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Chapman DM, Burton EKT, Hall JR, Rosenberger AT, Bandy DK. Characteristics of coexisting attractors and ghost orbiting in an optomechanical microresonator. CHAOS (WOODBURY, N.Y.) 2024; 34:043128. [PMID: 38587537 DOI: 10.1063/5.0201717] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/31/2024] [Accepted: 03/21/2024] [Indexed: 04/09/2024]
Abstract
We explore the nonlinear interactions of an optomechanical microresonator driven by two external optical signals. Optical whispering-gallery waves are coupled to acoustic surface waves of a fused silica medium in the equatorial plane of a generic microresonator. The system exhibits coexisting attractors whose behaviors include limit cycles, steady states, tori, quasi-chaos, and fully developed chaos with ghost orbits of a known attractor. Bifurcation diagrams demonstrate the existence of self-similarity, periodic windows, and coexisting attractors and show high-density lines within chaos that suggests a potential ghost orbit. In addition, the Lyapunov spectral components as a function of control parameter illuminate the dynamic nature of attractors and periodic windows with symmetric and asymmetric formations, their domains of existence, their bifurcations, and other nonlinear effects. We show that the power-shift method can access accurately and efficiently attractors in the optomechanical system as it does in other nonlinear systems. To test whether the ghost orbit is the link between two attractors interrupted by chaos, we examine the elements of the bifurcation diagrams as a function of control parameter. We also use detuning as a second control parameter to avoid the chaotic region and clarify that the two attractors are one.
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Affiliation(s)
- D M Chapman
- Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA
| | - E K T Burton
- Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA
| | - J R Hall
- Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA
| | - A T Rosenberger
- Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA
| | - D K Bandy
- Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA
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Bandy DK, Burton EKT, Hall JR, Chapman DM, Elrod JT. Predicting attractor characteristics using Lyapunov exponents in a laser with injected signal. CHAOS (WOODBURY, N.Y.) 2021; 31:013120. [PMID: 33754793 DOI: 10.1063/5.0018586] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/16/2020] [Accepted: 12/11/2020] [Indexed: 06/12/2023]
Abstract
Coexisting attractors are studied in a single-mode coherent model of a laser with an injected signal. We report that every attractor has a unique Lyapunov exponent (LE) pattern that is choreographed by the subtle variations in the attractor's dynamics and circumscribed by a common Lyapunov spectral pattern that begins and ends with two-zero LEs. Lyapunov spectra form symmetric-like and asymmetric bubbles; the former foreshadows an attractor's proximity to the cusp of an eminent change in dynamics and the latter indicates the presence of a bifurcation. We show that the peak values of the asymmetric bubbles are always associated with two-zero LEs; in fact, they are allied inseparably in forecasting period-doubling episodes. The two-zero LEs' predictor of torus dynamics is refined to include the convergence of three LEs to a triplet of zeros as a precursor to the two-zero spectra. We report that the long-standing two-zero LEs' signature is a necessary but not sufficient condition for predicting attractors and their dynamic conditions. The evolution of the attractor volume as a function of the injected signal is compared to the spectral formation of the attractor; we report slope changes and points of inflections in the volume trajectory where spectral changes indicate dynamic changes. Attractor viability is tested preliminarily by including random low-level noise in the frequency of the injected signal.
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Affiliation(s)
- D K Bandy
- Physics Department, Oklahoma State University, Stillwater, Oklahoma 74078, USA
| | - E K T Burton
- Physics Department, Oklahoma State University, Stillwater, Oklahoma 74078, USA
| | - J R Hall
- Physics Department, Oklahoma State University, Stillwater, Oklahoma 74078, USA
| | - D M Chapman
- Physics Department, Oklahoma State University, Stillwater, Oklahoma 74078, USA
| | - J T Elrod
- Physics Department, Oklahoma State University, Stillwater, Oklahoma 74078, USA
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Chávez CAT, Curilef S. Discontinuous spirals of stability in an optically injected semiconductor laser. CHAOS (WOODBURY, N.Y.) 2020; 30:053107. [PMID: 32491898 DOI: 10.1063/1.5119808] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/12/2019] [Accepted: 04/09/2020] [Indexed: 06/11/2023]
Abstract
We report a new kind of discontinuous spiral with stable periodic orbits in the parameter space of an optically injected semiconductor laser model, which is a combination of the intercalation of fish-like and cuspidal-like structures (the two normal forms of complex cubic dynamics). The spiral has a tridimensional structure that rolls up in at least three directions. A turn of approximately 2π radians along the spiral and toward the center increases the number of peaks in the laser intensity by one, which does not occur when traversing the discontinuities. We show that as we vary the linewidth enhancement factor (α), discontinuities are created (destroyed) through disaggregation (collapses) from (into) the so-called shrimp-like structures. Future experimental verification and applications, as well as theoretical studies to explain its origin and relation with homoclinic spirals that exist in its neighborhood, are needed.
