1
|
Dutta M. Fast high-quality numerical shadowing of chaotic maps using synchronization. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:056214. [PMID: 16383734 DOI: 10.1103/physreve.72.056214] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/31/2005] [Revised: 10/03/2005] [Indexed: 05/05/2023]
Abstract
For a chaotic, dynamical system, a typical noisy trajectory diverges exponentially from the true trajectory with the same initial conditions. Nonetheless, the noisy trajectory, as representative of the dynamical system, may have credibility if there exists a true trajectory, corresponding to a slightly different initial condition, that stays close to the noisy trajectory for long periods of time. For finding such shadowing trajectories, a synchronization based method is presented. We call it the synchronize-and-pullback algorithm. Several numerical examples are shown illustrating the method. Finally, an application of the proposed shadowing technique for noise filtration in the context of communication is presented.
Collapse
Affiliation(s)
- Mitrajit Dutta
- Department of Mathematics and Statistics, University of New Hampshire, Durham, New Hampshire 03824, USA.
| |
Collapse
|
2
|
Dutta M. Synchronizing chaotic maps using an arbitrary scalar signal. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:026217. [PMID: 16196694 DOI: 10.1103/physreve.72.026217] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/22/2004] [Revised: 05/31/2005] [Indexed: 05/04/2023]
Abstract
A method we call the slide-and-match algorithm is presented whereby any transmitter-receiver pair of identical, chaotic maps may be synchronized using almost any scalar function of the driving transmitter state, transmitted to the receiver as the synchronizing signal. No prior investigation of either the map or the signal generating function is required. In order to illustrate the concepts, numerical simulations of applications are presented.
Collapse
Affiliation(s)
- Mitrajit Dutta
- Department of Mathematics and Statistics, University of New Hampshire, Durham, New Hampshire 03824, USA
| |
Collapse
|
3
|
Perc M, Marhl M. Detecting and controlling unstable periodic orbits that are not part of a chaotic attractor. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:016204. [PMID: 15324149 DOI: 10.1103/physreve.70.016204] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/14/2003] [Indexed: 05/24/2023]
Abstract
A method for controlling unstable periodic orbits (UPOs) that have not been controllable before is presented. The method is based on detecting UPOs that are situated outside the skeleton of a chaotic attractor. The main idea is to exploit flexible parts of the attractor, which under weak external perturbations allow variable excursions of the trajectory away from its originally determined path. After the perturbation, the trajectory of the autonomous system seeks its path back to the chaotic attractor and reveals additional UPOs that are otherwise not used by the system. It is shown that these UPOs can be controlled as easily as the UPOs that form the basic chaotic attractor. The effectiveness of the proposed method is demonstrated on two different chaotic systems with very distinct response abilities to external perturbations. Additionally, some applications of the method in the fields of laser technology, information encoding, and biomedical engineering are discussed.
Collapse
Affiliation(s)
- Matjaz Perc
- Department of Physics, Faculty of Education, University of Maribor, Koroska cesta 160, SI-2000 Maribor, Slovenia
| | | |
Collapse
|
4
|
Huijberts H, Nijmeijer H, Willems R. System identification in communication with chaotic systems. ACTA ACUST UNITED AC 2000. [DOI: 10.1109/81.852932] [Citation(s) in RCA: 168] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
|
5
|
Shahverdiev EM. Boundedness of dynamical systems and chaos synchronization. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:3905-9. [PMID: 11970225 DOI: 10.1103/physreve.60.3905] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/04/1999] [Revised: 04/21/1999] [Indexed: 04/18/2023]
Abstract
Chaos synchronization in bounded dynamical systems is studied. We use boundedness of trajectories within the nonreplica approach to chaos synchronization and the Routh-Hurwitz criterion to propose a simple method to make negative conditional Lyapunov exponents. The method is tested on the classical Lorenz model.
Collapse
Affiliation(s)
- E M Shahverdiev
- Department of Information Science, Saga University, Saga 840-8502, Japan
| |
Collapse
|
6
|
Parekh N, Ravi Kumar V, Kulkarni BD. Synchronization and control of spatiotemporal chaos using time-series data from local regions. CHAOS (WOODBURY, N.Y.) 1998; 8:300-306. [PMID: 12779733 DOI: 10.1063/1.166310] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
In this paper we show that the analysis of the dynamics in localized regions, i.e., sub-systems can be used to characterize the chaotic dynamics and the synchronization ability of the spatiotemporal systems. Using noisy scalar time-series data for driving along with simultaneous self-adaptation of the control parameter representative control goals like suppressing spatiotemporal chaos and synchronization of spatiotemporally chaotic dynamics have been discussed. (c) 1998 American Institute of Physics.
