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Mali P, Radošević S, Gombar S, Tekić J, Pantić M, Pavkov-Hrvojević M. Largest Lyapunov exponent as a tool for detecting relative changes in the particle positions. Phys Rev E 2023; 107:064213. [PMID: 37464598 DOI: 10.1103/physreve.107.064213] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/10/2022] [Accepted: 06/09/2023] [Indexed: 07/20/2023]
Abstract
Dynamics of the driven Frenkel-Kontorova model with asymmetric deformable substrate potential is examined by analyzing response function, the largest Lyapunov exponent, and Poincaré sections for two neighboring particles. The obtained results show that the largest Lyapunov exponent, besides being used for investigating integral quantities, can be used for detecting microchanges in chain configuration of both damped Frenkel-Kontorova model with inertial term and its strictly overdamped limit. Slight changes in relative positions of the particles are registered through jumps of the largest Lyapunov exponent in the pinning regime. The occurrence of such jumps is highly dependent on type of commensurate structure and deformation of substrate potential. The obtained results also show that the minimal force required to initiate collective motion of the chain is not dependent on the number of Lyapunov exponent jumps in the pinning regime. These jumps are also registered in the sliding regime, where they are a consequence of a more complex structure of largest Lyapunov exponent on the step.
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Affiliation(s)
- Petar Mali
- Department of Physics, Faculty of Sciences, University of Novi Sad, 21000 Novi Sad, Serbia
| | - Slobodan Radošević
- Department of Physics, Faculty of Sciences, University of Novi Sad, 21000 Novi Sad, Serbia
| | - Sonja Gombar
- Department of Physics, Faculty of Sciences, University of Novi Sad, 21000 Novi Sad, Serbia
| | - Jasmina Tekić
- "Vinča" Institute of Nuclear Sciences, Laboratory for Theoretical and Condensed Matter Physics-020, University of Belgrade, 11001 Belgrade, Serbia
| | - Milan Pantić
- Department of Physics, Faculty of Sciences, University of Novi Sad, 21000 Novi Sad, Serbia
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Botha AE, Shukrinov YM, Tekić J, Kolahchi MR. Chaotic dynamics from coupled magnetic monodomain and Josephson current. Phys Rev E 2023; 107:024205. [PMID: 36932612 DOI: 10.1103/physreve.107.024205] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/19/2022] [Accepted: 01/27/2023] [Indexed: 06/18/2023]
Abstract
The ordinary (superconductor-insulator-superconductor) Josephson junction cannot exhibit chaos in the absence of an external ac drive, whereas in the superconductor-ferromagnet-superconductor Josephson junction, known as the φ_{0} junction, the magnetic layer effectively provides two extra degrees of freedom that can facilitate chaotic dynamics in the resulting four-dimensional autonomous system. In this work, we use the Landau-Lifshitz-Gilbert model for the magnetic moment of the ferromagnetic weak link, while the Josephson junction is described by the resistively capacitively shunted-junction model. We study the chaotic dynamics of the system for parameters surrounding the ferromagnetic resonance region, i.e., for which the Josephson frequency is reasonably close to the ferromagnetic frequency. We show that, due to the conservation of magnetic moment magnitude, two of the numerically computed full spectrum Lyapunov characteristic exponents are trivially zero. One-parameter bifurcation diagrams are used to investigate various transitions that occur between quasiperiodic, chaotic, and regular regions as the dc-bias current through the junction, I, is varied. We also compute two-dimensional bifurcation diagrams, which are similar to traditional isospike diagrams, to display the different periodicities and synchronization properties in the I-G parameter space, where G is the ratio between the Josephson energy and the magnetic anisotropy energy. We find that as I is reduced the onset of chaos occurs shortly before the transition to the superconducting state. This onset of chaos is signaled by a rapid rise in supercurrent (I_{S}⟶I) which corresponds, dynamically, to increasing anharmonicity in phase rotations of the junction.