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Affiliation(s)
- C Abraham T Chávez
- Instituto de Física, Universidade Federal do Rio Grande do Sul, 91501-970 Porto Alegre, Brazil
| | - Sergio Curilef
- Departamento de Física, Universidad Católica del Norte, Avenida Angamos 0610, 1270709 Antofagasta, Chile
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Rodrigues CDS, dos Santos CGP, de Miranda RCC, Parma E, Varela H, Nagao R. A numerical investigation of the effect of external resistance and applied potential on the distribution of periodicity and chaos in the anodic dissolution of nickel. Phys Chem Chem Phys 2020; 22:21823-21834. [DOI: 10.1039/d0cp04238b] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/24/2022]
Abstract
Low density, elongation, and suppression of the shrimp-like structures in the resistance-potential phase diagrams have been observed in the oscillatory dissolution of nickel.
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Affiliation(s)
| | | | | | - Eduardo Parma
- Institute of Chemistry
- University of Campinas
- 13083-970 Campinas
- Brazil
| | - Hamilton Varela
- Institute of Chemistry of São Carlos
- University of São Paulo
- 13560-970 São Carlos
- Brazil
- School of Earth Sciences and Environmental Engineering
| | - Raphael Nagao
- Institute of Chemistry
- University of Campinas
- 13083-970 Campinas
- Brazil
- Center for Innovation on New Energies
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da Silva RM, Manchein C, Beims MW. Optimal ratchet current for elastically interacting particles. CHAOS (WOODBURY, N.Y.) 2019; 29:111101. [PMID: 31779347 DOI: 10.1063/1.5127925] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/14/2019] [Accepted: 10/21/2019] [Indexed: 06/10/2023]
Abstract
In this work, we show that optimal ratchet currents of two interacting particles are obtained when stable periodic motion is present. By increasing the coupling strength between identical ratchet maps, it is possible to find, for some parametric combinations, current reversals, hyperchaos, multistability, and duplication of the periodic motion in the parameter space. Besides that, by setting a fixed value for the current of one ratchet, it is possible to induce a positive/negative/null current for the whole system in certain domains of the parameter space.
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Affiliation(s)
- Rafael M da Silva
- Departamento de Física, Universidade Federal da Paraíba, 58051-900 João Pessoa, PB, Brazil
| | - Cesar Manchein
- Departamento de Física, Universidade do Estado de Santa Catarina, 89219-710 Joinville, SC, Brazil
| | - Marcus W Beims
- Departamento de Física, Universidade Federal do Paraná, 81531-980 Curitiba, PR, Brazil
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Manchein C, da Silva RM, Beims MW. Proliferation of stability in phase and parameter spaces of nonlinear systems. CHAOS (WOODBURY, N.Y.) 2017; 27:081101. [PMID: 28863504 DOI: 10.1063/1.4994329] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
In this work, we show how the composition of maps allows us to multiply, enlarge, and move stable domains in phase and parameter spaces of discrete nonlinear systems. Using Hénon maps with distinct parameters, we generate many identical copies of isoperiodic stable structures (ISSs) in the parameter space and attractors in phase space. The equivalence of the identical ISSs is checked by the largest Lyapunov exponent analysis, and the multiplied basins of attraction become riddled. Our proliferation procedure should be applicable to any two-dimensional nonlinear system.
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Affiliation(s)
- Cesar Manchein
- Departamento de Física, Universidade do Estado de Santa Catarina, 89219-710 Joinville, Santa Catarina, Brazil
| | - Rafael M da Silva
- Departamento de Física, Universidade Federal do Paraná, 81531-980 Curitiba, Paraná, Brazil
| | - Marcus W Beims
- Departamento de Física, Universidade Federal do Paraná, 81531-980 Curitiba, Paraná, Brazil
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Savin DV, Kuznetsov AP, Savin AV, Feudel U. Different types of critical behavior in conservatively coupled Hénon maps. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:062905. [PMID: 26172770 DOI: 10.1103/physreve.91.062905] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/30/2014] [Indexed: 06/04/2023]
Abstract
We study the dynamics of two conservatively coupled Hénon maps at different levels of dissipation. It is shown that the decrease of dissipation leads to changes in the structure of the parameter plane and the scenarios of transition to chaos compared to the case of infinitely strong dissipation. Particularly, the Feigenbaum line becomes divided into several fragments. Some of these fragments have critical points of different types, namely, of C and H type, as their terminal points. Also the mechanisms of formation of these Feigenbaum line ruptures are described.
