1
|
Speakman E, Gunaratne GH. On a kneading theory for gene-splicing. CHAOS (WOODBURY, N.Y.) 2024; 34:043125. [PMID: 38579148 DOI: 10.1063/5.0199364] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/22/2024] [Accepted: 03/05/2024] [Indexed: 04/07/2024]
Abstract
Two well-known facets in protein synthesis in eukaryotic cells are transcription of DNA to pre-RNA in the nucleus and the translation of messenger-RNA (mRNA) to proteins in the cytoplasm. A critical intermediate step is the removal of segments (introns) containing ∼97% of the nucleic-acid sites in pre-RNA and sequential alignment of the retained segments (exons) to form mRNA through a process referred to as splicing. Alternative forms of splicing enrich the proteome while abnormal splicing can enhance the likelihood of a cell developing cancer or other diseases. Mechanisms for splicing and origins of splicing errors are only partially deciphered. Our goal is to determine if rules on splicing can be inferred from data analytics on nucleic-acid sequences. Toward that end, we represent a nucleic-acid site as a point in a plane defined in terms of the anterior and posterior sub-sequences of the site. The "point-set" representation expands analytical approaches, including the use of statistical tools, to characterize genome sequences. It is found that point-sets for exons and introns are visually different, and that the differences can be quantified using a family of generalized moments. We design a machine-learning algorithm that can recognize individual exons or introns with 91% accuracy. Point-set distributions and generalized moments are found to differ between organisms.
Collapse
Affiliation(s)
- Ethan Speakman
- Department of Physics, University of Houston, Houston, Texas 77204, USA
| | | |
Collapse
|
2
|
Chai M, Lan Y. Symbolic partition in chaotic maps. CHAOS (WOODBURY, N.Y.) 2021; 31:033144. [PMID: 33810756 DOI: 10.1063/5.0042705] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/03/2021] [Accepted: 03/02/2021] [Indexed: 06/12/2023]
Abstract
In this work, we only use data on the unstable manifold to locate the partition boundaries by checking folding points at different levels, which practically coincide with homoclinic tangencies. The method is then applied to the classic two-dimensional Hénon map and a well-known three-dimensional map. Comparison with previous results is made in the Hénon case, and Lyapunov exponents are computed through the metric entropy based on the partition to show the validity of the current scheme.
Collapse
Affiliation(s)
- Misha Chai
- School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
| | - Yueheng Lan
- School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
| |
Collapse
|
3
|
Gonchenko S, Gonchenko A, Kazakov A, Samylina E. On discrete Lorenz-like attractors. CHAOS (WOODBURY, N.Y.) 2021; 31:023117. [PMID: 33653031 DOI: 10.1063/5.0037621] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/14/2020] [Accepted: 01/20/2021] [Indexed: 06/12/2023]
Abstract
We study geometrical and dynamical properties of the so-called discrete Lorenz-like attractors. We show that such robustly chaotic (pseudohyperbolic) attractors can appear as a result of universal bifurcation scenarios, for which we give a phenomenological description and demonstrate certain examples of their implementation in one-parameter families of three-dimensional Hénon-like maps. We pay special attention to such scenarios that can lead to period-2 Lorenz-like attractors. These attractors have very interesting dynamical properties and we show that their crises can lead, in turn, to the emergence of discrete Lorenz shape attractors of new types.
Collapse
Affiliation(s)
- Sergey Gonchenko
- Mathematical Center of Lobachevsky State University, 23 Prospekt Gagarina, 603950 Nizhny Novgorod, Russia
| | - Alexander Gonchenko
- Mathematical Center of Lobachevsky State University, 23 Prospekt Gagarina, 603950 Nizhny Novgorod, Russia
| | - Alexey Kazakov
- National Research University Higher School of Economics, 25/12 Bolshaya Pecherskaya Ulitsa, 603155 Nizhny Novgorod, Russia
| | - Evgeniya Samylina
- National Research University Higher School of Economics, 25/12 Bolshaya Pecherskaya Ulitsa, 603155 Nizhny Novgorod, Russia
| |
Collapse
|
4
|
Li J, Tomsovic S. Asymptotic relationship between homoclinic points and periodic orbit stability exponents. Phys Rev E 2019; 100:052202. [PMID: 31870019 DOI: 10.1103/physreve.100.052202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/02/2019] [Indexed: 06/10/2023]
Abstract
The magnitudes of the terms in periodic orbit semiclassical trace formulas are determined by the orbits' stability exponents. In this paper, we demonstrate a simple asymptotic relationship between those stability exponents and the phase-space positions of particular homoclinic points.
Collapse
Affiliation(s)
- Jizhou Li
- Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164-2814, USA
| | - Steven Tomsovic
- Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164-2814, USA
| |
Collapse
|
5
|
Li J, Tomsovic S. Exact decomposition of homoclinic orbit actions in chaotic systems: Information reduction. Phys Rev E 2019; 99:032212. [PMID: 30999433 DOI: 10.1103/physreve.99.032212] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/09/2018] [Indexed: 11/07/2022]
Abstract
Homoclinic and heteroclinic orbits provide a skeleton of the full dynamics of a chaotic dynamical system and are the foundation of semiclassical sums for quantum wave packets, coherent states, and transport quantities. Here, the homoclinic orbits are organized according to the complexity of their phase-space excursions, and exact relations are derived expressing the relative classical actions of complicated orbits as linear combinations of those with simpler excursions plus phase-space cell areas bounded by stable and unstable manifolds. The total number of homoclinic orbits increases exponentially with excursion complexity, and the corresponding cell areas decrease exponentially in size as well. With the specification of a desired precision, the exponentially proliferating set of homoclinic orbit actions is expressible by a slower-than-exponentially increasing set of cell areas, which may present a means for developing greatly simplified semiclassical formulas.
