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Joe-Wong C, Ho TS, Rabitz H. Assessing the structure of classical molecular optimal control landscapes. Chem Phys 2019. [DOI: 10.1016/j.chemphys.2019.110504] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
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2
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Joe-Wong C, Ho TS, Rabitz H, Wu R. Topology of classical molecular optimal control landscapes for multi-target objectives. J Chem Phys 2015; 142:154115. [PMID: 25903874 DOI: 10.1063/1.4918274] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/13/2023] Open
Abstract
This paper considers laser-driven optimal control of an ensemble of non-interacting molecules whose dynamics lie in classical phase space. The molecules evolve independently under control to distinct final states. We consider a control landscape defined in terms of multi-target (MT) molecular states and analyze the landscape as a functional of the control field. The topology of the MT control landscape is assessed through its gradient and Hessian with respect to the control. Under particular assumptions, the MT control landscape is found to be free of traps that could hinder reaching the objective. The Hessian associated with an optimal control field is shown to have finite rank, indicating an inherent degree of robustness to control noise. Both the absence of traps and rank of the Hessian are shown to be analogous to the situation of specifying multiple targets for an ensemble of quantum states. Numerical simulations are presented to illustrate the classical landscape principles and further characterize the system behavior as the control field is optimized.
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Affiliation(s)
- Carlee Joe-Wong
- Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544-1000, USA
| | - Tak-San Ho
- Department of Chemistry, Princeton University, Princeton, New Jersey 08544-1009, USA
| | - Herschel Rabitz
- Department of Chemistry, Princeton University, Princeton, New Jersey 08544-1009, USA
| | - Rebing Wu
- Department of Automation, Tsinghua University, Beijing, People's Republic of China
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3
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Tamaševičiūtė E, Mykolaitis G, Bumelienė S, Tamaševičius A. Stabilizing saddles. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:060901. [PMID: 24483376 DOI: 10.1103/physreve.88.060901] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/25/2013] [Indexed: 06/03/2023]
Abstract
A synergetic control technique for stabilizing a priori unknown saddle steady states of dynamical systems is described. The method involves an unstable filter technique combined with a derivative feedback. The cut-off frequency of the filter is not limited by the damping of the system, and therefore can be set relatively high. This essentially increases the rate of convergence to the steady state. The synergetic technique is robust to the influence of unknown external forces, which change the coordinates of the steady state in the phase space.
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Affiliation(s)
- Elena Tamaševičiūtė
- Department of Electronics, Center for Physical Sciences and Technology, LT-01108 Vilnius, Lithuania
| | - Gytis Mykolaitis
- Department of Physics, Vilnius Gediminas Technical University, LT-10223 Vilnius, Lithuania
| | - Skaidra Bumelienė
- Department of Electronics, Center for Physical Sciences and Technology, LT-01108 Vilnius, Lithuania
| | - Arūnas Tamaševičius
- Department of Electronics, Center for Physical Sciences and Technology, LT-01108 Vilnius, Lithuania
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4
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Tamaševičius A, Tamaševičiūtė E, Mykolaitis G, Bumelienė S. Enhanced control of saddle steady states of dynamical systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:032904. [PMID: 24125322 DOI: 10.1103/physreve.88.032904] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/07/2013] [Indexed: 06/02/2023]
Abstract
An adaptive feedback technique for stabilizing a priori unknown saddle steady states of dynamical systems is described. The method is based on an unstable low-pass filter combined with a stable low-pass filter. The cutoff frequencies of both filters can be set relatively high. This allows considerable increase in the rate of convergence to the steady state. We demonstrate numerically and experimentally that the technique is robust to the influence of unknown external forces, which change the position of the steady state in the phase space. Experiments have been performed using electrical circuits imitating the damped Duffing-Holmes and chaotic Lindberg systems.
