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Chen YM, Liu JK. Circularly distributed multipliers with deterministic moduli assessing the stability of quasiperiodic response. Phys Rev E 2023; 107:014218. [PMID: 36797899 DOI: 10.1103/physreve.107.014218] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/31/2022] [Accepted: 01/05/2023] [Indexed: 01/31/2023]
Abstract
The stability and bifurcation of a periodic solution of a dynamical system can be handled well by using the Floquet multipliers of the perturbed system with periodic coefficients. However, for a quasiperiodic (QP) response as a natural extension of a periodic one, it is much more difficult to do it quantitatively. Therefore, proposed here is an approach for defining and obtaining effective multipliers for QP stability. The proposed approach is based on a series of auxiliary variables via which the perturbed system with QP coefficients is transformed approximately into a constant one, whereupon the multipliers are obtained efficiently by performing eigenvalue analysis on the constant coefficient matrix. The major finding involves circularly distributed multipliers with deterministic moduli, with the QP response being stable if all the moduli are less than or equal to unity; otherwise it is unstable. When the QP response degenerates to periodic due to the reducibility of fundamental frequencies, the proposed approach exactly provides the Floquet multipliers for the periodic solution. From this respect, the obtained multipliers can be considered to some extent as being a generalization for QP response of the Floquet multipliers for a periodic solution.
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Affiliation(s)
- Y M Chen
- Department of Mechanics, Sun Yat-sen University, 135 Xingang Road, Guangzhou 510275, China
| | - J K Liu
- Department of Mechanics, Sun Yat-sen University, 135 Xingang Road, Guangzhou 510275, China
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2
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Cubero D, Robb GRM, Renzoni F. Avoided Crossing and sub-Fourier-sensitivity in Driven Quantum Systems. PHYSICAL REVIEW LETTERS 2018; 121:213904. [PMID: 30517810 DOI: 10.1103/physrevlett.121.213904] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/27/2018] [Revised: 09/28/2018] [Indexed: 06/09/2023]
Abstract
The response of a linear system to an external perturbation is governed by the Fourier limit, with the inverse of the interaction time constituting a lower limit for the system bandwidth. This does not hold for nonlinear systems, which can thus exhibit sub-Fourier-behavior. The present Letter identifies a mechanism for sub-Fourier-sensitivity in driven quantum systems, which relies on avoided crossing between Floquet states. Features up to three orders of magnitude finer than the Fourier limit are presented.
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Affiliation(s)
- David Cubero
- Departamento de Física Aplicada I, EUP, Universidad de Sevilla, Calle Virgen de África 7, 41011 Sevilla, Spain
| | - Gordon R M Robb
- SUPA and Department of Physics, University of Strathclyde, John Anderson Building, 107 Rottenrow, Glasgow G4 0NG, United Kingdom
| | - Ferruccio Renzoni
- Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom
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Cubero D, Renzoni F. Asymptotic theory of quasiperiodically driven quantum systems. Phys Rev E 2018; 97:062139. [PMID: 30011550 DOI: 10.1103/physreve.97.062139] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/18/2018] [Indexed: 11/07/2022]
Abstract
The theoretical treatment of quasiperiodically driven quantum systems is complicated by the inapplicability of the Floquet theorem, which requires strict periodicity. In this work we consider a quantum system driven by a biharmonic driving and examine its asymptotic long-time limit, the limit in which features distinguishing systems with periodic and quasiperiodic driving occur. Also, in the classical case this limit is known to exhibit universal scaling, independent of the system details, with the system's reponse under quasiperiodic driving being described in terms of nearby periodically driven system results. We introduce a theoretical framework appropriate for the treatment of the quasiperiodically driven quantum system in the long-time limit and derive an expression, based on Floquet states for a periodically driven system approximating the different steps of the time evolution, for the asymptotic scaling of relevant quantities for the system at hand. These expressions are tested numerically, finding excellent agreement for the finite-time average velocity in a prototypical quantum ratchet consisting of a space-symmetric potential and a time-asymmetric oscillating force.
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Affiliation(s)
- David Cubero
- Departamento de Física Aplicada I, EUP, Universidad de Sevilla, Calle Virgen de África 7, 41011 Sevilla, Spain
| | - Ferruccio Renzoni
- Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom
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Casado-Pascual J. Directed motion of spheres induced by unbiased driving forces in viscous fluids beyond the Stokes' law regime. Phys Rev E 2018; 97:032219. [PMID: 29776079 DOI: 10.1103/physreve.97.032219] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/17/2017] [Indexed: 11/07/2022]
Abstract
The emergence of directed motion is investigated in a system consisting of a sphere immersed in a viscous fluid and subjected to time-periodic forces of zero average. The directed motion arises from the combined action of a nonlinear drag force and the applied driving forces, in the absence of any periodic substrate potential. Necessary conditions for the existence of such directed motion are obtained and an analytical expression for the average terminal velocity is derived within the adiabatic approximation. Special attention is paid to the case of two mutually perpendicular forces with sinusoidal time dependence, one with twice the period of the other. It is shown that, although neither of these two forces induces directed motion when acting separately, when added together, the resultant force generates directed motion along the direction of the force with the shortest period. The dependence of the average terminal velocity on the system parameters is analyzed numerically and compared with that obtained using the adiabatic approximation. Among other results, it is found that, for appropriate parameter values, the direction of the average terminal velocity can be reversed by varying the forcing strength. Furthermore, certain aspects of the observed phenomenology are explained by means of symmetry arguments.
