Detecting Unstable Periodic Orbits of Chaotic Dynamical Systems

Schmelcher, Peter; Diakonos, F. K.

doi:10.1103/physrevlett.78.4733

Cited by 136 publications

(98 citation statements)

References 12 publications

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“…We have presented a scheme for detecting UPOs in high dimensional chaotic systems based upon the stabilising transformations proposed in [2,5,19]. Due to the fact that one often wishes to study low dimensional dynamics embedded in a high dimensional phase space, it is possible to increase the efficiency of the stabilising transformations approach by restricting the construction of such transformations only to the low-dimensional unstable subspace.…”

Section: Conclusion and Further Workmentioning

confidence: 99%

“…(1.2). One example of such a set was proposed by Schmelcher and Diakonos (SD) [19]. It is the set C SD of orthogonal matrices such that only one entry {±1} per row or column is nonzero.…”

Section: Introductionmentioning

confidence: 99%

“…The method of detecting UPOs by stabilising transformations [2,5,19] aims at transforming the system in such a way that its UPOs become stable. Unlike in the Newton-type methods, the transformations are linear and thus do not suffer from spurious convergence.…”

Section: Introductionmentioning

confidence: 99%

“…The main difficulty of using this method is that the number of transformations, which were used in the past, becomes overwhelming as we move to higher dimensions [5,19,20]. We have recently proposed a set of stabilising transformations which is constructed from a small set of already found UPOs [2].…”

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confidence: 99%

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On the use of stabilizing transformations for detecting unstable periodic orbits in high-dimensional flows

Crofts

Davidchack

2009

Chaos: An Interdisciplinary Journal of Nonlinear Science

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In this paper we develop further a method for detecting unstable periodic orbits (UPOs) by stabilising transformations, where the strategy is to transform the system of interest in such a way that the orbits become stable. The main difficulty of using this method is that the number of transformations, which were used in the past, becomes overwhelming as we move to higher dimensions [5,19,20]. We have recently proposed a set of stabilising transformations which is constructed from a small set of already found UPOs [2]. The main benefit of using the proposed set is that its cardinality depends on the dimension of the unstable manifold at the UPO rather than the dimension of the system. In a typical situation the dimension of the unstable manifold is much smaller than the dimension of the system so the number of transformations is much smaller. Here we extend this approach to high-dimensional systems of ODEs and apply it to the model example of a chaotic spatially extended system -the KuramotoSivashinsky equation. A comparison is made between the performance of this new method against the competing methods of Newton-Armijo (NA) and Levernberg-Marquardt (LM). In the latter case, we take advantage of the fact that the LM algorithm is able to solve under-determined systems of equations, thus eliminating the need for any additional constraints.

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Section: Conclusion and Further Workmentioning

confidence: 99%

“…(1.2). One example of such a set was proposed by Schmelcher and Diakonos (SD) [19]. It is the set C SD of orthogonal matrices such that only one entry {±1} per row or column is nonzero.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

On the use of stabilizing transformations for detecting unstable periodic orbits in high-dimensional flows

Crofts

Davidchack

2009

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…Note that by using some numerical techniques, the fixed points of a chaotic system whose states are accessible can be calculated without using its dynamic equation (Ramesh andNarayanan 2001, Schmelcher andDiakonos, 1997) hence it is assumed that the fixed points of the system are obtained by a numerical algorithm without using the system parameters.…”

Section: Problem Statementmentioning

confidence: 99%

Adaptive Control of Chaos

Salarieh¹,

Shahrokhi²

2011

Chaotic Systems

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Fluctuational Escape and Related Phenomena in Nonlinear Optical Systems

Khovanov

Luchinsky

Mannella

et al.

Modern Nonlinear Optics, Part III

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ii FLUCTUATIONAL ESCAPE AND RELATED PHENOMENA been selected mainly for their own intrinsic scientific interest, but also in order to provide an indication of the power and utility of the simulation approach as a means of focusing on, and reaching an understanding of, the essential physics underlying the phenomena under investigation; they also provide examples of different theoretical approaches and situations where numerical and analogue simulations have led to the development of new experimental techniques and new ideas with potential technological significance. Although the different Sections all share the same general theme -of fluctuational escape phenomena in model nonlinear optical systems -they deal with quite different aspects of the subject; each of them is therefore to a considerable extent self-contained (with Secs. 1.4 and 1.5 being exceptions, because they should be read after Sec. 1.3) and thus can be read almost independently of the others. Before considering particular systems, we review briefly the scientific context of the work and discuss in a general way the significance of escape phenomena in nonlinear optics.The investigation of fluctuations by means of analog or digital simulation is usually found most useful for those systems where the fluctuations of the quantities of immediate physical interest can be assumed to be due to noise. The latter perception of fluctuations goes back to Einstein, Smoluchowski, and Langevin [2, 3, 4] and has often been used in optics (cf. Refs. [5,6,7,8]). In nonlinear optics, the noise can be regarded as arising from two main sources. First there are internal fluctuations in the macroscopic system itself. These arise because spontaneous emission of light by individual atoms occurs at random, and because of fluctuations in the populations of atomic energy levels. The physical characteristics of such noise are usually closely related to the physical characteristics of the model that describes the "regular" dynamics of the system, i.e. in the absence of noise. In particular, the power spectrum of thermal noise and its intensity can be expressed in terms of the dissipation characteristics via the fluctuation-dissipation relations (cf. Ref. [9]) and, if the dissipation is non-retarded so that the corresponding dissipative forces (e.g. the friction force) depend only on the instantaneous values of dynamical variables, the noise power spectrum is independent of frequency, i.e. the noise is white. The model of noise as being white and Gaussian is one of the most commonly used in optics because the quantities of physical interest often vary slowly compared with the fast random processes that give rise to the noise, like emission or absorption of a photon [5,6,7,8]. The second very important source of noise is external: for example, fluctuations of the pump power in a laser. The physical characteristics of such noise naturally vary from one particular system to another; its correlation time is often much longer than that of the internal noise, and its effects can be larg...

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Detecting Unstable Periodic Orbits of Chaotic Dynamical Systems

Cited by 136 publications

References 12 publications

On the use of stabilizing transformations for detecting unstable periodic orbits in high-dimensional flows

On the use of stabilizing transformations for detecting unstable periodic orbits in high-dimensional flows

Adaptive Control of Chaos

Fluctuational Escape and Related Phenomena in Nonlinear Optical Systems

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