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Affiliation(s)
- A E Botha
- Department of Physics, University of South Africa, Private Bag X6, Roodepoort, Johannesburg 1710, South Africa
| | - Yu M Shukrinov
- BLTP, JINR, Dubna, Moscow Region 141980, Russian Federation
- Department of Nanotechnology and New Materials, Dubna State University, Dubna, Moscow Region 141980, Russian Federation
- Moscow Institute of Physics and Technology (National Research University), Institutskii per. 9, 141701 Dolgoprudny, Moscow Region, Russian Federation
| | - J Tekić
- Laboratory for Theoretical and Condensed Matter Physics-020, "Vinča" Institute of Nuclear Sciences, National Institute of the Republic of Serbia, University of Belgrade, P.O. Box 522, 11001 Belgrade, Republic of Serbia
| | - M R Kolahchi
- Department of Physics, Institute for Advanced Studies in Basic Sciences, Zanjan 45137-66731, Iran
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Wei Y, Lei Y. Shapiro steps and chaos in the Frenkel-Kontorova model with substrate lateral vibration. Phys Rev E 2022; 106:044204. [PMID: 36397517 DOI: 10.1103/physreve.106.044204] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/22/2022] [Accepted: 09/27/2022] [Indexed: 06/16/2023]
Abstract
Numerical simulations are used to examine the dynamics of the dc-driven Frenkel-Kontorova model with an oscillation substrate subjected to lateral periodic excitations in overdamped and underdamped cases, respectively. The results reveal that the system exhibits frequency locking and chaotic behaviors due to the fact that the lateral vibration of the substrate potential introduces an additional frequency and degree of freedom into the system. In the overdamped case, we show that the appearance of subharmonic Shapiro steps can be attributed to the deformation of the substrate potential or inertia. The characteristics of the steps are significantly affected by the amplitude and frequency of the lateral vibration. When the vibration frequency is relatively high, the change of the width of the first harmonic Shapiro step with increasing amplitude exhibits a Bessel-type oscillation, but the oscillation deviates from the Bessel curve at lower frequencies. In the amplitude dependence, although the oscillatory behavior of the critical depinning force at the high frequency is anomalous, local maxima (minima) of the first step width correspond to local minima (maxima) of the critical depinning force, and the largest Lyapunov exponent obtained in the pinned state represents a mirror relationship of the critical depinning force. In contrast to the overdamped system, the underdamped one exhibits both subharmonic Shapiro steps and chaotic behaviors. We show the increased inertia of the latter system plays an important role in suppressing the emergence of subharmonic steps, which is opposite to the result of the former. When the dc force changes, chaos appears not only between adjacent subharmonic Shapiro steps but also on some specific steps where chaos should be avoided. The variation regularity of the first step width and the critical depinning force is thus annihilated in vibration amplitude and frequency dependence. However, superlubricity can be achieved by careful adjustment of vibration amplitude and frequency. The findings can serve as a theoretical guideline for technological applications such as device building and voltage standards.
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Affiliation(s)
- Yongfeng Wei
- School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710072, China
| | - Youming Lei
- School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710072, China
- Ministry of Industry and Information Technology Key Laboratory of Dynamics and Control of Complex Systems, Northwestern Polytechnical University, Xi'an, 710072, China
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Nashaat M, Sameh M, Botha AE, Kulikov KV, Shukrinov YM. Bifurcation structure and chaos in dynamics of nanomagnet coupled to Josephson junction. CHAOS (WOODBURY, N.Y.) 2022; 32:093142. [PMID: 36182372 DOI: 10.1063/5.0095009] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/07/2022] [Accepted: 08/26/2022] [Indexed: 06/16/2023]
Abstract
Irregular easy axis reorientation features are observed in numerical simulations of the nanomagnet coupled to the Josephson junction. We study magnetization bifurcations and chaos that appear in this system due to the interplay of superconductivity and magnetism. The bifurcation structure of magnetization under the variation of Josephson to magnetic energy ratio as a control parameter demonstrates several precessional motions that are related to chaotic behavior and orbits with different periodicities in the ferromagnetic resonance region. The effect of an external periodic signal on the bifurcation structure is also investigated. The results demonstrate high-frequency modes of a periodic motion and a chaotic response near resonance. Far from the ferromagnetic resonance, we observe a quasiperiodic behavior. The obtained results explain the irregular reorientation of the easy axis and the transitions between different types of motion.
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Affiliation(s)
- M Nashaat
- Department of Physics, Faculty of Science, Cairo University, 12613 Giza, Egypt
| | - M Sameh
- Department of Physics, Faculty of Science, Cairo University, 12613 Giza, Egypt
| | - A E Botha
- Department of Physics, Science Campus, University of South Africa, Johannesburg 1710, South Africa
| | - K V Kulikov
- BLTP, JINR, Dubna, Moscow Region 141980, Russia
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Mali P, Šakota A, Tekić J, Radošević S, Pantić M, Pavkov-Hrvojević M. Complexity of Shapiro steps. Phys Rev E 2020; 101:032203. [PMID: 32289931 DOI: 10.1103/physreve.101.032203] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/18/2019] [Accepted: 02/04/2020] [Indexed: 11/07/2022]
Abstract
We demonstrate, using the example of the dc+ac driven overdamped Frenkel-Kontorova model, that an easily calculable measure of complexity can be used for the examination of Shapiro steps in the presence of thermal noise. In real systems, thermal noise causes melting or even disappearance of Shapiro steps, which makes their analysis in the standard way from the response function difficult. Unlike in the conventional approach, here, by calculating the Kolmogorov complexity of certain areas in the response function, we were able to detect Shapiro steps, measure their size with the desired precision, and examine their temperature dependence. The aim of this work is to provide scientists, particularly experimentalists, with an unconventional, but practical and easy tool for examination of Shapiro steps in real systems.