Collapse
Affiliation(s)
- Jizhou Li
- Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164-2814, USA
| | - Steven Tomsovic
- Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164-2814, USA
| |
Collapse
|
6
|
Wan X, Xu L. A study for multiscale information transfer measures based on conditional mutual information. PLoS One 2018; 13:e0208423. [PMID: 30521578 PMCID: PMC6283631 DOI: 10.1371/journal.pone.0208423] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/15/2018] [Accepted: 11/17/2018] [Indexed: 11/28/2022] Open
Abstract
As the big data science develops, efficient methods are demanded for various data analysis. Granger causality provides the prime model for quantifying causal interactions. However, this theoretic model does not meet the requirement for real-world data analysis, because real-world time series are diverse whose models are usually unknown. Therefore, model-free measures such as information transfer measures are strongly desired. Here, we propose the multi-scale extension of conditional mutual information measures using MORLET wavelet, which are named the WM and WPM. The proposed measures are computational efficient and interpret information transfer by multi-scales. We use both synthetic data and real-world examples to demonstrate the efficiency of the new methods. The results of the new methods are robust and reliable. Via the simulation studies, we found the new methods outperform the wavelet extension of transfer entropy (WTE) in both computational efficiency and accuracy. The features and properties of the proposed measures are also discussed.
Collapse
Affiliation(s)
- Xiaogeng Wan
- Department of Mathematics, College of Science, Beijing University of Chemical Technology, Beijing, China
- * E-mail:
| | - Lanxi Xu
- Department of Mathematics, College of Science, Beijing University of Chemical Technology, Beijing, China
| |
Collapse
|
7
|
Li J, Tomsovic S. Exact relations between homoclinic and periodic orbit actions in chaotic systems. Phys Rev E 2018; 97:022216. [PMID: 29548081 DOI: 10.1103/physreve.97.022216] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/15/2017] [Indexed: 11/07/2022]
Abstract
Homoclinic and unstable periodic orbits in chaotic systems play central roles in various semiclassical sum rules. The interferences between terms are governed by the action functions and Maslov indices. In this article, we identify geometric relations between homoclinic and unstable periodic orbits, and derive exact formulas expressing the periodic orbit classical actions in terms of corresponding homoclinic orbit actions plus certain phase space areas. The exact relations provide a basis for approximations of the periodic orbit actions as action differences between homoclinic orbits with well-estimated errors. This enables an explicit study of relations between periodic orbits, which results in an analytic expression for the action differences between long periodic orbits and their shadowing decomposed orbits in the cycle expansion.
Collapse
Affiliation(s)
- Jizhou Li
- Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164-2814, USA
| | - Steven Tomsovic
- Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164-2814, USA
| |
Collapse
|
8
|
Rubido N, Grebogi C, Baptista MS. Entropy-based generating Markov partitions for complex systems. CHAOS (WOODBURY, N.Y.) 2018; 28:033611. [PMID: 29604645 DOI: 10.1063/1.5002097] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
Finding the correct encoding for a generic dynamical system's trajectory is a complicated task: the symbolic sequence needs to preserve the invariant properties from the system's trajectory. In theory, the solution to this problem is found when a Generating Markov Partition (GMP) is obtained, which is only defined once the unstable and stable manifolds are known with infinite precision and for all times. However, these manifolds usually form highly convoluted Euclidean sets, are a priori unknown, and, as it happens in any real-world experiment, measurements are made with finite resolution and over a finite time-span. The task gets even more complicated if the system is a network composed of interacting dynamical units, namely, a high-dimensional complex system. Here, we tackle this task and solve it by defining a method to approximately construct GMPs for any complex system's finite-resolution and finite-time trajectory. We critically test our method on networks of coupled maps, encoding their trajectories into symbolic sequences. We show that these sequences are optimal because they minimise the information loss and also any spurious information added. Consequently, our method allows us to approximately calculate the invariant probability measures of complex systems from the observed data. Thus, we can efficiently define complexity measures that are applicable to a wide range of complex phenomena, such as the characterisation of brain activity from electroencephalogram signals measured at different brain regions or the characterisation of climate variability from temperature anomalies measured at different Earth regions.
Collapse
Affiliation(s)
- Nicolás Rubido
- Instituto de Física de Facultad de Ciencias (IFFC), Universidad de la República (UdelaR), Iguá 4225, Montevideo, Uruguay
| | - Celso Grebogi
- Institute for Complex Systems and Mathematical Biology (ICSMB), King's College, University of Aberdeen (UoA), AB24 3UE Aberdeen, United Kingdom
| | - Murilo S Baptista
- Institute for Complex Systems and Mathematical Biology (ICSMB), King's College, University of Aberdeen (UoA), AB24 3UE Aberdeen, United Kingdom
| |
Collapse
|
9
|
Lorimer T, Gomez F, Stoop R. Two universal physical principles shape the power-law statistics of real-world networks. Sci Rep 2015; 5:12353. [PMID: 26202858 PMCID: PMC4512011 DOI: 10.1038/srep12353] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/23/2015] [Accepted: 06/26/2015] [Indexed: 11/15/2022] Open
Abstract
The study of complex networks has pursued an understanding of macroscopic behaviour by focusing on power-laws in microscopic observables. Here, we uncover two universal fundamental physical principles that are at the basis of complex network generation. These principles together predict the generic emergence of deviations from ideal power laws, which were previously discussed away by reference to the thermodynamic limit. Our approach proposes a paradigm shift in the physics of complex networks, toward the use of power-law deviations to infer meso-scale structure from macroscopic observations.
