# Earthquake Processes: A View from Synergetics and the Theory of Catastrophes

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## Abstract

The theory of catastrophes and synergetics, developed by mathematicians in the second half of the twentieth century (Haken in Synergetics. Introduction and advances topics. Springer, Berlin, 1978; Arnol’d VI in Catastrophic theory. Springer, Berlin 2004), is used to analyze earthquake processes in active faults. It is shown that the oscillators of the seismic radiation are vibrating blocks at the border of the fault, and the source of the energy supply is the margin of the fault moving with the velocity *V*_{0}. The input or loss of energy is regulated by a nonlinear element, the frictional force *F*(*V*), which depends on the difference *V* between the velocity of the fault margin *V*_{0} and the velocity of the seismic block vibration: *V* = *V*_{0} − d*x/*d*t*. Precise conditions for the occurrence of an earthquake, its attenuation, or the generation of a tectonic tremor in its source are formulated.

## Keywords

Earthquake mechanisms nonlinear friction oscillations of rock blocks in faults## List of symbols

*A*Steepness of linear friction function

*F*(*V*) in Eq. (33)*a*Side of a block

*b*Dimensionless empirical constant in Eq. (3)

*C*Constant in Eq. (34)

- Co
Control parameter in Eq. (22)

*D*Effective coefficient of the frictional damping in Eq. (34)

- Da
Damping of oscillation

- Dis
Dissipative function in Eq. (35)

*E*Total energy (Lagrange’s function)

*e*Dimensionless constant in Eq. (23)

*F**F*^{*}Scale of the strength

*f*Polynomial function in Eq. (21)

*G*Block loading in Eq. (2)

*g*Gravity acceleration

*h*Coefficient of the lateral friction

- In
Integral in Eq. (25)

*k*Coefficient of elastic stiffness in Eq. (1)

*M*Young’s modulus

*m*Mass of a block

*N*Dimensionless load

*N*_{o}Shear stress at the upper boundary of the ocean slab

*p*Empirical dimensionless constant in Eq. (3)

*q*Dimensionless constant in Eq. (42)

*q*_{1}Dimensionless constant in Eq. (42)

*r*Empirical dimensionless constant in Eq. (5)

*S*Actual touch area of the contact zone

*s*Dimensionless velocity of the flank surface of the fault

*T*Temperature of the contact zone

*T*_{i}Tilt of the dropping characteristic of the friction

*t*Time

*t*_{e}Time of the catastrophe

*U*Potential in Eq. (53)

*U*^{/}First derivative of the potential

*U*by amplitude*α**U*^{//}Second derivative of the potential

*U*by amplitude*α**U*^{///}Third derivative of the potential

*U*by amplitude*α**V*Relative velocity

*V*_{o}Velocity of the lower margin of a fault

*V*_{L}Velocity when the friction

*F*is absent*V*^{*}Characteristic relative velocity in Eq. (3)

*W*Dimensionless relative velocity

*x*Displacement of the block

*y*Dimensionless coordinate in Eq. (6)

*Y*Coordinate in Eq. (34)

*z*Dimensionless coordinate in Eq. (16)

*z*_{0}The height of roughness protrusions on the contact surface

## Greek symbols

*α*Dimensionless amplitude of oscillations

*α*_{0}Initial amplitude

*α*_{c}Extreme points for the potential

*β*Coefficient in Eq. (3)

*δ*Dimensionless attenuation of the oscillations

*ε*Dimensionless constant in Eq. (15)

- Θ
_{1,2,3,4} Dimensionless integrative constant in Eqs. (26), (28), (30), (32)

- Θ
_{5} Constant value in Eq. (46)

- Θ
_{6} Constant value in Eq. (50)

*θ*Phase of oscillations in Eq. (18)

*κ**λ*Dimensionless constant in Eq. (23)

*μ*Dimensionless coefficient of the friction in law (2)

*ξ*Dimensionless coordinate in Eq. (5)

*ρ*Density of a basaltic block

*σ*Electrical conductivity of the contact zone

*τ*Dimensionless time

*ω*_{0}Frequency of the block self-oscillations

## Subscripts

*t*Differentiation by time

*τ*Differentiation by dimensionless time

## Notes

### Acknowledgements

The authors would like to thank two anonymous reviewers who thoroughly reviewed the manuscript; their critical comments and valuable suggestions were very helpful in preparing this paper.

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