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Affiliation(s)
- Jesús Casado-Pascual
- Física Teórica, Universidad de Sevilla, Apartado de Correos 1065, 41080 Sevilla, Spain
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Lindner JF, Kohar V, Kia B, Hippke M, Learned JG, Ditto WL. Strange nonchaotic stars. PHYSICAL REVIEW LETTERS 2015; 114:054101. [PMID: 25699444 DOI: 10.1103/physrevlett.114.054101] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/20/2014] [Indexed: 06/04/2023]
Abstract
The unprecedented light curves of the Kepler space telescope document how the brightness of some stars pulsates at primary and secondary frequencies whose ratios are near the golden mean, the most irrational number. A nonlinear dynamical system driven by an irrational ratio of frequencies generically exhibits a strange but nonchaotic attractor. For Kepler's "golden" stars, we present evidence of the first observation of strange nonchaotic dynamics in nature outside the laboratory. This discovery could aid the classification and detailed modeling of variable stars.
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Affiliation(s)
- John F Lindner
- Department of Physics and Astronomy, University of Hawai'i at Mānoa, Honolulu, Hawai'i 96822, USA and Physics Department, The College of Wooster, Wooster, Ohio 44691, USA
| | - Vivek Kohar
- Department of Physics and Astronomy, University of Hawai'i at Mānoa, Honolulu, Hawai'i 96822, USA
| | - Behnam Kia
- Department of Physics and Astronomy, University of Hawai'i at Mānoa, Honolulu, Hawai'i 96822, USA
| | - Michael Hippke
- Institute for Data Analysis, Luiter Straße 21b, 47506 Neukirchen-Vluyn, Germany
| | - John G Learned
- Department of Physics and Astronomy, University of Hawai'i at Mānoa, Honolulu, Hawai'i 96822, USA
| | - William L Ditto
- Department of Physics and Astronomy, University of Hawai'i at Mānoa, Honolulu, Hawai'i 96822, USA
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Casado-Pascual J, Cuesta JA, Quintero NR, Alvarez-Nodarse R. General approach for dealing with dynamical systems with spatiotemporal periodicities. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:022905. [PMID: 25768567 DOI: 10.1103/physreve.91.022905] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/30/2014] [Indexed: 06/04/2023]
Abstract
Dynamical systems often contain oscillatory forces or depend on periodic potentials. Time or space periodicity is reflected in the properties of these systems through a dependence on the parameters of their periodic terms. In this paper we provide a general theoretical framework for dealing with these kinds of systems, regardless of whether they are classical or quantum, stochastic or deterministic, dissipative or nondissipative, linear or nonlinear, etc. In particular, we are able to show that simple symmetry considerations determine, to a large extent, how their properties depend functionally on some of the parameters of the periodic terms. For the sake of illustration, we apply this formalism to find the functional dependence of the expectation value of the momentum of a Bose-Einstein condensate, described by the Gross-Pitaewskii equation, when it is exposed to a sawtooth potential whose amplitude is periodically modulated in time. We show that, by using this formalism, a small set of measurements is enough to obtain the functional form for a wide range of parameters. This can be very helpful when characterizing experimentally the response of systems for which performing measurements is costly or difficult.
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Affiliation(s)
- Jesús Casado-Pascual
- Física Teórica, Universidad de Sevilla, Apartado de Correos 1065, 41080 Sevilla, Spain
| | - José A Cuesta
- Grupo Interdisciplinar de Sistemas Complejos (GISC), Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Spain
- Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), Universidad de Zaragoza, 50009 Zaragoza, Spain
| | - Niurka R Quintero
- Instituto de Matemáticas de la Univesidad de Sevilla (IMUS)
- Departamento de Física Aplicada I, E.P.S., Universidad de Sevilla, Virgen de África 7, 41011 Sevilla, Spain
| | - Renato Alvarez-Nodarse
- Instituto de Matemáticas de la Univesidad de Sevilla (IMUS)
- Departamento de Análisis Matemático, Universidad de Sevilla, Apdo 1160, 41080 Sevilla, Spain
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