ALFVÉN INTERIOR CRISIS

2004 ◽  
Vol 14 (07) ◽  
pp. 2375-2380 ◽  
Author(s):  
F. A. BOROTTO ◽  
A. C.-L. CHIAN ◽  
E. L. REMPEL
Keyword(s):  
Periodic Orbits ◽  
Large Amplitude ◽  
Numerical Study ◽  
Alfven Wave ◽  
Unstable Periodic Orbits ◽  
Stable And Unstable Manifolds ◽  
Laboratory Plasmas ◽  
Derivative Nonlinear Schrödinger Equation ◽  
Unstable Manifolds ◽  
Low Dimensional

A numerical study of an interior crisis of a large-amplitude Alfvén wave described by the driven-dissipative derivative nonlinear Schrödinger equation, in the low-dimensional limit, is reported. An example of Alfvén interior crisis is characterized using the unstable periodic orbits and their associated invariant stable and unstable manifolds in the Poincaré plane. We suggest that this type of chaotic transition can be observed in space and laboratory plasmas.

scholarly journals Chaos in driven Alfvén systems: unstable periodic orbits and chaotic saddles

2007 ◽  
Vol 14 (1) ◽  
pp. 17-29 ◽  
Author(s):  
A. C.-L. Chian ◽  
W. M. Santana ◽  
E. L. Rempel ◽  
F. A. Borotto ◽  
T. Hada ◽  
...  
Keyword(s):  
Periodic Orbits ◽  
Chaotic Attractor ◽  
Unstable Periodic Orbits ◽  
Gap Filling ◽  
Intermittent Turbulence ◽  
Dynamical Processes ◽  
Nonlinear Dynamical ◽  
Derivative Nonlinear Schrödinger Equation ◽  
Low Dimensional ◽  
Chaotic Saddle

Abstract. The chaotic dynamics of Alfvén waves in space plasmas governed by the derivative nonlinear Schrödinger equation, in the low-dimensional limit described by stationary spatial solutions, is studied. A bifurcation diagram is constructed, by varying the driver amplitude, to identify a number of nonlinear dynamical processes including saddle-node bifurcation, boundary crisis, and interior crisis. The roles played by unstable periodic orbits and chaotic saddles in these transitions are analyzed, and the conversion from a chaotic saddle to a chaotic attractor in these dynamical processes is demonstrated. In particular, the phenomenon of gap-filling in the chaotic transition from weak chaos to strong chaos via an interior crisis is investigated. A coupling unstable periodic orbit created by an explosion, within the gaps of the chaotic saddles embedded in a chaotic attractor following an interior crisis, is found numerically. The gap-filling unstable periodic orbits are responsible for coupling the banded chaotic saddle (BCS) to the surrounding chaotic saddle (SCS), leading to crisis-induced intermittency. The physical relevance of chaos for Alfvén intermittent turbulence observed in the solar wind is discussed.

ALFVÉN CHAOTIC SADDLES

2004 ◽  
Vol 14 (11) ◽  
pp. 4009-4017 ◽  
Author(s):  
E. L. REMPEL ◽  
A. C.-L. CHIAN
Keyword(s):  
Periodic Orbits ◽  
Chaotic Attractor ◽  
Schrodinger Equation ◽  
Alfven Wave ◽  
Wave System ◽  
Unstable Periodic Orbits ◽  
Saddle Node Bifurcation ◽  
Onset Of Chaos ◽  
Saddle Node ◽  
Derivative Nonlinear Schrödinger Equation

We examine the dynamical roles of nonattracting chaotic sets known as chaotic saddles in an Alfvén wave system described by the driven-dissipative derivative nonlinear Schrödinger equation. These Alfvén chaotic saddles have gaps which are filled at the onset of chaos via a saddle-node bifurcation and at a chaotic transition via an interior crisis. It is shown that after an interior crisis an Alfvén chaotic attractor consists of two chaotic saddles connected by a set of coupling unstable periodic orbits.

