Photos From Dynamics Days 2012
Ernie Barreto
This is a contribution to the Dynamical Systems Magazine issue of January, 2012.
Towards a Dynamics of Seizure Mechanics
Steven J. Schiff, John. R. Cressman, Ernest Barreto, and J.
Ziburkus
This chapter appears in Computational Neuroscience in Epilepsy,
I. Soltesz and K. Stanley, eds., Academic Press, January 14, 2008.
Shadowing
Ernest Barreto (2008) Shadowing.
Scholarpedia,
3(1):2243.The Onset of Synchronism in Globally Coupled
Ensembles of Chaotic and Periodic Dynamical Units
Edward Ott, Paul So, Ernest Barreto, and Thomas Antonsen
This appears in Chaotic Dynamics and Transport in
Classical and Quantum Systems,
Proceedings of the NATO Advanced Study Institute on Chaotic Dynamics
and Transport
in Classical and Quantum Systems, Cargese, Corsica, 18-30 August 2003
Series: NATO Science Series II: Mathematics, Physics and Chemistry,
Vol. 182,
P. Collet, M. Courbage, S. Mtens, A. Neishtadt, and G. Zalsavsky, eds.
Springer, 2005.
For further information, click
here to send me an email
at ebarreto_AT_gmu_DOT_edu.
Synchrony in Globally Coupled Chaotic,
Periodic, and Mixed Ensembles of
Dynamical Units
Edward Ott, Paul So, Ernest Barreto, and Thomas Antonsen
The onset of collective synchronous behavior in globally coupled ensembles of oscillators is discussed. We present a formalism that is applicable to general ensembles of heterogeneous, continuous time dynamical units that, when uncoupled, are chaotic, periodic, or a mixture of both. A discussion of convergence issues, important for the proper implementation of our method, is included. Our work leads to a quantitative prediction for the critical coupling value at the onset of collective synchrony and for the growth rate of the resulting coherent state.
This appears in Synchronization: Theory and
Application,
Proceedings of the NATO Advanced Study Institute, Yalta Region, Crimea,
20-21 May 2002
Series: NATO Science Series II: Mathematics, Physics and Chemistry,
Vol. 109,
A. Pikovsky, Y.L. Maistrenko, eds. Springer, 2003.
For further information, click
here to send me an email
at ebarreto_AT_gmu_DOT_edu.
Cajal and Today's Consciousness Research
Ernest Barreto
This is a review of Scientific Approaches to Consciousness on the Centennial of Ramón y Cajal’s Textura, Annals of the New York Academy of Sciences, Vol. 929, 2001.
This appears in Complexity, Vol. 7 #3,
pp. 14-16 (2002).
For further information, click
here to send me an email
at ebarreto_AT_gmu_DOT_edu.
The Breakdown on Synchronization and
Shadowing in Coupled Chaotic
Systems:
Analysis via Subsystem Decomposition
Ernest Barreto and Paul So
Abstract
We present a powerful method of analyzing coupled chaotic systems in terms of a "subsystem" decomposition based on unstable periodic orbits. We focus on two important phenomena that have received a great deal of attention: synchronization and shadowability. This formalism forms the foundation for discussing these subjects in systems of non-identical coupled elements without special symmetries or invariant manifolds.
This appears in Space Time Chaos:
Characterization, Control and Synchronization (S. Boccaletti
et
al., eds.),
World Scientific, 2001.
For further information, click
here to send me an email
at ebarreto_AT_gmu_DOT_edu.
Control Of Chaos: Impact Oscillators and
Targeting
Ernest Barreto, Fernando Casas, Celso Grebogi, and Eric J.
Kostelich
Abstract
We present two applications of chaos control techniques that can be of importance in mechanical systems. First, we apply chaos control to select a desired sequence of impacts in a map that captures the universal properties of impact oscillators near grazing. Next we describe a targeting method that can significantly reduce the chaotic transients that precede stabilization when these control methods are used.
This appears in IUTAM Symposium on Interaction
between
Dynamics and
Control in Advanced Mechanical Systems, ed. by D. H. van
Campen.
Dordrecht, The Netherlands: Kluwer Academic Publishers, 1997,
pp. 17-26.
For further information, click
here to send me an email
at ebarreto_AT_gmu_DOT_edu.
Controlling Chaos in Mechanical Systems
Ernest Barreto, Ying-Cheng Lai, and Celso Grebogi
Abstract
This chapter addresses the issue of controlling chaos by applying small perturbations to accessible parameters of a system. The key observation is that a chaotic attractor typically has embedded within it a dense and infinite number of unstable periodic orbits. Thus, rather than eliminating chaos by making large scale changes to the system, we seek to stabilize these existing unstable periodic orbits with only small control perturbations. We present and discuss the method of Ott, Grebogi and Yorke. We also discuss a global control method that, when coupled with the OGY method, permits faster and more efficient stabilization. We illustrate the method with an application to a mechanical system, the kicked double rotor.
This is Chapter 11 in Dynamics and Chaos in
Manufacturing Processes , Francis C. Moon, editor. New York:
John Wiley & Sons, Inc.,
1997.
For further information, click
here to send me an email
at ebarreto_AT_gmu_DOT_edu.
Targeting and Control of Chaos
Eric J. Kostelich and Ernest Barreto
Abstract
"Control of chaos" refers to a procedure in which a saddle fixed point in a chaotic attractor is stabilized by means of small time dependent perturbations. Control may be switched between different saddle periodic orbits, but it is necessary to wait for the trajectory to enter a small neighborhood of the saddle point before the control algorithm can be applied. This paper describes an extension of the control idea, called "targeting". By targeting, we mean a process in which a typical initial condition can be steered to a prespecified point on a chaotic attractor using a sequence of small, time dependent changes to a convenient parameter. We show, using a 4-dimensional mapping describing a kicked double rotor, that points on a chaotic attractor with two positive Lyapunov exponents can be steered between typical saddle periodic points extremely rapidly--in as little as 12 iterations on the average. Without targeting, typical trajectories require 10,000 or more iterations to reach a small neighborhood of saddle periodic points of interest.
This chapter appears in Control and Chaos,
edited by K.
Judd,
A. Mees, K. L. Teo and T. L. Vincent. Boston: Birkhäuser,
1997,
pp. 158-169.
For further information, click
here to send me an email
at ebarreto_AT_gmu_DOT_edu.
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