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Affiliation(s)
- Dmitry V Savin
- Department of Nonlinear Processes, Chernyshevsky Saratov State University, Astrakhanskaya Street 83, 410012, Saratov, Russia
| | - Alexander P Kuznetsov
- Department of Nonlinear Processes, Chernyshevsky Saratov State University, Astrakhanskaya Street 83, 410012, Saratov, Russia
- Kotel'nikov Institute of Radioengineering and Electronics of RAS, Saratov Branch, Zelenaya Street 38, 410019, Saratov, Russia
| | - Alexey V Savin
- Department of Nonlinear Processes, Chernyshevsky Saratov State University, Astrakhanskaya Street 83, 410012, Saratov, Russia
| | - Ulrike Feudel
- Institute for Chemistry and Biology of the Marine Environment, Carl von Ossietzky University Oldenburg, Carl von Ossietzky Street 9-11, D-26111, Oldenburg, Germany
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Nascimento MA, Nagao R, Eiswirth M, Varela H. Coupled slow and fast surface dynamics in an electrocatalytic oscillator: Model and simulations. J Chem Phys 2014; 141:234701. [DOI: 10.1063/1.4903172] [Citation(s) in RCA: 24] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Affiliation(s)
- Melke A. Nascimento
- Institute of Chemistry of São Carlos, University of São Paulo, PO Box 780, 13560-970, São Carlos, SP, Brazil
- Fritz Haber Institute of the Max Planck Society, Department of Physical Chemistry, Faradayweg 4-6, D-14195 Berlin, Germany
| | - Raphael Nagao
- Institute of Chemistry of São Carlos, University of São Paulo, PO Box 780, 13560-970, São Carlos, SP, Brazil
| | - Markus Eiswirth
- Fritz Haber Institute of the Max Planck Society, Department of Physical Chemistry, Faradayweg 4-6, D-14195 Berlin, Germany
- Ertl Center for Electrochemistry and Catalysis, GIST, Cheomdan-gwagiro 261, Buk-gu, Gwangju 500-712, South Korea
| | - Hamilton Varela
- Institute of Chemistry of São Carlos, University of São Paulo, PO Box 780, 13560-970, São Carlos, SP, Brazil
- Fritz Haber Institute of the Max Planck Society, Department of Physical Chemistry, Faradayweg 4-6, D-14195 Berlin, Germany
- Ertl Center for Electrochemistry and Catalysis, GIST, Cheomdan-gwagiro 261, Buk-gu, Gwangju 500-712, South Korea
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Doedel EJ, Pando L CL. Multiparameter bifurcations and mixed-mode oscillations in Q-switched CO2 lasers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:052904. [PMID: 25353858 DOI: 10.1103/physreve.89.052904] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/10/2013] [Indexed: 06/04/2023]
Abstract
We study the origin of mixed-mode oscillations and related bifurcations in a fully molecular laser model that describes CO2 monomode lasers with a slow saturable absorber. Our study indicates that the presence of isolas of periodic mixed-mode oscillations, as the pump parameter and the cavity-frequency detuning change, is inherent to Q-switched CO2 monomode lasers. We compare this model, known as the dual four-level model, to the more conventional 3:2 model and to a CO2 laser model for fast saturable absorbers. In these models, we find similarities as well as qualitative differences, such as the different nature of the homoclinic tangency to a relevant unstable periodic orbit, where the Gavrilov-Shilnikov theory and its extensions may hold. We also show that there are isolas of periodic mixed-mode oscillations in a model for CO2 lasers with modulated losses, as the pump parameter varies. The coarse-grained bifurcation diagrams of the periodic mixed-mode oscillations in these models suggest that these oscillations belong to similar classes.