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Affiliation(s)
- Arūnas Tamaševičius
- Department of Electronics, Center for Physical Sciences and Technology, LT-01108 Vilnius, Lithuania
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5
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Joe-Wong C, Ho TS, Long R, Rabitz H, Wu R. Topology of classical molecular optimal control landscapes in phase space. J Chem Phys 2013; 138:124114. [PMID: 23556716 DOI: 10.1063/1.4797498] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/28/2022] Open
Abstract
Optimal control of molecular dynamics is commonly expressed from a quantum mechanical perspective. However, in most contexts the preponderance of molecular dynamics studies utilize classical mechanical models. This paper treats laser-driven optimal control of molecular dynamics in a classical framework. We consider the objective of steering a molecular system from an initial point in phase space to a target point, subject to the dynamic constraint of Hamilton's equations. The classical control landscape corresponding to this objective is a functional of the control field, and the topology of the landscape is analyzed through its gradient and Hessian with respect to the control. Under specific assumptions on the regularity of the control fields, the classical control landscape is found to be free of traps that could hinder reaching the objective. The Hessian associated with an optimal control field is shown to have finite rank, indicating the presence of an inherent degree of robustness to control noise. Extensive numerical simulations are performed to illustrate the theoretical principles on (a) a model diatomic molecule, (b) two coupled Morse oscillators, and (c) a chaotic system with a coupled quartic oscillator, confirming the absence of traps in the classical control landscape. We compare the classical formulation with the mathematically analogous quantum state-to-state transition probability control landscape.
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Affiliation(s)
- Carlee Joe-Wong
- Department of Mathematics, Princeton University, Princeton, New Jersey 08544-1000, USA
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6
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de Sousa MC, Caldas IL, Rizzato FB, Pakter R, Steffens FM. Controlling chaos in wave-particle interactions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:016217. [PMID: 23005517 DOI: 10.1103/physreve.86.016217] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/25/2012] [Revised: 05/11/2012] [Indexed: 06/01/2023]
Abstract
We analyze the behavior of a relativistic particle moving under the influence of a uniform magnetic field and a stationary electrostatic wave. We work with a set of pulsed waves that allows us to obtain an exact map for the system. We also use a method of control for near-integrable Hamiltonians that consists of the addition of a small and simple control term to the system. This control term creates invariant tori in phase space that prevent chaos from spreading to large regions, making the controlled dynamics more regular. We show numerically that the control term just slightly modifies the system but is able to drastically reduce chaos with a low additional cost of energy. Moreover, we discuss how the control of chaos and the consequent recovery of regular trajectories in phase space are useful to improve regular particle acceleration.
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Affiliation(s)
- M C de Sousa
- Instituto de Física, Universidade de São Paulo, São Paulo, São Paulo, Brazil
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7
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Tamasevicius A, Tamaseviciūte E, Mykolaitis G, Bumeliene S, Kirvaitis R. Stabilization of saddle steady states of conservative and weakly damped dissipative dynamical systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:026205. [PMID: 20866891 DOI: 10.1103/physreve.82.026205] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/08/2009] [Revised: 07/02/2010] [Indexed: 05/29/2023]
Abstract
An adaptive feedback method for tracking and stabilizing unknown and/or slowly varying saddle-type steady states of conservative and weakly damped dissipative dynamical systems is proposed. We demonstrate that a conservative saddle point can be stabilized with neither unstable nor stable filter technique. The proposed controller involves both filters working in parallel. As a specific example, the Lagrange point L2 of the Sun-Earth system is discussed and the second-order saddle model is considered. Analog simulations have been performed using an inclusive nonlinear electrical circuit, imitating dynamics of a body along the Sun-Earth line. External chaotic perturbations have been used to check the robustness of the control technique.
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Affiliation(s)
- Arūnas Tamasevicius
- Semiconductor Physics Institute, Center for Physical Sciences and Technology, LT-01108 Vilnius, Lithuania.
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8
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Macau EEN, Grebogi C. Control of chaos and its relevancy to spacecraft steering. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2006; 364:2463-81. [PMID: 16893798 DOI: 10.1098/rsta.2006.1835] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/11/2023]
Abstract
In 1990, a seminal work named controlling chaos showed that not only the chaotic evolution could be controlled, but also the complexity inherent in the chaotic dynamics could be exploited to provide a unique level of flexibility and efficiency in technological uses of this phenomenon. Control of chaos is also making substantial contribution in the field of astrodynamics, especially related to the exciting issue of low-energy transfer. The purpose of this work is to bring up the main ideas regarding the control of chaos and targeting, and to show how these techniques can be extended to Hamiltonian situations. We give realistic examples related to astrodynamics problems, in which these techniques are unique in terms of efficiency related to low-energy spacecraft transfer and in-orbit stabilization.
