scholarly journals Unstable periodic orbit detection for ODEs with periodic forcing

2006 ◽  
Vol 359 (6) ◽  
pp. 555-558 ◽  
Author(s):  
J.H.B. Deane ◽  
L. Marsh
Keyword(s):  
Periodic Orbit ◽  
Unstable Periodic Orbit ◽  
Periodic Forcing

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.

CHAOS CONTROL IMPLEMENTATION BY INTERRUPTED ELECTRIC CIRCUIT WITH SWITCHING DELAY

2009 ◽  
Vol 19 (07) ◽  
pp. 2359-2362
Author(s):  
TAKUJI KOUSAKA ◽  
TETSUSHI UETA ◽  
YUE MA
Keyword(s):  
Periodic Orbit ◽  
Chaos Control ◽  
Electric Circuit ◽  
Unstable Periodic Orbit ◽  
Chaotic Circuit ◽  
Return Map ◽  
Switching Delay ◽  
Suppression Of Chaos ◽  
Control Implementation

We have demonstrated that the chaotic circuit with a switching delay is modeled by a return map, and a controller for the suppression of chaos is proposed. A circuit representing a controller stabilizing a period-1 unstable periodic orbit in an interrupted electric circuit with a certain switching delay is also discussed.

Identification of Unstable Periodic Orbit in Inter–Edge-Localized-Mode Intervals in JT-60U

1999 ◽  
Vol 83 (7) ◽  
pp. 1339-1342 ◽  
Author(s):  
P. E. Bak ◽  
R. Yoshino ◽  
N. Asakura ◽  
T. Nakano
Keyword(s):  
Periodic Orbit ◽  
Unstable Periodic Orbit ◽  
Edge Localized Mode ◽  
Localized Mode

Improved Estimations of Unstable Periodic Orbits Extracted From Chaotic Sets

2001 ◽  
Author(s):  
Z. Al-Zamel ◽  
B. F. Feeny
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Duffing Oscillator ◽  
Unstable Periodic Orbits ◽  
Unstable Periodic Orbit ◽  
Affine Map ◽  
Saddle Type ◽  
Recurrence Method

Abstract Unstable periodic orbits of the saddle type are often extracted from chaotic sets. We use the recurrence method of extracting segments of the chaotic data to approximate the true unstable periodic orbit. Then nearby trajectories are then examined to obtain the dynamics local to the extracted orbit, in terms of an affine map. The affine map is then used to estimate the true orbit. Accuracy is evaluated in examples including well known maps and the Duffing oscillator.

Experimental Stabilization of Unstable Periodic Orbit in Magneto-Elastic Chaos by Delayed Feedback Control

1997 ◽  
Vol 07 (12) ◽  
pp. 2837-2846 ◽  
Author(s):  
Takashi Hikihara ◽  
Masato Touno ◽  
Toshiaki Kawagoshi
Keyword(s):  
Feedback Control ◽  
Periodic Orbit ◽  
Chaotic Attractor ◽  
Control Method ◽  
Elastic Beam ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
Unstable Periodic Orbit ◽  
Beam System ◽  
Onset Timing

In our previous paper, it was confirmed that the unstable periodic orbit embedded in the chaotic attractor in magneto-elastic beam system can be stabilized by delayed feedback control experimentally. It seems an advantage that the control method does not require any exact model of the system. However, the application of the control raises the problem that we cannot predict the stabilized unstable periodic orbit until it converges. In this paper, an "onset window" is introduced to determine the onset timing for targeting the desired orbit embedded in the chaotic attractor experimentally. Moreover, the dependence of the stabilization on the delay and the gain parameters is also discussed based on the experimental results.

Chaos in a model of forced quasi-geostrophic flow over topography: an application of Melnikov's method

1991 ◽  
Vol 226 ◽  
pp. 511-547 ◽  
Author(s):  
J. S. Allen ◽  
R. M. Samelson ◽  
P. A. Newberger
Keyword(s):  
Periodic Orbit ◽  
Saddle Point ◽  
Wind Stress ◽  
Homoclinic Orbits ◽  
Periodic Forcing ◽  
Melnikov's Method ◽  
Geostrophic Flow ◽  
Melnikov’S Method ◽  
Parameter Values ◽  
Time Periodic

We demonstrate the existence of a chaotic invariant set of solutions of an idealized model for wind-forced quasi-geostrophic flow over a continental margin with variable topography. The model (originally formulated to investigate mean flow generation by topographic wave drag) has bottom topography that slopes linearly offshore and varies sinusoidally alongshore. The alongshore topographic scales are taken to be short compared to the cross-shelf scale, allowing Hart's (1979) quasi-two-dimensional approximation, and the governing equations reduce to a non-autonomous system of three coupled nonlinear ordinary differential equations. For weak (constant plus time-periodic) forcing and weak friction, we apply a recent extension (Wiggins & Holmes 1987) of the method of Melnikov (1963) to test for the existence of transverse homoclinic orbits in the model. The inviscid unforced equations have two constants of motion, corresponding to energy E and enstrophy M, and reduce to a one-degree-of-freedom Hamiltonian system which, for a range of values of the constant G = E − M, has a pair of homoclinic orbits to a hyperbolic saddle point. Weak forcing and friction cause slow variations in G, but for a range of parameter values one saddle point is shown to persist as a hyperbolic periodic orbit and Melnikov's method may be applied to study the perturbations of the associated homoclinic orbits. In the absence of time-periodic forcing, the hyperbolic periodic orbit reduces to the unstable fixed point that occurs with steady forcing and friction. The method yields analytical expressions for the parameter values for which sets of chaotic solutions exist for sufficiently weak time-dependent forcing and friction. The predictions of the perturbation analysis are verified numerically with computations of Poincaré sections for solutions in the stable and unstable manifolds of the hyperbolic periodic orbit and with computations of solutions for general initial-value problems. In the presence of constant positive wind stress τ0 (equatorward on eastern ocean boundaries), chaotic solutions exist when the ratio of the oscillatory wind stress τ1 to the bottom friction parameter r is above a critical value that depends on τ0/r and the bottom topographic height. The analysis complements a previous study of this model (Samelson & Allen 1987), in which chaotic solutions were observed numerically for weak near-resonant forcing and weak friction.

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