scholarly journals On the escape from potentials with two exit channels

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Juan F. Navarro
Keyword(s):  
Hamiltonian System ◽  
Periodic Orbits ◽  
Unstable Periodic Orbits ◽  
Stable And Unstable Manifolds ◽  
Surface Of Section ◽  
Escape Dynamics ◽  
Unstable Manifolds

Abstract The aim of this paper is to investigate the escape dynamics in a Hamiltonian system describing the motion of stars in a galaxy with two exit channels through the analysis of the successive intersections of the stable and unstable manifolds to the main unstable periodic orbits with an adequate surface of section. We describe in detail the origin of the spirals shapes of the windows through which stars escape.

scholarly journals Time averaged properties along unstable periodic orbits and chaotic orbits in two map systems

2008 ◽  
Vol 15 (4) ◽  
pp. 675-680 ◽  
Author(s):  
Y. Saiki ◽  
M. Yamada
Keyword(s):  
Fluid Dynamics ◽  
Periodic Orbit ◽  
Periodic Orbits ◽  
Space Physics ◽  
Unstable Periodic Orbits ◽  
Unstable Periodic Orbit ◽  
Geophysical Fluid Dynamics ◽  
Chaotic Orbits ◽  
Low Dimensional ◽  
Complex Phenomena

Abstract. Unstable periodic orbit (UPO) recently has become a keyword in analyzing complex phenomena in geophysical fluid dynamics and space physics. In this paper, sets of UPOs in low dimensional maps are theoretically or systematically found, and time averaged properties along UPOs are studied, in relation to those of chaotic orbits.

scholarly journals Locating oscillatory orbits of the parametrically-excited pendulum

1996 ◽  
Vol 37 (3) ◽  
pp. 309-319 ◽  
Author(s):  
M. J. Clifford ◽  
S. R. Bishop
Keyword(s):  
Periodic Solutions ◽  
Periodic Orbits ◽  
Numerical Techniques ◽  
Stable And Unstable Manifolds ◽  
Parametrically Excited ◽  
Countable Infinity ◽  
Unstable Manifolds

AbstractA method is considered for locating oscillating, nonrotating solutions for the parametrically-excited pendulum by inferring that a particular horseshoe exists in the stable and unstable manifolds of the local saddles. In particular, odd-periodic solutions are determined which are difficult to locate by alternative numerical techniques. A pseudo-Anosov braid is also located which implies the existence of a countable infinity of periodic orbits without the horseshoe assumption being necessary.

PSEUDO-DETERMINISTIC CHAOTIC SYSTEMS

2003 ◽  
Vol 13 (11) ◽  
pp. 3235-3253 ◽  
Author(s):  
R. L. VIANA ◽  
S. E. DE S. PINTO ◽  
J. R. R. BARBOSA ◽  
C. GREBOGI
Keyword(s):  
Chaotic Systems ◽  
Critical Value ◽  
Time Model ◽  
Stable And Unstable Manifolds ◽  
Chaotic Dynamical System ◽  
Discrete Time Model ◽  
Finite Time Lyapunov Exponents ◽  
Unstable Manifolds ◽  
Model Equations ◽  
Low Dimensional

We call a chaotic dynamical system pseudo-deterministic when it does not produce numerical, or pseudo-trajectories that stay close, or shadow chaotic true trajectories, even though the model equations are strictly deterministic. In this case, single chaotic trajectories may not be meaningful, and only statistical predictions, at best, could be drawn on the model, like in a stochastic system. The dynamical reason for this behavior is nonhyperbolicity characterized either by tangencies of stable and unstable manifolds or by the presence of periodic orbits embedded in a chaotic invariant set with a different number of unstable directions. We emphasize herewith the latter by studying a low-dimensional discrete-time model in which the phenomenon appears due to a saddle-repeller bifurcation. We also investigate the behavior of the finite-time Lyapunov exponents for the system, which quantifies this type of nonhyperbolicity as a system parameter evolves past a critical value. We argue that the effect of unstable dimension variability is more intense when the invariant chaotic set of the system loses transversal stability through a blowout bifurcation.

scholarly journals Relative periodic orbits form the backbone of turbulent pipe flow