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Affiliation(s)
- Elbert E N Macau
- Laboratório Associado de Matemática Aplicada e Computação (LAC), Instituto Nacional de Pesquisas Espaciais (INPE), 12227-010 São José dos Campos, São Paulo, Brazil
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Kulp CW, Tracy ER. Control of multidimensional integrable Hamiltonian systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:036213. [PMID: 16241554 DOI: 10.1103/physreve.72.036213] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/12/2004] [Indexed: 05/05/2023]
Abstract
In this paper, we study the controllability of a four-dimensional integrable Hamiltonian system that arises as a low-mode truncation of the nonlinear Schrödinger equation [Bishop, Phys. Lett. A 144, 17 (1990)]. The controller targets a solution of the uncontrolled dynamics. We show that in the limit of small control coupling, a Takens-Bogdanov bifurcation occurs at the control target. These results support our earlier claim that Takens-Bogdanov bifurcations will generically occur when dissipative control is applied to integrable Hamiltonian sytems. The presence of the Takens-Bogdanov bifurcation causes the control to be extremely sensitive to noise. Here, we implement an algorithm first developed in Kulp and Tracy [Phys. Rev. E 70, 016205 (2004)] to extract a subcritical noise threshold for the four-dimensional system.
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Affiliation(s)
- C W Kulp
- Department of Physics and Astronomy, Eastern Kentucky University, Richmond, Kentucky 40475, USA
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10
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Abstract
▪ Abstract Coherent control of atomic and molecular processes has been a rapidly developing field. Applications of coherent control to large and complex molecular systems are expected to encounter the effects of chaos in the underlying classical dynamics, i.e., quantum chaos. Hence, recent work has focused on examining control in model chaotic systems. This work is reviewed, with an emphasis on a variety of new quantum phenomena that are of interest to both areas of quantum chaos and coherent control.
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Affiliation(s)
- Jiangbin Gong
- Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois 60637, USA.
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11
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Ciraolo G, Briolle F, Chandre C, Floriani E, Lima R, Vittot M, Pettini M, Figarella C, Ghendrih P. Control of Hamiltonian chaos as a possible tool to control anomalous transport in fusion plasmas. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:056213. [PMID: 15244910 DOI: 10.1103/physreve.69.056213] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/15/2003] [Indexed: 05/24/2023]
Abstract
It is shown that a relevant control of Hamiltonian chaos is possible through suitable small perturbations whose form can be explicitly computed. In particular, it is possible to control (reduce) the chaotic diffusion in the phase space of a Hamiltonian system with 1.5 degrees of freedom which models the diffusion of charged test particles in a turbulent electric field across the confining magnetic field in controlled thermonuclear fusion devices. Though still far from practical applications, this result suggests that some strategy to control turbulent transport in magnetized plasmas, in particular, tokamaks, is conceivable. The robustness of the control is investigated in terms of a departure from the optimum magnitude, of a varying cutoff at large wave vectors, and of random errors on the phases of the modes. In all three cases, there is a significant region of maximum efficiency in the vicinity of the optimum control term.
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Affiliation(s)
- Guido Ciraolo
- Facoltà di Ingegneria, Università di Firenze, via S. Marta, I-50129 Florence, Italy.
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12
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Do Y, Lai YC. Statistics of shadowing time in nonhyperbolic chaotic systems with unstable dimension variability. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:016213. [PMID: 14995699 DOI: 10.1103/physreve.69.016213] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/17/2003] [Indexed: 05/24/2023]
Abstract
Severe obstruction to shadowing of computer-generated trajectories can occur in nonhyperbolic chaotic systems with unstable dimension variability. That is, when the dimension of the unstable eigenspace changes along a trajectory in the invariant set, no true trajectory of reasonable length can be found to exist near any numerically generated trajectory. An important quantity characterizing the shadowability of numerical trajectories is the shadowing time, which measures for how long a trajectory remains valid. This time depends sensitively on initial condition. Here we show that the probability distribution of the shadowing time contains two distinct scaling behaviors: an algebraic scaling for short times and an exponential scaling for long times. The exponential behavior depends on system details but the small-time algebraic behavior appears to be universal. We describe the computational procedure for computing the shadowing time and give a physical analysis for the observed scaling behaviors.