Collapse
Affiliation(s)
- Nita Parekh
- Chemical Engineering Division, National Chemical Laboratory, Pune - 411 008, India
| | | | | |
Collapse
|
7
|
Pecora LM, Carroll TL, Johnson GA, Mar DJ, Heagy JF. Fundamentals of synchronization in chaotic systems, concepts, and applications. CHAOS (WOODBURY, N.Y.) 1997; 7:520-543. [PMID: 12779679 DOI: 10.1063/1.166278] [Citation(s) in RCA: 195] [Impact Index Per Article: 7.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
The field of chaotic synchronization has grown considerably since its advent in 1990. Several subdisciplines and "cottage industries" have emerged that have taken on bona fide lives of their own. Our purpose in this paper is to collect results from these various areas in a review article format with a tutorial emphasis. Fundamentals of chaotic synchronization are reviewed first with emphases on the geometry of synchronization and stability criteria. Several widely used coupling configurations are examined and, when available, experimental demonstrations of their success (generally with chaotic circuit systems) are described. Particular focus is given to the recent notion of synchronous substitution-a method to synchronize chaotic systems using a larger class of scalar chaotic coupling signals than previously thought possible. Connections between this technique and well-known control theory results are also outlined. Extensions of the technique are presented that allow so-called hyperchaotic systems (systems with more than one positive Lyapunov exponent) to be synchronized. Several proposals for "secure" communication schemes have been advanced; major ones are reviewed and their strengths and weaknesses are touched upon. Arrays of coupled chaotic systems have received a great deal of attention lately and have spawned a host of interesting and, in some cases, counterintuitive phenomena including bursting above synchronization thresholds, destabilizing transitions as coupling increases (short-wavelength bifurcations), and riddled basins. In addition, a general mathematical framework for analyzing the stability of arrays with arbitrary coupling configurations is outlined. Finally, the topic of generalized synchronization is discussed, along with data analysis techniques that can be used to decide whether two systems satisfy the mathematical requirements of generalized synchronization. (c) 1997 American Institute of Physics.
Collapse
Affiliation(s)
- Louis M. Pecora
- Code 6343, U.S. Naval Research Laboratory, Washington, District of Columbia 20375
| | | | | | | | | |
Collapse
|
8
|
Brown R, Rulkov NF. Synchronization of chaotic systems: Transverse stability of trajectories in invariant manifolds. CHAOS (WOODBURY, N.Y.) 1997; 7:395-413. [PMID: 12779668 DOI: 10.1063/1.166213] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
We examine synchronization of identical chaotic systems coupled in a drive/response manner. A rigorous criterion is presented which, if satisfied, guarantees that synchronization to the driving trajectory is linearly stable to perturbations. An easy to use approximate criterion for estimating linear stability is also presented. One major advantage of these criteria is that, for simple systems, many of the calculations needed to implement them can be performed analytically. Geometrical interpretations of the criterion are discussed, as well as how they may be used to investigate synchronization between mutual coupled systems and the stability of invariant manifolds within a dynamical system. Finally, the relationship between our criterion and results from control theory are discussed. Analytical and numerical results from tests of these criteria on four different dynamical systems are presented. (c) 1997 American Institute of Physics.
Collapse
Affiliation(s)
- Reggie Brown
- Institute for Nonlinear Science, University of California, San Diego, La Jolla, California 92093-0402
| | | |
Collapse
|
9
|
|
10
|
Sun H, Wang L. Comment on "Chaotic Masking Scheme with a Linear Inverse System". PHYSICAL REVIEW LETTERS 1996; 77:2842. [PMID: 10062062 DOI: 10.1103/physrevlett.77.2842] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
|
11
|
Peng JH, Ding EJ, Ding M, Yang W. Synchronizing hyperchaos with a scalar transmitted signal. PHYSICAL REVIEW LETTERS 1996; 76:904-907. [PMID: 10061581 DOI: 10.1103/physrevlett.76.904] [Citation(s) in RCA: 26] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
|
12
|
Parekh N, Kumar VR, Kulkarni BD. Control of self-replicating patterns in a model reaction-diffusion system. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:5100-5104. [PMID: 9964007 DOI: 10.1103/physreve.52.5100] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
|
13
|
Pérez G, Cerdeira HA. Extracting messages masked by chaos. PHYSICAL REVIEW LETTERS 1995; 74:1970-1973. [PMID: 10057809 DOI: 10.1103/physrevlett.74.1970] [Citation(s) in RCA: 35] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
|
14
|
Rulkov NF, Sushchik MM, Tsimring LS, Abarbanel HD. Generalized synchronization of chaos in directionally coupled chaotic systems. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 51:980-994. [PMID: 9962737 DOI: 10.1103/physreve.51.980] [Citation(s) in RCA: 548] [Impact Index Per Article: 18.9] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/12/2023]
|
15
|
Brown R, Rulkov NF, Tufillaro NB. Synchronization of chaotic systems: The effects of additive noise and drift in the dynamics of the driving. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 50:4488-4508. [PMID: 9962527 DOI: 10.1103/physreve.50.4488] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
|
16
|
Lai YC, Grebogi C. Synchronization of spatiotemporal chaotic systems by feedback control. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 50:1894-1899. [PMID: 9962191 DOI: 10.1103/physreve.50.1894] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
|