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Affiliation(s)
- Petar Mali
- Department of Physics, Faculty of Science, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia
| | - Anđela Šakota
- Department of Physics, Faculty of Science, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia
| | - Jasmina Tekić
- "Vinča" Institute of Nuclear Sciences, Laboratory for Theoretical and Condensed Matter Physics - 020, University of Belgrade, P.O. Box 522, 11001 Belgrade, Serbia
| | - Slobodan Radošević
- Department of Physics, Faculty of Science, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia
| | - Milan Pantić
- Department of Physics, Faculty of Science, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia
| | - Milica Pavkov-Hrvojević
- Department of Physics, Faculty of Science, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia
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Tekić J, Botha AE, Mali P, Shukrinov YM. Inertial effects in the dc+ac driven underdamped Frenkel-Kontorova model: Subharmonic steps, chaos, and hysteresis. Phys Rev E 2019; 99:022206. [PMID: 30934229 DOI: 10.1103/physreve.99.022206] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/28/2018] [Indexed: 06/09/2023]
Abstract
The effects of inertial terms on the dynamics of the dc+ac driven Frenkel-Kontorova model were examined. As the mass of particles was varied, the response of the system to the driving forces and appearance of the Shapiro steps were analyzed in detail. Unlike in the overdamped case, the increase of mass led to the appearance of the whole series of subharmonic steps in the staircase of the average velocity as a function of average driving force in any commensurate structure. At certain values of parameters, the subharmonic steps became separated by chaotic windows while the whole structure retained scaling similar to the original staircase. The mass of the particles also determined their sensitivity to the forces governing their dynamics. Depending on their mass, they were found to exhibit three types of dynamics, from dynamical mode-locking with chaotic windows, through to a typical dc response, to essentially a free-particle response. Examination of this dynamics in both the upforce and downforce directions showed that the system may not only exhibit hysteresis, but also that large Shapiro steps may appear in the downforce direction, even in cases for which no dynamical mode-locking occurred in the upforce direction.
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Affiliation(s)
- J Tekić
- "Vinca" Institute of Nuclear Sciences, Laboratory for Theoretical and Condensed Matter Physics-020, University of Belgrade, P.O. Box 522, 11001 Belgrade, Serbia
| | - A E Botha
- Department of Physics, University of South Africa, Science Campus, Private Bag X6, Florida Park 1710, South Africa
| | - P Mali
- Department of Physics, Faculty of Science, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia
| | - Yu M Shukrinov
- BLTP, JINR, Dubna, Moscow Region, 141980, Russian Federation
- Department of Nanotechnology and New Materials, Dubna State University, Dubna, Moscow Region, 141980, Russian Federation
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Rahmonov I, Shukrinov Y, Atanasova P, Zemlyanaya E, Streltsova O, Zuev M, Plecenik A, Irie A. Simulation of Collective Excitations in Long Josephson Junction Stacks. EPJ WEB OF CONFERENCES 2018. [DOI: 10.1051/epjconf/201817306011] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The phase dynamics of a stack of long Josephson junctions has been studied. Both inductive and capacitive couplings between Josephson junctions have been taken into account in the calculations. The IV-curve, the dependence on the bias current of the radiation power and dynamics of each JJs of the stack have been investigated. The coexistence of the charge traveling wave and fluxon states has been observed. This state can be considered as a new collective excitation in the system of coupled Josephson junctions. We demonstrate that the observed collective excitation leads to the decrease of radiation power from the system.