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Affiliation(s)
- Younghae Do
- Department of Mathematics and Statistics, Arizona State University, Tempe, Arizona 85287, USA
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13
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Kwon OJ, Lee H. Controlling chaos to solutions with complex eigenvalues. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 67:026201. [PMID: 12636770 DOI: 10.1103/physreve.67.026201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/14/2002] [Indexed: 05/24/2023]
Abstract
We derive formulas for parameter and variable perturbations to control chaos using linearized dynamics. They are available irrespective of the dimension of the system, the number of perturbed parameters or variables, and the kinds of eigenvalues of the linearized dynamics. We illustrate this using the two coupled Duffing oscillators and the two coupled standard maps.
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Affiliation(s)
- Oh-Jong Kwon
- Department of Science Education, Gongju National University of Education, Gongju 314-711, Republic of Korea.
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14
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Cartwright JHE, Magnasco MO, Piro O. Bailout embeddings, targeting of invariant tori, and the control of Hamiltonian chaos. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:045203. [PMID: 12005907 DOI: 10.1103/physreve.65.045203] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/22/2001] [Revised: 12/20/2001] [Indexed: 05/23/2023]
Abstract
We introduce a technique, which we term bailout embedding, that can be used to target orbits having particular properties out of all orbits in a flow or map. We explicitly construct a bailout embedding for Hamiltonian systems so as to target invariant tori. We show how the bailout dynamics are able to lock onto extremely small regular islands in a chaotic sea.
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15
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Macau EEN, Caldas IL. Driving trajectories in chaotic scattering. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:026215. [PMID: 11863640 DOI: 10.1103/physreve.65.026215] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/22/2001] [Indexed: 05/23/2023]
Abstract
In this work we introduce a general approach for targeting in chaotic scattering that can be used to find a transfer trajectory between any two points located inside the scattering region. We show that this method can be used in association with a control of chaos strategy to drive around and keep a particle inside the scattering region. As an illustration of how powerful this approach is, we use it in a case of practical interest in celestial mechanics in which it is desired to control the evolution of two satellites that evolve around a large central body.
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Affiliation(s)
- Elbert E N Macau
- Laboratório de Integraçāo e Testes (LIT), Instituto Nacional de Pesquisas Espaciais (INPE), São José das Campos, São Paulo, Brazil.
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16
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Bolotin YL, Gonchar VY, Krokhin AA, Hernández-Tejeda PH, Tur A, Yanovsky VV. Local and global control of high-period unstable orbits in reversible maps. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:026218. [PMID: 11497688 DOI: 10.1103/physreve.64.026218] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/13/2000] [Revised: 05/07/2001] [Indexed: 05/23/2023]
Abstract
We study the nonlinear dynamics of a complex system, described by a two-dimensional reversible map. The phase space of this map exhibits elements typical of Hamiltonian systems (stability islands) as well as of dissipative systems (attractor). Due to the interaction between the stability islands and the attractor, the transition to chaos in this system occurs through the collapse of the stability island and stochastization of the limiting-cycles orbits. We show how to apply the method of discrete parametric control to stabilize unstable high-period orbits. To achieve highly efficient control we introduce the concepts of local and global control. These concepts are useful in situations where there are "dangerous" points on the target orbit, i.e., the points where the probability of breakdown of control is high. As a result, the dangerous points turn out to be much more sensitive to external noise than other points on the orbit, and only the dangerous points determine how effective the control is.
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Affiliation(s)
- Y L Bolotin
- National Science Center, Kharkov Institute of Physics and Technology, Ukraine
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17
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Xu H, Wang G, Chen S. Controlling dissipative and Hamiltonian chaos by a constant periodic pulse method. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:016201. [PMID: 11461361 DOI: 10.1103/physreve.64.016201] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/17/2000] [Revised: 03/01/2001] [Indexed: 05/23/2023]
Abstract
A constant periodic pulse method is proposed to control dissipative and Hamiltonian chaos. Using the convergence of the chaotic orbit in finite time, the stable segment of the chaotic orbit that satisfies the desired dynamical features can be made to form a closed orbit by the action of a proper perturbation on the system variables. A way to determine the intensity of the perturbation and the corresponding fixed points is presented. The method is robust against the presence of external noise.