Collapse
Affiliation(s)
- Tom Lorimer
- Institute of Neuroinformatics and Institute of Computational Science, University of Zurich and ETH Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
| | - Florian Gomez
- Institute of Neuroinformatics and Institute of Computational Science, University of Zurich and ETH Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
| | - Ruedi Stoop
- Institute of Neuroinformatics and Institute of Computational Science, University of Zurich and ETH Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
| |
Collapse
|
10
|
Martín JC. Encoding by control of the symbolic dynamics emitted by a chaotic laser. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:022914. [PMID: 25768576 DOI: 10.1103/physreve.91.022914] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/21/2014] [Indexed: 06/04/2023]
Abstract
Application to a chaotic erbium-doped fiber laser of the digital encoding technique by control of its emitted symbolic dynamics is numerically tested. Criteria to select the better working conditions and the perturbation to be introduced in any control parameter are proposed. Once they are chosen, the procedure to prepare the system for control and the way to carry it out are described. It is shown that the general method cannot be blindly applied, but it must be adapted to the particular case under analysis for a good performance. Finally, in relation to a possible experimental implementation, influence of noise in the bit error rate of the communication system is discussed.
Collapse
Affiliation(s)
- Juan Carlos Martín
- Department of Applied Physics and I3A, University of Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain
| |
Collapse
|
11
|
Roy S, Hua JC, Barnhill W, Gunaratne GH, Gord JR. Deconvolution of reacting-flow dynamics using proper orthogonal and dynamic mode decompositions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:013001. [PMID: 25679702 DOI: 10.1103/physreve.91.013001] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/27/2014] [Indexed: 06/04/2023]
Abstract
Analytical and computational studies of reacting flows are extremely challenging due in part to nonlinearities of the underlying system of equations and long-range coupling mediated by heat and pressure fluctuations. However, many dynamical features of the flow can be inferred through low-order models if the flow constituents (e.g., eddies or vortices) and their symmetries, as well as the interactions among constituents, are established. Modal decompositions of high-frequency, high-resolution imaging, such as measurements of species-concentration fields through planar laser-induced florescence and of velocity fields through particle-image velocimetry, are the first step in the process. A methodology is introduced for deducing the flow constituents and their dynamics following modal decomposition. Proper orthogonal (POD) and dynamic mode (DMD) decompositions of two classes of problems are performed and their strengths compared. The first problem involves a cellular state generated in a flat circular flame front through symmetry breaking. The state contains two rings of cells that rotate clockwise at different rates. Both POD and DMD can be used to deconvolve the state into the two rings. In POD the contribution of each mode to the flow is quantified using the energy. Each DMD mode can be associated with an energy as well as a unique complex growth rate. Dynamic modes with the same spatial symmetry but different growth rates are found to be combined into a single POD mode. Thus, a flow can be approximated by a smaller number of POD modes. On the other hand, DMD provides a more detailed resolution of the dynamics. Two classes of reacting flows behind symmetric bluff bodies are also analyzed. In the first, symmetric pairs of vortices are released periodically from the two ends of the bluff body. The second flow contains von Karman vortices also, with a vortex being shed from one end of the bluff body followed by a second shedding from the opposite end. The way in which DMD can be used to deconvolve the second flow into symmetric and von Karman vortices is demonstrated. The analyses performed illustrate two distinct advantages of DMD: (1) Unlike proper orthogonal modes, each dynamic mode is associated with a unique complex growth rate. By comparing DMD spectra from multiple nominally identical experiments, it is possible to identify "reproducible" modes in a flow. We also find that although most high-energy modes are reproducible, some are not common between experimental realizations; in the examples considered, energy fails to differentiate between reproducible and nonreproducible modes. Consequently, it may not be possible to differentiate reproducible and nonreproducible modes in POD. (2) Time-dependent coefficients of dynamic modes are complex. Even in noisy experimental data, the dynamics of the phase of these coefficients (but not their magnitude) are highly regular. The phase represents the angular position of a rotating ring of cells and quantifies the downstream displacement of vortices in reacting flows. Thus, it is suggested that the dynamical characterizations of complex flows are best made through the phase dynamics of reproducible DMD modes.
Collapse
Affiliation(s)
- Sukesh Roy
- Spectral Energies, LLC, Dayton, Ohio 45431, USA
| | - Jia-Chen Hua
- Department of Physics, University of Houston, Houston, Texas 77204, USA
| | - Will Barnhill
- Department of Physics, University of Houston, Houston, Texas 77204, USA
| | | | - James R Gord
- Aerospace Systems Directorate, Air Force Research Laboratory, WPAFB, Ohio 45433, USA
| |
Collapse
|
12
|
Gonzalez F, Jung C. A development scenario connecting the ternary symmetric horseshoe with the binary horseshoe. CHAOS (WOODBURY, N.Y.) 2014; 24:043141. [PMID: 25554061 DOI: 10.1063/1.4905007] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
It is explained in which way the ternary symmetric horseshoe can be obtained along a development scenario starting with a binary horseshoe. We explain the case of a complete ternary horseshoe in all detail and then give briefly some further incomplete cases. The key idea is to start with a three degrees of freedom system with a rotational symmetry, reduce the system with the help of the conserved angular momentum to one with two degrees of freedom where the value of the conserved angular momentum acts as a parameter and then let its value go to zero.
Collapse
Affiliation(s)
- F Gonzalez
- Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Av. Universidad s/n, 62251 Cuernavaca, Mexico
| | - C Jung
- Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Av. Universidad s/n, 62251 Cuernavaca, Mexico
| |
Collapse
|
13
|
Baesens C, MacKay RS. Analysis of a scenario for chaotic quantal slowing down of inspiration. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2013; 3:18. [PMID: 24040967 PMCID: PMC3848870 DOI: 10.1186/2190-8567-3-18] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 05/13/2013] [Accepted: 08/24/2013] [Indexed: 06/02/2023]
Abstract
On exposure to opiates, preparations from rat brain stems have been observed to continue to produce regular expiratory signals, but to fail to produce some inspiratory signals. The numbers of expirations between two successive inspirations form an apparently random sequence. Here, we propose an explanation based on the qualitative theory of dynamical systems. A relatively simple scenario for the dynamics of interaction between the generators of expiratory and inspiratory signals produces pseudo-random behaviour of the type observed.