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Affiliation(s)
- Eusebius J Doedel
- Department of Computer Science, Concordia University, 1455 Boulevard de Maisonneuve Ouest, Montréal, Québec, Canada H3G 1M8
| | - Carlos L Pando L
- Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla 72570, México
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10
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Manchein C, Celestino A, Beims MW. Temperature resistant optimal ratchet transport. PHYSICAL REVIEW LETTERS 2013; 110:114102. [PMID: 25166540 DOI: 10.1103/physrevlett.110.114102] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/08/2012] [Indexed: 06/03/2023]
Abstract
Stable periodic structures containing optimal ratchet transport, recently found in the parameter space dissipation versus ratchet parameter by [A. Celestino et al. Phys. Rev. Lett. 106, 234101 (2011)], are shown to be resistant to reasonable temperatures, reinforcing the expectation that they are essential to explain the optimal ratchet transport in nature. Critical temperatures for their destruction, valid from the overdamping to close to the conservative limits, are obtained numerically and shown to be connected to the current efficiency, given here analytically. A region where thermal activation of the rachet current takes place is also found, and its underlying mechanism is unveiled. Results are demonstrated for a discrete ratchet model and generalized to the Langevin equation with an additional external oscillating force.
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Affiliation(s)
- C Manchein
- Departamento de Física, Universidade do Estado de Santa Catarina, 89219-710 Joinville, Brazil
| | - A Celestino
- Departamento de Física, Universidade Federal do Paraná, 81531-980 Curitiba, Brazil
| | - M W Beims
- Departamento de Física, Universidade Federal do Paraná, 81531-980 Curitiba, Brazil and Max-Planck-Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany
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11
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Toomey JP, Nichkawde C, Kane DM, Schires K, Henning ID, Hurtado A, Adams MJ. Stability of the nonlinear dynamics of an optically injected VCSEL. OPTICS EXPRESS 2012; 20:10256-10270. [PMID: 22535116 DOI: 10.1364/oe.20.010256] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
Automated protocols have been developed to characterize time series data in terms of stability. These techniques are applied to the output power time series of an optically injected vertical cavity surface emitting laser (VCSEL) subject to varying injection strength and optical frequency detuning between master and slave lasers. Dynamic maps, generated from high resolution, computer controlled experiments, identify regions of dynamic instability in the parameter space.
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Affiliation(s)
- J P Toomey
- Department of Physics and Astronomy, Macquarie University, Sydney NSW 2109 Australia.
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12
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Pereira FAC, Colli E, Sartorelli JC. Period adding cascades: experiment and modeling in air bubbling. CHAOS (WOODBURY, N.Y.) 2012; 22:013135. [PMID: 22463011 DOI: 10.1063/1.3695345] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
Period adding cascades have been observed experimentally/numerically in the dynamics of neurons and pancreatic cells, lasers, electric circuits, chemical reactions, oceanic internal waves, and also in air bubbling. We show that the period adding cascades appearing in bubbling from a nozzle submerged in a viscous liquid can be reproduced by a simple model, based on some hydrodynamical principles, dealing with the time evolution of two variables, bubble position and pressure of the air chamber, through a system of differential equations with a rule of detachment based on force balance. The model further reduces to an iterating one-dimensional map giving the pressures at the detachments, where time between bubbles come out as an observable of the dynamics. The model has not only good agreement with experimental data, but is also able to predict the influence of the main parameters involved, like the length of the hose connecting the air supplier with the needle, the needle radius and the needle length.
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Doedel EJ, Pando CLL. Isolas of periodic passive Q-switching self-pulsations in the three-level:two-level model for a laser with a saturable absorber. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:056207. [PMID: 22181484 DOI: 10.1103/physreve.84.056207] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/05/2011] [Revised: 07/07/2011] [Indexed: 05/31/2023]
Abstract
We show that a fundamental feature of the three-level:two-level model, used to describe molecular monomode lasers with a saturable absorber, is the existence of isolas of periodic passive Q-switching (PQS) self-pulsations. A common feature of these closed families of periodic solutions is that they contain regions of stability of the PQS self-pulsation bordered by period-doubling and fold bifurcations, when the control parameter is either the incoherent external pump or the cavity frequency detuning. These findings unveil the fundamental solution structure that is at the origin of the phenomenon known as "period-adding cascades" in our system. Using numerical continuation techniques we determine these isolas systematically, as well as the changes they undergo as secondary parameters are varied.
