Complex Dynamics of Spring-Block Earthquake Model Under Periodic Parameter Perturbations
Abstract
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.
Keywords:
spring-block model / coupled oscillations / spring constant / stress ratio / deterministic chaosSource:
Journal of Computational and Nonlinear Dynamics, 2014, 9, 3Publisher:
- Asme, New York
Funding / projects:
- Magmatism and geodynamics of the Balkan Peninsula from Mesozoic to present day: significance for the formation of metallic and non-metallic mineral deposits (RS-MESTD-Basic Research (BR or ON)-176016)
DOI: 10.1115/1.4026259
ISSN: 1555-1423
WoS: 000337047100019
Scopus: 2-s2.0-84896502522
Collections
Institution/Community
PharmacyTY - JOUR AU - Kostić, Srdan AU - Vasović, Nebojša AU - Franović, Igor AU - Todorović, Kristina PY - 2014 UR - https://farfar.pharmacy.bg.ac.rs/handle/123456789/2193 AB - 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. PB - Asme, New York T2 - Journal of Computational and Nonlinear Dynamics T1 - Complex Dynamics of Spring-Block Earthquake Model Under Periodic Parameter Perturbations VL - 9 IS - 3 DO - 10.1115/1.4026259 ER -
@article{ author = "Kostić, Srdan and Vasović, Nebojša and Franović, Igor and Todorović, Kristina", year = "2014", abstract = "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.", publisher = "Asme, New York", journal = "Journal of Computational and Nonlinear Dynamics", title = "Complex Dynamics of Spring-Block Earthquake Model Under Periodic Parameter Perturbations", volume = "9", number = "3", doi = "10.1115/1.4026259" }
Kostić, S., Vasović, N., Franović, I.,& Todorović, K.. (2014). Complex Dynamics of Spring-Block Earthquake Model Under Periodic Parameter Perturbations. in Journal of Computational and Nonlinear Dynamics Asme, New York., 9(3). https://doi.org/10.1115/1.4026259
Kostić S, Vasović N, Franović I, Todorović K. Complex Dynamics of Spring-Block Earthquake Model Under Periodic Parameter Perturbations. in Journal of Computational and Nonlinear Dynamics. 2014;9(3). doi:10.1115/1.4026259 .
Kostić, Srdan, Vasović, Nebojša, Franović, Igor, Todorović, Kristina, "Complex Dynamics of Spring-Block Earthquake Model Under Periodic Parameter Perturbations" in Journal of Computational and Nonlinear Dynamics, 9, no. 3 (2014), https://doi.org/10.1115/1.4026259 . .