Collapse
Affiliation(s)
- C Baesens
- Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK
| | - RS MacKay
- Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK
| |
Collapse
|
14
|
Hua JC, Gunaratne GH, Kostka S, Jiang N, Kiel BV, Gord JR, Roy S. Dynamical-systems analysis and unstable periodic orbits in reacting flows behind symmetric bluff bodies. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:033011. [PMID: 24125348 DOI: 10.1103/physreve.88.033011] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/21/2013] [Indexed: 06/02/2023]
Abstract
Dynamical systems analysis is performed for reacting flows stabilized behind four symmetric bluff bodies to determine the effects of shape on the nature of flame stability, acoustic coupling, and vortex shedding. The task requires separation of regular, repeatable aspects of the flow from experimental noise and highly irregular, nonrepeatable small-scale structures caused primarily by viscous-mediated energy cascading. The experimental systems are invariant under a reflection, and symmetric vortex shedding is observed throughout the parameter range. As the equivalence ratio-and, hence, acoustic coupling-is reduced, a symmetry-breaking transition to von Karman vortices is initiated. Combining principal-components analysis with a symmetry-based filtering, we construct bifurcation diagrams for the onset and growth of von Karman vortices. We also compute Lyapunov exponents for each flame holder to help quantify the transitions. Furthermore, we outline changes in the phase-space orbits that accompany the onset of von Karman vortex shedding and compute unstable periodic orbits (UPOs) embedded in the complex flows prior to and following the bifurcation. For each flame holder, we find a single UPO in flows without von Karman vortices and a pair of UPOs in flows with von Karman vortices. These periodic orbits organize the dynamics of the flow and can be used to reduce or control flow irregularities. By subtracting them from the overall flow, we are able to deduce the nature of irregular facets of the flows.
Collapse
Affiliation(s)
- Jia-Chen Hua
- Department of Physics, University of Houston, Houston, Texas 77204, USA
| | | | | | | | | | | | | |
Collapse
|
15
|
Narayanan S, Gunaratne GH, Hussain F. A dynamical systems approach to the control of chaotic dynamics in a spatiotemporal jet flow. CHAOS (WOODBURY, N.Y.) 2013; 23:033133. [PMID: 24089969 DOI: 10.1063/1.4820819] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/02/2023]
Abstract
We present a strategy for control of chaos in open flows and provide its experimental validation in the near field of a transitional jet flow system. The low-dimensional chaotic dynamics studied here results from vortex ring formation and their pairings over a spatially extended region of the flow that was excited by low level periodic forcing of the primary instability. The control method utilizes unstable periodic orbits (UPO) embedded within the chaotic attractor. Since hydrodynamic instabilities in the open flow system are convective, both monitoring and control can be implemented at a few locations, resulting in a simple and effective control algorithm. Experiments were performed in an incompressible, initially laminar, 4 cm diameter circular air jet, at a Reynolds number of 23,000, housed in a low-noise, large anechoic chamber. Distinct trajectory bundles surrounding the dominant UPOs were found from experimentally derived, time-delayed embedding of the chaotic attractor. Velocity traces from a pair of probes placed at the jet flow exit and farther downstream were used to empirically model the UPOs and compute control perturbations to be applied at the jet nozzle lip. Open loop control was used to sustain several nearly periodic states.
Collapse
Affiliation(s)
- Satish Narayanan
- Systems & Controls Engineering, United Technologies Corporation, East Hartford, Connecticut 06108, USA
| | | | | |
Collapse
|
16
|
Ma H, Lin W, Lai YC. Detecting unstable periodic orbits in high-dimensional chaotic systems from time series: reconstruction meeting with adaptation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:050901. [PMID: 23767476 DOI: 10.1103/physreve.87.050901] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/08/2013] [Indexed: 06/02/2023]
Abstract
Detecting unstable periodic orbits (UPOs) in chaotic systems based solely on time series is a fundamental but extremely challenging problem in nonlinear dynamics. Previous approaches were applicable but mostly for low-dimensional chaotic systems. We develop a framework, integrating approximation theory of neural networks and adaptive synchronization, to address the problem of time-series-based detection of UPOs in high-dimensional chaotic systems. An example of finding UPOs from the classic Mackey-Glass equation is presented.
Collapse
Affiliation(s)
- Huanfei Ma
- School of Mathematical Sciences, Soochow University, Suzhou 215006, China
| | | | | |
Collapse
|
17
|
beim Graben P, Potthast R. Inverse problems in dynamic cognitive modeling. CHAOS (WOODBURY, N.Y.) 2009; 19:015103. [PMID: 19335007 DOI: 10.1063/1.3097067] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
Inverse problems for dynamical system models of cognitive processes comprise the determination of synaptic weight matrices or kernel functions for neural networks or neural/dynamic field models, respectively. We introduce dynamic cognitive modeling as a three tier top-down approach where cognitive processes are first described as algorithms that operate on complex symbolic data structures. Second, symbolic expressions and operations are represented by states and transformations in abstract vector spaces. Third, prescribed trajectories through representation space are implemented in neurodynamical systems. We discuss the Amari equation for a neural/dynamic field theory as a special case and show that the kernel construction problem is particularly ill-posed. We suggest a Tikhonov-Hebbian learning method as regularization technique and demonstrate its validity and robustness for basic examples of cognitive computations.