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Sokolović I, Mali P, Odavić J, Radošević S, Medvedeva SY, Botha AE, Shukrinov YM, Tekić J. Devil's staircase and the absence of chaos in the dc- and ac-driven overdamped Frenkel-Kontorova model. Phys Rev E 2017; 96:022210. [PMID: 28950628 DOI: 10.1103/physreve.96.022210] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/27/2017] [Indexed: 06/07/2023]
Abstract
The devil's staircase structure arising from the complete mode locking of an entirely nonchaotic system, the overdamped dc+ac driven Frenkel-Kontorova model with deformable substrate potential, was observed. Even though no chaos was found, a hierarchical ordering of the Shapiro steps was made possible through the use of a previously introduced continued fraction formula. The absence of chaos, deduced here from Lyapunov exponent analyses, can be attributed to the overdamped character and the Middleton no-passing rule. A comparative analysis of a one-dimensional stack of Josephson junctions confirmed the disappearance of chaos with increasing dissipation. Other common dynamic features were also identified through this comparison. A detailed analysis of the amplitude dependence of the Shapiro steps revealed that only for the case of a purely sinusoidal substrate potential did the relative sizes of the steps follow a Farey sequence. For nonsinusoidal (deformed) potentials, the symmetry of the Stern-Brocot tree, depicting all members of particular Farey sequence, was seen to be increasingly broken, with certain steps being more prominent and their relative sizes not following the Farey rule.
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Affiliation(s)
- I Sokolović
- Department of Physics, Faculty of Science, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia
| | - P Mali
- Department of Physics, Faculty of Science, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia
| | - J Odavić
- Institut für Theorie der Statistishen Physik - RWTH Aachen University, Peter-Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich, Germany
| | - S Radošević
- Department of Physics, Faculty of Science, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia
| | - S Yu Medvedeva
- BLTP, JINR, Dubna, Moscow Region, 141980, Russian Federation
- Moscow Institute of Physics and Technology, Institutsky Lane 9, Dolgoprudny, Moscow Region, 141700, Russian Federation
| | - A E Botha
- Department of Physics, University of South Africa, Science Campus, Private Bag X6, Florida Park 1710, South Africa
| | - Yu M Shukrinov
- BLTP, JINR, Dubna, Moscow Region, 141980, Russian Federation
- Department of Nanotechnology and New Materials, Dubna State University, Dubna, Moscow Region, 141980, Russian Federation
| | - J Tekić
- "Vinča" Institute of Nuclear Sciences, Laboratory for Theoretical and Condensed Matter Physics - 020, University of Belgrade, P.O. Box 522, 11001 Belgrade, Serbia
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Abstract
For a chaotic system pairs of initially close-by trajectories become eventually fully uncorrelated on the attracting set. This process of decorrelation can split into an initial exponential decrease and a subsequent diffusive process on the chaotic attractor causing the final loss of predictability. Both processes can be either of the same or of very different time scales. In the latter case the two trajectories linger within a finite but small distance (with respect to the overall extent of the attractor) for exceedingly long times and remain partially predictable. Standard tests for chaos widely use inter-orbital correlations as an indicator. However, testing partially predictable chaos yields mostly ambiguous results, as this type of chaos is characterized by attractors of fractally broadened braids. For a resolution we introduce a novel 0–1 indicator for chaos based on the cross-distance scaling of pairs of initially close trajectories. This test robustly discriminates chaos, including partially predictable chaos, from laminar flow. Additionally using the finite time cross-correlation of pairs of initially close trajectories, we are able to identify laminar flow as well as strong and partially predictable chaos in a 0–1 manner solely from the properties of pairs of trajectories.
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Odavić J, Mali P, Tekić J. Farey sequence in the appearance of subharmonic Shapiro steps. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:052904. [PMID: 26066226 DOI: 10.1103/physreve.91.052904] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/30/2015] [Indexed: 06/04/2023]
Abstract
The largest Lyapunov exponent has been examined in the dynamical-mode locking phenomena of the ac+dc driven dissipative Frenkel-Kontorova model with deformable substrate potential. Due to deformation, large fractional and higher order subharmonic steps appear in the response function of the system. Computation of the largest Lyapunov exponent as a way to verify their presence led to the observation of the Farey sequence. In the standard regime, the appearance of half-integer and other subharmonic steps between the large harmonic steps, and their relative sizes follow the Farey construction. In the nonstandard regime, however, the half-integer steps are larger than harmonic ones, and Farey construction is only present in the appearance of higher order subharmonic steps. The examination of Lyapunov exponents has also shown that regardless of the substrate potential or deformation, there was no chaos in the system.
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Affiliation(s)
- Jovan Odavić
- Institut für Theorie der Statistishen Physik-RWTH Aachen University, Peter-Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich, Germany
| | - Petar Mali
- Department of Physics, Faculty of Science, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia
| | - Jasmina Tekić
- "Vinča" Institute of Nuclear Sciences, Laboratory for Theoretical and Condensed Matter Physics 020, University of Belgrade, P.O. Box 522, 11001 Belgrade, Serbia
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