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Research Papers

Complex Dynamics of Spring-Block Earthquake Model Under Periodic Parameter Perturbations

[+] Author and Article Information
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 9(3), 031019 (Mar 13, 2014) (10 pages) Paper No: CND-13-1134; doi: 10.1115/1.4026259 History: Received June 10, 2013; Revised December 11, 2013

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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References

Figures

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Fig. 1

The Burridge–Knopoff block and spring model, represented by a slider coupled through a spring to a loader plate, which moves with velocity V

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Fig. 2

Bifurcations of the system (2) (or (3) if δi = 0 is set) under the variation of one of the parameters ε (1) and ξ (2). Orbital diagram is constructed for the section with plane θ = 1, and calculation step 0.01, showing the asymptotic dynamics after 8 × 106 time units. At each instance, the parameters held constant are awarded values that admit the equilibrium point, ε = 0.2, ξ = 0.6, and γ = 0.8.

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Fig. 3

Attractors of the system (3) (or (4) with δi = 0) in parameter plane ε-ξ. The remaining parameters are held fixed at values admitting the equilibrium point, as in Fig. 2 Corresponding time series and phase portraits for points 1 and 2 are shown in Fig. 4 EQ and PM are abbreviations for equilibrium state and periodic motion, respectively.

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Fig. 4

Temporal evolution of variable V and the appropriate phase portraits for: (1) ε = 0.1, ξ = 1.0, and γ = 0.8 (equilibrium state); (2) ε = 0.5, ξ = 1, and γ = 0.8 (periodic motion)

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Fig. 5

Single peak in power spectrum indicates the oscillatory behavior of the model. Parameter values are the same as in Fig. 4(2).

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Fig. 6

Bifurcations of the system (3) under the variation of the parameter ε. Orbital diagram is constructed for the section with plane θ = 1, and calculation step ωε = 0.01, showing the asymptotic dynamics after 8 × 106 time units. At each instance, the parameters held constant are awarded the values near the equilibrium point but admitting the limit cycle, ε = 0.4 (δε = 0.4), ξ = 0.5, and γ = 0.8. Corresponding time series and phase portraits for periodic and chaotic motion are shown in Fig. 7.

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Fig. 7

Temporal evolution of variable V and the appropriate phase portraits for: (1) ωε = 0.9 (periodic motion); (2) ωε = 0.2 (deterministic chaos). At each instance, the parameters held constant are awarded the values near the equilibrium point, but admitting the limit cycle, ε = 0.4 (δε = 0.4), ξ = 0.5, and γ = 0.8.

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Fig. 8

(1) Fourier power spectrum of periodic motion (first peak is for fundamental frequency, other peaks represent harmonics); (2) the broadband noise in the Fourier power spectrum indicates the onset of deterministic chaos. Parameter values are the same as in Figs. 7(1) and 7(2), respectively.

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Fig. 9

Maximal Lyapunov exponent converges well to λmax = 0.072, confirming the onset of deterministic chaos. The parameter values are identical to those in Fig. 7(2).

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Fig. 10

Bifurcations of the system (3) under the variation of the parameter ξ. Orbital diagram is constructed for the section with plane θ = 1, and calculation step ωξ = 0.01, showing the asymptotic dynamics after 4 × 106 time units. At each instance, the parameters held constant are awarded values near the equilibrium point, but admitting the limit cycle, ε = 0.4, ξ = 0.5 (δξ = 0.4), and γ = 0.8. Corresponding time series and phase portraits for quasi-periodic motion are shown in Fig. 11.

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Fig. 11

Temporal evolution of variable V (1) and the appropriate phase portrait (2) for ωξ = 0.65 (quasi-periodic motion). At each instance, the parameters held constant are awarded the values near the equilibrium point, but admitting the limit cycle, ε = 0.4 (δε = 0.4), ξ = 0.5, and γ = 0.8.

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Fig. 12

Several incommensurate frequencies in Fourier power spectrum indicate quasi-periodic motion. Parameter values are the same as in Fig. 11.

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Fig. 13

Bifurcations of the system (4), for different perturbation amplitudes δε and δξ: (1) ωε=0.2; (2) ωξ=0.3. In both cases, orbital diagram is constructed for the section with plane θ = 1, and calculation step ωε = ωξ = 0.01, showing the dynamics after 8 × 106 and 4 × 106 time units, respectively. At each instance, the parameters held constant are awarded values near the equilibrium point, but admitting the limit cycle, ε = 0.4, ξ = 0.5, and γ = 0.8.

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Fig. 14

Bifurcations of the system (3), for different perturbation amplitudes δε and δξ: (1) ωε=0.9; (2) ωξ=0.9. In both cases, orbital diagram is constructed for the section with plane θ = 1, and calculation step ωε = ωξ = 0.01 showing the dynamics after 8 × 106 time units. At each instance, the parameters held constant are awarded values near the equilibrium point but admitting the limit cycle, ε = 0.4, ξ = 0.5, and γ = 0.8.

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Fig. 15

Attractors of the system (3) in parameter plane δε - δξ, for the frequencies near the frequency of the block in unperturbed state, admitting chaos due to a single parameter perturbation (ωε=0.2, ωξ=0.3). Diagram is constructed for the grid 0.01 × 0.01. At each instance, the parameters held constant are awarded the values near the equilibrium point but admitting the limit cycle, ε = 0.4, ξ = 0.5, and γ = 0.8. P and C are abbreviations for periodic motion and deterministic chaos, respectively. Corresponding time series and phase portraits for points 1 and 2 are shown in Fig. 16.

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Fig. 16

Temporal evolution of variable V and the appropriate phase portraits for points 1 and 2 in Fig. 15: (1) δε = 0.1, δξ = 0.05 (periodic motion); (2) δε = 0.2, δξ = 0.2 (deterministic chaos). At each instance, the parameters held constant are awarded values near the equilibrium point but admitting the limit cycle, ε = 0.4, ξ = 0.5, and γ = 0.8. Values of the angular frequency are chosen to be near the angular frequency of the block in an unperturbed state, admitting the onset of deterministic chaos for a single parameter perturbation: ωε=0.2, ωξ=0.3.

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Fig. 17

(1) Two commensurate peaks in power spectrum (first peak for the fundamental frequency, and the second peak for the harmonic) imply the periodic motion (2). The broadband noise in the Fourier power spectrum indicates the chaotic behavior of the system. The parameter values are the same as in Figs. 16(1) and 16(2), respectively.

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Fig. 18

Calculation of maximal Lyapunov exponent for a time series in Fig. 17(2): (1) λmax = 0.009 (method of Wolf et al. [37]); (2) λmax ≈ 0.009 (method of Rosenstein et al. [38]). Effective expansion rate S(Δn) represents the average of the logarithm of Di(Δn), defined as the average distance of all nearby trajectories to the reference trajectory as a function of the relative time Δn. The slope of dashed lines indicating the predominant slope of S(Δn) in dependence on Δndt presents a robust estimate for the maximal Lyapunov exponent. The results are determined for 1000 reference points and neighboring distance ε = 0.1–0.5.

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