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Affiliation(s)
- H Xu
- Graduate School, China Academy of Engineering Physics, P.O. Box 2101, Beijing 100088, People's Republic of China.
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18
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Abdullaev SS. Structure of motion near saddle points and chaotic transport in hamiltonian systems. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 62:3508-3528. [PMID: 11088851 DOI: 10.1103/physreve.62.3508] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/30/2000] [Indexed: 05/23/2023]
Abstract
Generic symmetry and transport properties of near separatrix motion in 11 / 2-degree-of-freedom Hamiltonian systems are studied. First the rescaling invariance of motion near saddle points, with respect to the transformation epsilon-->lambdaepsilon, chi-->chi+pi of the amplitude epsilon and phase chi, of the time-periodic perturbation, is recalled. The rescaling parameter lambda depends only on the frequency of the perturbation, and the behavior of an unperturbed Hamiltonian near a saddle point. Additional rescaling symmetry of the motion with respect to transformation epsilon-->lambda(1/2)epsilon, chi-->chi+/-pi/2 is found for some Hamiltonian systems possessing symmetry in the phase space. It is shown that these rescaling invariance properties of motion lead to strong periodic (or quasiperiodic) dependencies of all statistical characteristics of the chaotic motion near the separatrix on log(10)epsilon with the period log(10)lambda. These properties are examined for different models of chaotic motion by direct numerical integrations of equations of motion, as by well as using a computationally efficient method of the separatrix mapping.
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Affiliation(s)
- SS Abdullaev
- Institut fur Plasmaphysik, Forschungszentrum Julich GmbH, EURATOM Association, Trilateral Euregio Cluster, D-52425 Julich, Germany
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19
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Zhang Y, Chen S, Yao Y. Controlling hamiltonian chaos by adaptive integrable mode coupling. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 62:2135-9. [PMID: 11088679 DOI: 10.1103/physreve.62.2135] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/12/1999] [Revised: 04/14/2000] [Indexed: 11/07/2022]
Abstract
The adaptive integrable mode coupling method is proposed to control two-dimensional Hamiltonian chaos. We demonstrate that this control method can stabilize chaotic motions into regular ones in a model of the standard map. Global stochasticity can be removed from the phase space by the control being switched on and off.
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Affiliation(s)
- Y Zhang
- LCP, Institute of Applied Physics and Computational Mathematics, P.O. Box 8009(26), Beijing 100088, China
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20
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Triandaf I, Schwartz IB. Quality factor control in a lasing microcavity model. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 61:3601-9. [PMID: 11088138 DOI: 10.1103/physreve.61.3601] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/25/1999] [Revised: 12/05/1999] [Indexed: 11/07/2022]
Abstract
We consider a dynamics model of lasing microcavities, a class of optical resonators (1-10 &mgr;m in diameter) used in microlasers and for optical coupling of optical fibers. Inside such a cavity light circulates around the perimeter and is trapped by internal reflection. This is known as "whispering gallery" or high-Q modes. The cavity is a deformable cylindrical (or spherical) dielectric and at certain deformations light can escape by refraction. The quality of the resonator or Q factor, is defined as Q=omegatau, where tau is the escape time and omega is the frequency of light. We show that by appropriately deforming the cavity, the Q factor can be controlled by prolonging or shortening the average length of time spent by light trajectories inside the cavity.
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Affiliation(s)
- I Triandaf
- Special Project in Nonlinear Science, U.S. Naval Research Laboratory, Code 6700.3, Plasma Physics Division, Washington, D.C. 20375-5000, USA
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21
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Oloumi A, Teychenné D. Controlling Hamiltonian chaos via Gaussian curvature. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:R6279-82. [PMID: 11970614 DOI: 10.1103/physreve.60.r6279] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/03/1999] [Revised: 07/14/1999] [Indexed: 11/07/2022]
Abstract
We present a method allowing one to partly stabilize some chaotic Hamiltonians which have two degrees of freedom. The purpose of the method is to avoid the regions of V(q(1),q(2)) where its Gaussian curvature becomes negative. We show the stabilization of the Hénon-Heiles system, over a wide area, for the critical energy E=1/6. Total energy of the system varies only by a few percent.
