model constants at the rapid distortion limit. This model is tested for a range of plane strain and elliptic flows and compared to direct numerical simulation (DNS) results. The present model shows agreement with DNS and improvements over the earlier models like those by Launder et al. (1975, “Progress in the Development of a Reynolds-Stress Turbulence Closure,” J. Fluid Mech., 68(3), pp. 537–566.) and Speziale et al. (1991, “Modelling the Pressure–Strain Correlation of Turbulence: An Invariant Dynamical Systems Approach,” J. Fluid Mech., 227(1), pp. 245–272.) that have been reported to give satisfactory performance for hyperbolic flows but not satisfactory for elliptic flows.

Lalley's theorem on periodic orbits of hyperbolic flows

1998 ◽  
Vol 18 (1) ◽  
pp. 17-39 ◽  
Author(s):  
MARTINE BABILLOT ◽  
FRANÇOIS LEDRAPPIER
Keyword(s):  
Periodic Orbits ◽  
Hyperbolic Flows

Heteroclinics for a reversible Hamiltonian system

1994 ◽  
Vol 14 (4) ◽  
pp. 817-829 ◽  
Author(s):  
Paul H. Rabinowitz
Keyword(s):  
Hamiltonian System ◽  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Existence Of Solutions

AbstractThis paper uses an elementary variational argument to establish the existence of solutions heteroclinic to a pair of periodic orbits for a class of Hamiltonian systems including Hamiltonians of multiple pendulum type.

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