Collapse
Affiliation(s)
- Peter beim Graben
- School of Psychology and Clinical Language Sciences, University of Reading, Reading, Berkshire, United Kingdom.
| | | |
Collapse
|
18
|
Beim Graben P, Gerth S, Vasishth S. Towards dynamical system models of language-related brain potentials. Cogn Neurodyn 2008; 2:229-55. [PMID: 19003488 PMCID: PMC2518748 DOI: 10.1007/s11571-008-9041-5] [Citation(s) in RCA: 32] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/12/2007] [Accepted: 03/24/2008] [Indexed: 11/25/2022] Open
Abstract
Event-related brain potentials (ERP) are important neural correlates of cognitive processes. In the domain of language processing, the N400 and P600 reflect lexical-semantic integration and syntactic processing problems, respectively. We suggest an interpretation of these markers in terms of dynamical system theory and present two nonlinear dynamical models for syntactic computations where different processing strategies correspond to functionally different regions in the system's phase space.
Collapse
Affiliation(s)
- Peter Beim Graben
- School of Psychology and Clinical Language Sciences, University of Reading, Whiteknights, PO Box 217, Reading, RG6 6AH, UK,
| | | | | |
Collapse
|
19
|
Lan Y, Cvitanović P. Unstable recurrent patterns in Kuramoto-Sivashinsky dynamics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:026208. [PMID: 18850922 DOI: 10.1103/physreve.78.026208] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/26/2008] [Indexed: 05/26/2023]
Abstract
We undertake an exploration of recurrent patterns in the antisymmetric subspace of the one-dimensional Kuramoto-Sivashinsky system. For a small but already rather "turbulent" system, the long-time dynamics takes place on a low-dimensional invariant manifold. A set of equilibria offers a coarse geometrical partition of this manifold. The Newton descent method enables us to determine numerically a large number of unstable spatiotemporally periodic solutions. The attracting set appears surprisingly thin-its backbone consists of several Smale horseshoe repellers, well approximated by intrinsic local one-dimensional return maps, each with an approximate symbolic dynamics. The dynamics appears decomposable into chaotic dynamics within such local repellers, interspersed by rapid jumps between them.
Collapse
Affiliation(s)
- Yueheng Lan
- Department of Mechanical and Environmental Engineering, University of California, Santa Barbara, California 93106, USA.
| | | |
Collapse
|
20
|
Carroll TL. Optimizing chaos-based signals for complex radar targets. CHAOS (WOODBURY, N.Y.) 2007; 17:033103. [PMID: 17902985 DOI: 10.1063/1.2751392] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/17/2023]
Abstract
There has been interest in the use of chaotic signals for radar, but most researchers consider only a few chaotic systems and how these signals perform for the detection of point targets. The range of possible chaotic signals is far greater than what most of these researchers consider, so to demonstrate this, I use a chaotic map whose parameters may be adjusted by a numerical optimization routine, producing different chaotic signals that are modulated onto a carrier and optimized for different situations. It is also suggested that any advantage for these chaos-based signals may come in the detection of complex targets, not point targets, and I compare the performance of chaos-based signals to a standard radar signal, the linear frequency modulated chirp. I find that I can optimize a chaos-based signal to increase the cross-correlation with the reflection from one complex target compared to the cross-correlation with the reflection from a different target, thus allowing the identification of a complex target. I am also able to increase the cross-correlation of the reflection from a complex target compared with the cross-correlation with the reflection from spatially extended clutter. I show that a larger output signal-to-noise ratio is possible if I cross-correlate with a reference signal that is different from the transmitted signal, and I justify my results by showing how the ambiguity diagram for a chaos-based signal can be different than the ambiguity diagram for a noise signal.
Collapse
Affiliation(s)
- T L Carroll
- U.S. Naval Research Laboratory, 4555 Overlook Avenue, SW, Washington, DC 20375, USA.
| |
Collapse
|
21
|
Pethel SD, Corron NJ, Bollt E. Symbolic dynamics of coupled map lattices. PHYSICAL REVIEW LETTERS 2006; 96:034105. [PMID: 16486708 DOI: 10.1103/physrevlett.96.034105] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/28/2005] [Indexed: 05/06/2023]
Abstract
We present a method to reduce the [FORMULA: SEE TEXT] dynamics of coupled map lattices (CMLs) of N invertibly coupled unimodal maps to a sequence of N-bit symbols. We claim that the symbolic description is complete and provides for the identification of all fixed points, periodic orbits, and dense orbits as well as an efficient representation for studying pattern formation in CMLs. We give our results for CMLs in terms of symbolic dynamical concepts well known for one-dimensional chaotic maps, including generating partitions, Gray orderings, and kneading sequences. An example utilizing coupled quadratic maps is given.
Collapse
Affiliation(s)
- Shawn D Pethel
- U.S. Army Aviation and Missile Command, AMSRD-AMR-WS-ST, Redstone Arsenal, Alabama 35898, USA
| | | | | |
Collapse
|
22
|
Emmanouilidou A, Jung C. Partitioning the phase space in a natural way for scattering systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:016219. [PMID: 16486270 DOI: 10.1103/physreve.73.016219] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/12/2005] [Indexed: 05/06/2023]
Abstract
In this paper, we demonstrate a recent procedure for the construction of a symbolic dynamics for open systems by applying it to a model potential, the driven inverted Gaussian, which has proven very useful in describing laser-atom interaction. The symbolic dynamics and the corresponding partition of the Poincaré map are natural from the point of view of an asymptotic observer since the resulting branching tree coincides with the one extracted from the scattering functions. In general, the whole procedure is approximate because it only describes the globally unstable part of the chaotic invariant set, that is, the part that can be seen by an asymptotic observer in scattering data. It ignores Kolmogorov-Arnold-Moser islands and their fractal surroundings.