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Affiliation(s)
- A Oloumi
- National Biocomputation Center, Stanford University Medical Center, 701 Welch Road, Suite 1128, Palo Alto, California 94304, USA
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22
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Abdullaev SS, Spatschek KH. Rescaling invariance and anomalous transport in a stochastic layer. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:R6287-90. [PMID: 11970616 DOI: 10.1103/physreve.60.r6287] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/04/1999] [Indexed: 04/18/2023]
Abstract
The anomalous chaotic transport in a one-degree-of-freedom Hamiltonian system subjected to a small time-periodic perturbation is investigated. Strong quasiperiodic dependencies of the statistical properties of the motion on log epsilon are found, where epsilon is a perturbation parameter. The period log lambda depends on the rescaling parameter lambda, which is determined only by the frequency of perturbation and behavior of unperturbed Hamiltonian near a saddle point. The results confirm and generalize a recently established new universal rescaling property of perturbed motion near a saddle point.
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Affiliation(s)
- S S Abdullaev
- Institut für Theoretische Physik I, Heinrich-Heine-Universität Düsseldorf, D-40225 Düsseldorf, Germany
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23
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Ding M, Ding EJ, Ditto WL, Gluckman B, In V, Peng JH, Spano ML, Yang W. Control and synchronization of chaos in high dimensional systems: Review of some recent results. CHAOS (WOODBURY, N.Y.) 1997; 7:644-652. [PMID: 12779690 DOI: 10.1063/1.166284] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
Controlling chaos and synchronization of chaos have evolved for a number of years as essentially two separate areas of research. Only recently it has been realized that both subjects share a common root in control theory. In addition, as limitations of low dimensional chaotic systems in modeling real world phenomena become increasingly apparent, investigations into the control and synchronization of high dimensional chaotic systems are beginning to attract more interest. We review some recent advances in control and synchronization of chaos in high dimensional systems. Efforts will be made to stress the common origins of the two subjects. (c) 1997 American Institute of Physics.
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Affiliation(s)
- Mingzhou Ding
- Center for Complex Systems and Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, Florida 33431
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Konishi K, Ishii M, Kokame H. Stabilizing unstable periodic points of one-dimensional nonlinear systems using delayed-feedback signals. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:3455-3460. [PMID: 9965489 DOI: 10.1103/physreve.54.3455] [Citation(s) in RCA: 22] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Gauthier DJ, Bienfang JC. Intermittent Loss of Synchronization in Coupled Chaotic Oscillators: Toward a New Criterion for High-Quality Synchronization. PHYSICAL REVIEW LETTERS 1996; 77:1751-1754. [PMID: 10063162 DOI: 10.1103/physrevlett.77.1751] [Citation(s) in RCA: 25] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Ding M, Yang W, In V, Ditto WL, Spano ML, Gluckman B. Controlling chaos in high dimensions: Theory and experiment. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 53:4334-4344. [PMID: 9964766 DOI: 10.1103/physreve.53.4334] [Citation(s) in RCA: 59] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Nagai Y, Lai YC. Selection of a desirable chaotic phase using small feedback control. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 51:3842-3848. [PMID: 9963094 DOI: 10.1103/physreve.51.3842] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Newell TC, Alsing PM, Gavrielides A, Kovanis V. Synchronization of chaotic resonators based on control theory. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 51:2963-2973. [PMID: 9962974 DOI: 10.1103/physreve.51.2963] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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John JK, Amritkar RE. Synchronization of unstable orbits using adaptive control. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:4843-4848. [PMID: 9961801 DOI: 10.1103/physreve.49.4843] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Lai YC, Grebogi C. Converting transient chaos into sustained chaos by feedback control. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:1094-1098. [PMID: 9961317 DOI: 10.1103/physreve.49.1094] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Newell TC, Alsing PM, Gavrielides A, Kovanis V. Synchronization of chaos using proportional feedback. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:313-318. [PMID: 9961219 DOI: 10.1103/physreve.49.313] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Alvarez-Ramírez J. Using nonlinear saturated feedback to control chaos: The Hénon map. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 48:3165-3167. [PMID: 9960954 DOI: 10.1103/physreve.48.3165] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Lai YC, Tél T, Grebogi C. Stabilizing chaotic-scattering trajectories using control. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 48:709-717. [PMID: 9960650 DOI: 10.1103/physreve.48.709] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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