2017 ◽  
Vol 833 ◽  
pp. 274-301 ◽  
Author(s):  
N. B. Budanur ◽  
K. Y. Short ◽  
M. Farazmand ◽  
A. P. Willis ◽  
P. Cvitanović
Keyword(s):  
Periodic Orbits ◽  
Pipe Flow ◽  
Chaotic Dynamics ◽  
Stokes Equations ◽  
Turbulent Pipe Flow ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Infinite Dimensional ◽  
Unstable Manifolds ◽  
Low Dimensional

The chaotic dynamics of low-dimensional systems, such as Lorenz or Rössler flows, is guided by the infinity of periodic orbits embedded in their strange attractors. Whether this is also the case for the infinite-dimensional dynamics of Navier–Stokes equations has long been speculated, and is a topic of ongoing study. Periodic and relative periodic solutions have been shown to be involved in transitions to turbulence. Their relevance to turbulent dynamics – specifically, whether periodic orbits play the same role in high-dimensional nonlinear systems like the Navier–Stokes equations as they do in lower-dimensional systems – is the focus of the present investigation. We perform here a detailed study of pipe flow relative periodic orbits with energies and mean dissipations close to turbulent values. We outline several approaches to reduction of the translational symmetry of the system. We study pipe flow in a minimal computational cell at $Re=2500$, and report a library of invariant solutions found with the aid of the method of slices. Detailed study of the unstable manifolds of a sample of these solutions is consistent with the picture that relative periodic orbits are embedded in the chaotic saddle and that they guide the turbulent dynamics.

Nonlinear excitation of slow shear Alfvén mode by electromagnetic waves

1987 ◽  
Vol 38 (3) ◽  
pp. 453-459 ◽  
Author(s):  
H. Saleem
Keyword(s):  
Growth Rates ◽  
Large Amplitude ◽  
Electromagnetic Waves ◽  
Alfven Wave ◽  
Hybrid Wave ◽  
Nonlinear Excitation ◽  
Laboratory Plasmas

Nonlinear decay of large-amplitude ordinary and extraordinary electromagnetic waves into an electrostatic upper-hybrid wave and a slow shear Alfvén wave in a low-β plasma is considered. Expressions for the growth rates and thresholds are obtained. Applications of this investigation to both space and laboratory plasmas are pointed out.

scholarly journals Locating and Stabilizing Unstable Periodic Orbits Embedded in the Horseshoe Map

2021 ◽  
Vol 31 (04) ◽  
pp. 2150110
Author(s):  
Yuu Miino ◽  
Daisuke Ito ◽  
Tetsushi Ueta ◽  
Hiroshi Kawakami
Keyword(s):  
Dynamical System ◽  
Periodic Orbits ◽  
Ad Hoc ◽  
Computation Method ◽  
Control Input ◽  
Unstable Periodic Orbits ◽  
Stable And Unstable Manifolds ◽  
Controlling Chaos ◽  
Dimensional Dynamical System ◽  
The Given

Based on the theory of symbolic dynamical systems, we propose a novel computation method to locate and stabilize the unstable periodic points (UPPs) in a two-dimensional dynamical system with a Smale horseshoe. This method directly implies a new framework for controlling chaos. By introducing the subset based correspondence between a planar dynamical system and a symbolic dynamical system, we locate regions sectioned by stable and unstable manifolds comprehensively and identify the specified region containing a UPP with the particular period. Then Newton’s method compensates the accurate location of the UPP with the regional information as an initial estimation. On the other hand, the external force control (EFC) is known as an effective method to stabilize the UPPs. By applying the EFC to the located UPPs, robust controlling chaos is realized. In this framework, we never use ad hoc approaches to find target UPPs in the given chaotic set. Moreover, the method can stabilize UPPs with the specified period regardless of the situation where the targeted chaotic set is attractive. As illustrative numerical experiments, we locate and stabilize UPPs and the corresponding unstable periodic orbits in a horseshoe structure of the Duffing equation. In spite of the strong instability of UPPs, the controlled orbit is robust and the control input retains being tiny in magnitude.

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