Collapse
Affiliation(s)
- A Emmanouilidou
- Center for Nonlinear Science, School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430, USA
| | | |
Collapse
|
23
|
Jung C, Emmanouilidou A. Construction of a natural partition of incomplete horseshoes. CHAOS (WOODBURY, N.Y.) 2005; 15:23101. [PMID: 16035877 DOI: 10.1063/1.1859111] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/03/2023]
Abstract
We present a method for constructing a partition of an incomplete horseshoe in a Poincare map. The partition is based only on the unstable manifolds of the outermost fixed points and eventually their limits. Consequently, this partition becomes natural from the point of view of asymptotic scattering observations. The symbolic dynamics derived from this partition coincides with the one derived from the hierarchical structure of the singularities of the scattering functions.
Collapse
Affiliation(s)
- C Jung
- Centro de Ciencias Fisicas, UNAM, Apdo postal 48-3, 62251 Cuernavaca, Mexico
| | | |
Collapse
|
24
|
|
25
|
Emmanouilidou A, Jung C, Reichl LE. Classical scattering for a driven inverted Gaussian potential in terms of the chaotic invariant set. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:046207. [PMID: 14683035 DOI: 10.1103/physreve.68.046207] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/08/2003] [Indexed: 11/07/2022]
Abstract
We study the classical electron scattering from a driven inverted Gaussian potential, an open system, in terms of its chaotic invariant set. This chaotic invariant set is described by a ternary horseshoe construction on an appropriate Poincaré surface of section. We find the development parameters that describe the hyperbolic component of the chaotic invariant set. In addition, we show that the hierarchical structure of the fractal set of singularities of the scattering functions is the same as the structure of the chaotic invariant set. Finally, we construct a symbolic encoding of the hierarchical structure of the set of singularities of the scattering functions and use concepts from the thermodynamical formalism to obtain one of the measures of chaos of the fractal set of singularities, the topological entropy.
Collapse
Affiliation(s)
- A Emmanouilidou
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany
| | | | | |
Collapse
|
26
|
Onishi T, Shudo A, Ikeda KS, Takahashi K. Semiclassical study on tunneling processes via complex-domain chaos. ACTA ACUST UNITED AC 2003; 68:056211. [PMID: 14682875 DOI: 10.1103/physreve.68.056211] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/15/2002] [Revised: 05/20/2003] [Indexed: 11/07/2022]
Abstract
We investigate the semiclassical mechanism of tunneling processes in nonintegrable systems. The significant role of complex-phase-space chaos in the description of the tunneling processes is elucidated by studying a kicked scattering model. Behaviors of tunneling orbits are encoded into symbolic sequences based on the structure of a complex homoclinic tangle. By means of the symbolic coding, the phase space itineraries of tunneling orbits are related with the amounts of imaginary parts of actions gained by the orbits, so that the systematic search of dominant tunneling orbits becomes possible.
Collapse
Affiliation(s)
- T Onishi
- Department of Physics, Tokyo Metropolitan University, Minami-Ohsawa, Hachioji 192-0397, Japan.
| | | | | | | |
Collapse
|
27
|
Kennel MB, Buhl M. Estimating good discrete partitions from observed data: symbolic false nearest neighbors. PHYSICAL REVIEW LETTERS 2003; 91:084102. [PMID: 14525241 DOI: 10.1103/physrevlett.91.084102] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/01/2002] [Revised: 05/14/2003] [Indexed: 05/23/2023]
Abstract
A symbolic analysis of observed time series requires a discrete partition of a continuous state space containing the dynamics. A particular kind of partition, called "generating," preserves all deterministic dynamical information in the symbolic representation, but such partitions are not obvious beyond one dimension. Existing methods to find them require significant knowledge of the dynamical evolution operator. We introduce a statistic and algorithm to refine empirical partitions for symbolic state reconstruction. This method optimizes an essential property of a generating partition, avoiding topological degeneracies, by minimizing the number of "symbolic false nearest neighbors." It requires only the observed time series and is sensible even in the presence of noise when no truly generating partition is possible.
Collapse
Affiliation(s)
- Matthew B Kennel
- Institute For Nonlinear Science, University of California-San Diego, La Jolla, CA 92093-0402, USA.
| | | |
Collapse
|
28
|
Tirnakli U. Two-dimensional maps at the edge of chaos: numerical results for the Henon map. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:066212. [PMID: 12513389 DOI: 10.1103/physreve.66.066212] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/26/2002] [Revised: 08/06/2002] [Indexed: 05/24/2023]
Abstract
The mixing properties (or sensitivity to initial conditions) of the two-dimensional Henon map have been explored numerically at the edge of chaos. Three independent methods, which have been developed and used so far for one-dimensional maps, have been used to accomplish this task. These methods are (i) the measure of the divergence of initially nearby orbits, (ii) analysis of the multifractal spectrum, and (iii) computation of nonextensive entropy increase rates. The results obtained closely agree with those of the one-dimensional cases and constitute a verification of this scenario in two-dimensional maps. This obviously makes the idea of weak chaos even more robust.
Collapse
Affiliation(s)
- Ugur Tirnakli
- Department of Physics, Faculty of Science, Ege University, 35100 Izmir, Turkey.
| |
Collapse
|
29
|
Collins P, Krauskopf B. Entropy and bifurcations in a chaotic laser. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:056201. [PMID: 12513580 DOI: 10.1103/physreve.66.056201] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/21/2002] [Indexed: 05/24/2023]
Abstract
We compute bounds on the topological entropy associated with a chaotic attractor of a semiconductor laser with optical injection. We consider the Poincaré return map to a fixed plane, and are able to compute the stable and unstable manifolds of periodic points globally, even though it is impossible to find a plane on which the Poincaré map is globally smoothly defined. In this way, we obtain the information that forms the input of the entropy calculations, and characterize the boundary crisis in which the chaotic attractor is destroyed. This boundary crisis involves a periodic point with negative eigenvalues, and the entropy associated with the chaotic attractor persists in a chaotic saddle after the bifurcation.