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Affiliation(s)
- Eusebius J Doedel
- Department of Computer Science, Concordia University, Montréal, Québec, Canada
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15
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Freire JG, Gallas JAC. Stern–Brocot trees in the periodicity of mixed-mode oscillations. Phys Chem Chem Phys 2011; 13:12191-8. [DOI: 10.1039/c0cp02776f] [Citation(s) in RCA: 74] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
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16
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Zou Y, Donner RV, Donges JF, Marwan N, Kurths J. Identifying complex periodic windows in continuous-time dynamical systems using recurrence-based methods. CHAOS (WOODBURY, N.Y.) 2010; 20:043130. [PMID: 21198100 DOI: 10.1063/1.3523304] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
The identification of complex periodic windows in the two-dimensional parameter space of certain dynamical systems has recently attracted considerable interest. While for discrete systems, a discrimination between periodic and chaotic windows can be easily made based on the maximum Lyapunov exponent of the system, this remains a challenging task for continuous systems, especially if only short time series are available (e.g., in case of experimental data). In this work, we demonstrate that nonlinear measures based on recurrence plots obtained from such trajectories provide a practicable alternative for numerically detecting shrimps. Traditional diagonal line-based measures of recurrence quantification analysis as well as measures from complex network theory are shown to allow an excellent classification of periodic and chaotic behavior in parameter space. Using the well-studied Rössler system as a benchmark example, we find that the average path length and the clustering coefficient of the resulting recurrence networks are particularly powerful discriminatory statistics for the identification of complex periodic windows.
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Affiliation(s)
- Yong Zou
- Potsdam Institute for Climate Impact Research, P.O. Box 601203, 14412 Potsdam, Germany
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Freire JG, Gallas JAC. Non-Shilnikov cascades of spikes and hubs in a semiconductor laser with optoelectronic feedback. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:037202. [PMID: 21230214 DOI: 10.1103/physreve.82.037202] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/05/2010] [Indexed: 05/30/2023]
Abstract
Incomplete homoclinic scenarios were recently measured in a semiconductor laser with optoelectronic feedback. We show here that such a laser contains cascades of spirals of periodic oscillations and hubs which look identical to the familiar ones observed in complete homoclinic scenarios. This means that hubs are far more general than presumed so far, being not limited by Shilnikov's theorem. Laser hubs open the possibility of measuring complex distributions of non-Shilnikov laser oscillations, and we briefly discuss how to do it.
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Affiliation(s)
- Joana G Freire
- Instituto de Física, Universidade Federal do Rio Grande do Sul, 91501-970 Porto Alegre, RS , Brazil
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Stegemann C, Albuquerque HA, Rech PC. Some two-dimensional parameter spaces of a Chua system with cubic nonlinearity. CHAOS (WOODBURY, N.Y.) 2010; 20:023103. [PMID: 20590299 DOI: 10.1063/1.3374677] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/29/2023]
Abstract
In this paper we investigate three two-dimensional parameter spaces of a three-parameter set of autonomous differential equations used to model the Chua oscillator, where the piecewise-linear function usually taken to describe the nonlinearity of the Chua diode has been replaced by a cubic polynomial. It is made by using three independent two-dimensional cross sections of the three-dimensional parameter space generated by the model, which contains three parameters. We show that, independent of the parameter set considered in plots, all the diagrams present periodic structures embedded in a large chaotic region, and we also show that these structures organize themselves in period-adding cascades. We argue that these selected two-dimensional cross sections can be representative of the three-dimensional parameter space as a whole, in the range of parameters here investigated.
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Affiliation(s)
- Cristiane Stegemann
- Departamento de Fisica, Universidade do Estado de Santa Catarina, 89223-100 Joinville, Brazil.
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Freire JG, Field RJ, Gallas JAC. Relative abundance and structure of chaotic behavior: The nonpolynomial Belousov–Zhabotinsky reaction kinetics. J Chem Phys 2009; 131:044105. [DOI: 10.1063/1.3168400] [Citation(s) in RCA: 42] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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20
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Wieczorek S. Stochastic bifurcation in noise-driven lasers and Hopf oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:036209. [PMID: 19392037 DOI: 10.1103/physreve.79.036209] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/08/2008] [Revised: 01/23/2009] [Indexed: 05/27/2023]
Abstract
This paper considers nonlinear dynamics in an ensemble of uncoupled lasers, each being a limit-cycle oscillator, which are driven by the same external white Gaussian noise. As the external-noise strength increases, there is an onset of synchronization and then subsequent loss of synchrony. Local analysis of the laser equations shows that synchronization becomes unstable via stochastic bifurcation to chaos, defined as a passing of the largest Lyapunov exponent through zero. The locus of this bifurcation is calculated in the three-dimensional parameter space defined by the Hopf parameter, amount of amplitude-phase coupling, and external-noise strength. Numerical comparison between the laser system and the normal form of Hopf bifurcation uncovers a square-root law for this stochastic bifurcation as well as strong enhancement in noise-induced chaos due to the laser's relaxation oscillation.
