A simple model of earthquake nucleation that may account for the onset of chaotic dynamics is proposed and analyzed. It represents a generalization of the Burridge–Knopoff single-block model with Dieterich–Ruina's rate- and state-dependent friction law. It is demonstrated that deterministic chaos may emerge when some of the parameters are assumed to undergo small oscillations about their equilibrium values. Implementing the standard numerical methods from the theory of dynamical systems, the analysis is carried out for the cases having one or two periodically variable parameters, such that the appropriate bifurcation diagrams, phase portraits, power spectra, and the Lyapunov exponents are obtained. The results of analysis indicate two different scenarios to chaos. On one side, the Ruelle–Takens–Newhouse route to chaos is observed for the cases of limit amplitude perturbations. On the other side, when the angular frequency is assumed constant for the value near the periodic motion of the block in an unperturbed case, variation of oscillation amplitudes probably gives rise to global bifurcations, with immediate occurrence of chaotic behavior. Further analysis shows that chaotic behavior emerges only for small oscillation frequencies and higher perturbation amplitudes when two perturbed parameters are brought into play. If higher oscillation frequencies are assumed, no bifurcation occurs, and the system under study exhibits only the periodic motion. In contrast to the previous research, the onset of chaos is observed for much smaller values of the stress ratio parameter. In other words, even the relatively small perturbations of the control parameters could lead to deterministic chaos and, thus, to instabilities and earthquakes.
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University of Belgrade Faculty
of Mining and Geology,
Đušina 7,
e-mail: srdjan.kostic@rgf.bg.ac.rs
University of Belgrade Faculty
of Mining and Geology,
Đušina 7,
e-mail: nebojsa.vasovic@rgf.bg.ac.rs
Statistical Physics, and Electrodynamics,
University of Belgrade Faculty of Physics,
Studentski Trg 12,
e-mal: igor.franovic@gmail.com
University of Belgrade Faculty of Pharmacy,
Vojvode Stepe 450,
e-mail: kisi@pharmacy.bg.ac.rs
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July 2014
Research-Article
Complex Dynamics of Spring-Block Earthquake Model Under Periodic Parameter Perturbations
Srđan Kostić,
University of Belgrade Faculty
of Mining and Geology,
Đušina 7,
e-mail: srdjan.kostic@rgf.bg.ac.rs
Srđan Kostić
1
Department of Geology
,University of Belgrade Faculty
of Mining and Geology,
Đušina 7,
Belgrade 11000 Serbia
e-mail: srdjan.kostic@rgf.bg.ac.rs
1Corresponding author.
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Nebojša Vasović,
University of Belgrade Faculty
of Mining and Geology,
Đušina 7,
e-mail: nebojsa.vasovic@rgf.bg.ac.rs
Nebojša Vasović
Department of Applied Mathematics
,University of Belgrade Faculty
of Mining and Geology,
Đušina 7,
Belgrade 11000 Serbia
e-mail: nebojsa.vasovic@rgf.bg.ac.rs
Search for other works by this author on:
Igor Franović,
Statistical Physics, and Electrodynamics,
University of Belgrade Faculty of Physics,
Studentski Trg 12,
e-mal: igor.franovic@gmail.com
Igor Franović
Department of Theoretical Mechanics
,Statistical Physics, and Electrodynamics,
University of Belgrade Faculty of Physics,
Studentski Trg 12,
Belgrade 11000 Serbia
e-mal: igor.franovic@gmail.com
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Kristina Todorović
University of Belgrade Faculty of Pharmacy,
Vojvode Stepe 450,
e-mail: kisi@pharmacy.bg.ac.rs
Kristina Todorović
Department of Physics and Mathematics
,University of Belgrade Faculty of Pharmacy,
Vojvode Stepe 450,
Belgrade 11000 Serbia
e-mail: kisi@pharmacy.bg.ac.rs
Search for other works by this author on:
Srđan Kostić
Department of Geology
,University of Belgrade Faculty
of Mining and Geology,
Đušina 7,
Belgrade 11000 Serbia
e-mail: srdjan.kostic@rgf.bg.ac.rs
Nebojša Vasović
Department of Applied Mathematics
,University of Belgrade Faculty
of Mining and Geology,
Đušina 7,
Belgrade 11000 Serbia
e-mail: nebojsa.vasovic@rgf.bg.ac.rs
Igor Franović
Department of Theoretical Mechanics
,Statistical Physics, and Electrodynamics,
University of Belgrade Faculty of Physics,
Studentski Trg 12,
Belgrade 11000 Serbia
e-mal: igor.franovic@gmail.com
Kristina Todorović
Department of Physics and Mathematics
,University of Belgrade Faculty of Pharmacy,
Vojvode Stepe 450,
Belgrade 11000 Serbia
e-mail: kisi@pharmacy.bg.ac.rs
1Corresponding author.
Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received June 10, 2013; final manuscript received December 11, 2013; published online March 13, 2014. Assoc. Editor: Ahmet S. Yigit.
J. Comput. Nonlinear Dynam. Jul 2014, 9(3): 031019 (10 pages)
Published Online: March 13, 2014
Article history
Received:
June 10, 2013
Revision Received:
December 11, 2013
Citation
Kostić, S., Vasović, N., Franović, I., and Todorović, K. (March 13, 2014). "Complex Dynamics of Spring-Block Earthquake Model Under Periodic Parameter Perturbations." ASME. J. Comput. Nonlinear Dynam. July 2014; 9(3): 031019. https://doi.org/10.1115/1.4026259
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