Collapse
Affiliation(s)
- Pieter Collins
- Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, United Kingdom.
| | | |
Collapse
|
30
|
Pingel D, Schmelcher P, Diakonos FK. Detecting unstable periodic orbits in chaotic continuous-time dynamical systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:026214. [PMID: 11497684 DOI: 10.1103/physreve.64.026214] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/19/2001] [Indexed: 05/23/2023]
Abstract
We extend the recently developed method for detecting unstable periodic points of chaotic time-discrete dynamical systems to find unstable periodic orbits in time-continuous systems, given by a set of ordinary differential equations. This is achieved by the reduction of the continuous flow to a Poincaré map which is then searched for periodic points. The algorithm has global convergence properties and needs no a priori knowledge of the system. It works well for both dissipative and Hamiltonian dynamical systems which is demonstrated by exploring the Lorenz system and the hydrogen atom in a strong magnetic field. The advantages and general features of the approach are discussed in detail.
Collapse
Affiliation(s)
- D Pingel
- Theoretical Chemistry, Institute for Physical Chemistry, Im Neuenheimer Feld 229, University of Heidelberg, Germany.
| | | | | |
Collapse
|
31
|
Buljan H, Paar V. Many-hole interactions and the average lifetimes of chaotic transients that precede controlled periodic motion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:066205. [PMID: 11415204 DOI: 10.1103/physreve.63.066205] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/15/2000] [Indexed: 05/23/2023]
Abstract
We consider n small regions (referred to as the holes) on a chaotic attractor and study the average lifetime it takes for a randomly initiated trajectory to land in their union. The holes are thought of as n possible escape routes for the trajectory. The escape route through one of the holes may be considerably reduced by other holes, depending on their positions. This effect, referred to as shadowing, can significantly prolong the average lifetime. The main result of this paper is the construction and analysis (numerical and theoretical) of the many-hole interactions. They are interpreted as the amount of shadowing between the holes. The "effective range" of these interactions is associated with the largest Lyapunov exponent. The shadowing effect is shown to be very large when the holes are located on n points of an unstable periodic orbit. Considerable attention is paid to this case since it is of interest to the field of controlling chaos.
Collapse
Affiliation(s)
- H Buljan
- Department of Physics, Faculty of Science, University of Zagreb, 10000 Zagreb, Croatia
| | | |
Collapse
|
32
|
Kobes R, Liu J, Peles S. Analysis of a parametrically driven pendulum. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:036219. [PMID: 11308753 DOI: 10.1103/physreve.63.036219] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/18/2000] [Revised: 11/13/2000] [Indexed: 05/23/2023]
Abstract
We study in this paper the behavior of a periodically driven nonlinear mechanical system. Bifurcation diagrams are found which locate regions of quasiperiodic, periodic, and chaotic behavior within the parameter space of the system. We also conduct a symbolic analysis of the model which demonstrates that the symbolic dynamics of two-dimensional maps can be applied effectively to the study of ordinary differential equations in order to gain global knowledge about them.
Collapse
Affiliation(s)
- R Kobes
- Department of Physics, University of Winnipeg, Winnipeg, Manitoba, Canada R3B 2E9.
| | | | | |
Collapse
|
33
|
Bollt EM, Stanford T, Lai YC, Zyczkowski K. Validity of threshold-crossing analysis of symbolic dynamics from chaotic time series. PHYSICAL REVIEW LETTERS 2000; 85:3524-3527. [PMID: 11030937 DOI: 10.1103/physrevlett.85.3524] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/04/2000] [Revised: 08/07/2000] [Indexed: 05/23/2023]
Abstract
A practical and popular technique to extract the symbolic dynamics from experimentally measured chaotic time series is the threshold-crossing method, by which an arbitrary partition is utilized for determining the symbols. We address to what extent the symbolic dynamics so obtained can faithfully represent the phase-space dynamics. Our principal result is that such a practice can lead to a severe misrepresentation of the dynamical system. The measured topological entropy is a Devil's staircase-like, but surprisingly nonmonotone, function of a parameter characterizing the amount of misplacement of the partition.
Collapse
Affiliation(s)
- EM Bollt
- Mathematics Department, 572 Holloway Road, U.S. Naval Academy, Annapolis, Maryland 21402-5002, USA
| | | | | | | |
Collapse
|
34
|
Paar V, Buljan H. Bursts in the chaotic trajectory lifetimes preceding controlled periodic motion. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 62:4869-72. [PMID: 11089032 DOI: 10.1103/physreve.62.4869] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/09/2000] [Indexed: 11/07/2022]
Abstract
The average lifetime [tau(H)] it takes for a randomly started trajectory to land in a small region (H) on a chaotic attractor is studied. tau(H) is an important issue for controlling chaos. We point out that if the region H is visited by a short periodic orbit, the lifetime tau(H) strongly deviates from the inverse of the naturally invariant measure contained within that region [&mgr;(N)(H)(-1)]. We introduce the formula that relates tau(H)/&mgr;(N)(H)(-1) to the expanding eigenvalue of the short periodic orbit visiting H.
Collapse
Affiliation(s)
- V Paar
- Department of Physics, Faculty of Science, University of Zagreb, 10000 Zagreb, Croatia
| | | |
Collapse
|
35
|
Pingel D, Schmelcher P, Diakonos FK, Biham O. Theory and applications of the systematic detection of unstable periodic orbits in dynamical systems. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 62:2119-2134. [PMID: 11088678 DOI: 10.1103/physreve.62.2119] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/26/2000] [Indexed: 05/23/2023]
Abstract
A topological approach and understanding to the detection of unstable periodic orbits based on a recently proposed method [Phys. Rev. Lett. 78, 4733 (1997)] is developed. This approach provides a classification of the set of transformations necessary for finding the orbits. Applications to the Ikeda and Henon map are performed, allowing a study of the distributions of Lyapunov exponents for high periods. In particular, the properties of the least unstable orbits up to period 36 are investigated and discussed.