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Affiliation(s)
- Sebastian Wieczorek
- Mathematics Research Institute, University of Exeter, EX4 4QF, United Kingdom
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Freire JG, Bonatto C, DaCamara CC, Gallas JAC. Multistability, phase diagrams, and intransitivity in the Lorenz-84 low-order atmospheric circulation model. CHAOS (WOODBURY, N.Y.) 2008; 18:033121. [PMID: 19045459 DOI: 10.1063/1.2953589] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
We report phase diagrams detailing the intransitivity observed in the climate scenarios supported by a prototype atmospheric general circulation model, namely, the Lorenz-84 low-order model. So far, this model was known to have a pair of coexisting climates described originally by Lorenz. Bifurcation analysis allows the identification of a remarkably wide parameter region where up to four climates coexist simultaneously. In this region the dynamical behavior depends crucially on subtle and minute tuning of the model parameters. This strong parameter sensitivity makes the Lorenz-84 model a promising candidate of testing ground to validate techniques of assessing the sensitivity of low-order models to perturbations of parameters.
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Affiliation(s)
- Joana G Freire
- Instituto de Física, Universidade Federal do Rio Grande do Sul, 91501-970 Porto Alegre, Brazil
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Bonatto C, Gallas JAC. Periodicity hub and nested spirals in the phase diagram of a simple resistive circuit. PHYSICAL REVIEW LETTERS 2008; 101:054101. [PMID: 18764395 DOI: 10.1103/physrevlett.101.054101] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/14/2008] [Indexed: 05/26/2023]
Abstract
We report the discovery of a remarkable "periodicity hub" inside the chaotic phase of an electronic circuit containing two diodes as a nonlinear resistance. The hub is a focal point from where an infinite hierarchy of nested spirals emanates. By suitably tuning two reactances simultaneously, both current and voltage may have their periodicity increased continuously without bound and without ever crossing the surrounding chaotic phase. Familiar period-adding current and voltage cascades are shown to be just restricted one-parameter slices of an exceptionally intricate and very regular onionlike parameter surface centered at the focal hub which organizes all the dynamics.
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Affiliation(s)
- Cristian Bonatto
- Instituto de Física, Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil
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Bonatto C, Gallas JAC. Accumulation boundaries: codimension-two accumulation of accumulations in phase diagrams of semiconductor lasers, electric circuits, atmospheric and chemical oscillators. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2008; 366:505-17. [PMID: 17698470 DOI: 10.1098/rsta.2007.2107] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/16/2023]
Abstract
We report high-resolution phase diagrams for several familiar dynamical systems described by sets of ordinary differential equations: semiconductor lasers; electric circuits; Lorenz-84 low-order atmospheric circulation model; and Rössler and chemical oscillators. All these systems contain chaotic phases with highly complicated and interesting accumulation boundaries, curves where networks of stable islands of regular oscillations with ever-increasing periodicities accumulate systematically. The experimental exploration of such codimension-two boundaries characterized by the presence of infinite accumulation of accumulations is feasible with existing technology for some of these systems.
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Affiliation(s)
- Cristian Bonatto
- Instituto de Física, Universidade Federal do Rio Grande do Sul, 91501-970 Porto Alegre, Brazil
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Bonatto C, Gallas JAC, Ueda Y. Chaotic phase similarities and recurrences in a damped-driven Duffing oscillator. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:026217. [PMID: 18352113 DOI: 10.1103/physreve.77.026217] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/07/2007] [Revised: 11/04/2007] [Indexed: 05/26/2023]
Abstract
We report strong evidence of remarkably close periodic repetitions of the structuring of the parameter space of a damped-driven Duffing oscillator as the amplitude of the drive increases. Families of period-adding cascades and some intricate networks of periodic oscillations embedded in chaotic phases are also found to recur closely as the driving force grows. Such surprising regularities suggest that some hitherto unknown renormalization mechanism may be operating in higher codimension, controlling the alternation of chaos and order in parameter space of certain flows.
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Affiliation(s)
- Cristian Bonatto
- Instituto de Física, Universidade Federal do Rio Grande do Sul, 91501-970 Porto Alegre, Brazil
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