Collapse
Affiliation(s)
- D Pingel
- Theoretical Chemistry, Institute for Physical Chemistry, INF 229, University of Heidelberg, 69120 Heidelberg, Germany
| | | | | | | |
Collapse
|
36
|
Davidchack RL, Lai YC, Bollt EM, Dhamala M. Estimating generating partitions of chaotic systems by unstable periodic orbits. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 61:1353-1356. [PMID: 11046413 DOI: 10.1103/physreve.61.1353] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/27/1999] [Indexed: 05/23/2023]
Abstract
An outstanding problem in chaotic dynamics is to specify generating partitions for symbolic dynamics in dimensions larger than 1. It has been known that the infinite number of unstable periodic orbits embedded in the chaotic invariant set provides sufficient information for estimating the generating partition. Here we present a general, dimension-independent, and efficient approach for this task based on optimizing a set of proximity functions defined with respect to periodic orbits. Our algorithm allows us to obtain the approximate location of the generating partition for the Ikeda-Hammel-Jones-Moloney map.
Collapse
Affiliation(s)
- RL Davidchack
- Department of Physics and Astronomy, University of Kansas, Lawrence, Kansas 66045, USA
| | | | | | | |
Collapse
|
37
|
Lipp C, Jung C. From scattering singularities to the partition of a horseshoe. CHAOS (WOODBURY, N.Y.) 1999; 9:706-714. [PMID: 12779867 DOI: 10.1063/1.166445] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
In a chaotic scattering system there are two different approaches to construct a symbolic dynamics. One comes from the branching tree obtained from a scattering function. The other comes from a Markov partition based on the line of primary homoclinic tangencies in the Poincare map taken in the interaction region. In general the two results only coincide for a complete horseshoe. We show how to make a different choice for the partition in the internal Poincare section based on scattering behavior and not on homoclinic tangencies. Then the corresponding symbolic dynamics coincides also for the incomplete case with the one obtained naturally from the scattering functions. The scattering based partition lines of the horseshoe are constructed by an iterative procedure. (c) 1999 American Institute of Physics.
Collapse
Affiliation(s)
- C. Lipp
- Institut fur theoretische Physik, Universitat Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland
| | | |
Collapse
|
38
|
|
39
|
Lipp C, Jung C. A degenerate bifurcation to chaotic scattering in a multicentre potential. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/28/23/029] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
|
40
|
Ruckerl B, Jung C. Scaling properties of a scattering system with an incomplete horseshoe. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/27/1/005] [Citation(s) in RCA: 57] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
|
41
|
|
42
|
Ruckerl B, Jung C. Hierarchical structure in the chaotic scattering off a magnetic dipole. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/27/20/014] [Citation(s) in RCA: 33] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
|
43
|
Bäcker A, Dullin HR. Symbolic dynamics and periodic orbits for the cardioid billiard. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/30/6/023] [Citation(s) in RCA: 28] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
|
44
|
|
45
|
Hansen KT, Kohler A. Chaotic scattering through potentials with rainbow singularities. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:6214-6225. [PMID: 9965841 DOI: 10.1103/physreve.54.6214] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
|
46
|
Letellier C, Gouesbet G, Soufi F, Buchler JR, Kollath Z. Chaos in variable stars: Topological analysis of W Vir model pulsations. CHAOS (WOODBURY, N.Y.) 1996; 6:466-476. [PMID: 12780277 DOI: 10.1063/1.166189] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
The topological characterization of chaos is applied to the irregular pulsations of a model for a star of the W Virginis type, computed with a state-of-the-art numerical hydrodynamical code. The banded W Vir attractor is found to possess an additional twist when compared to the Rossler band. It is shown that the stellar light-curve contains the same dynamical information about the attractor as the stellar radius or as the radial velocity variations. (c) 1996 American Institute of Physics.
Collapse
Affiliation(s)
- C. Letellier
- LESP, URA CNRS 230, INSA de Rouen, BP 08, 76 130 Mont-Saint-Aignan, FranceDASGAL, Observatoire de Paris, Place Jules Jansen, 92190 Meudon, FrancePhysics Department, University of Florida, Gainesville, Florida 32611
| | | | | | | | | |
Collapse
|
47
|
Saraceno M, Vallejos RO. The quantized D-transformation. CHAOS (WOODBURY, N.Y.) 1996; 6:193-199. [PMID: 12780247 DOI: 10.1063/1.166164] [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
We construct a new example of a quantum map, the quantized version of the D-transformation, which is the natural extension to two dimensions of the tent map. The classical, quantum and semiclassical behavior is studied. We also exhibit some relationships between the quantum versions of the D-map and the parity projected baker's map. The method of construction allows a generalization to dissipative maps which includes the quantization of a horseshoe. (c) 1996 American Institute of Physics.
Collapse
Affiliation(s)
- M. Saraceno
- Departamento de Fisica, Comision Nacional de Energia Atomica, Av. del Libertador 8250, 1429 Buenos Aires, Argentina
| | | |
Collapse
|
48
|
Pollner P, Vattay G. New method for computing topological pressure. PHYSICAL REVIEW LETTERS 1996; 76:4155-4158. [PMID: 10061215 DOI: 10.1103/physrevlett.76.4155] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
|
49
|
Dana I, Kalisky T. Symbolic dynamics for strong chaos on stochastic webs: General quasisymmetry. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 53:R2025-R2028. [PMID: 9964598 DOI: 10.1103/physreve.53.r2025] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
|
50
|
Wu Z. Symbolic dynamics analysis of chaotic time series with a driven frequency. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 53:1446-1452. [PMID: 9964405 DOI: 10.1103/physreve.53.1446] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
|