The complex oscillatory behavior of a spring-block model is analyzed
via the Hopf bifurcation mechanism. The mathematical spring-block model
includes Dieterich–Ruina's friction law and Stribeck's effect. The existence
of self-sustained oscillations in the transition zone – where slow earthquakes
are generated within the frictionally unstable region – is determined. An
upper limit for this region is proposed as a function of seismic parameters
and frictional coefficients which are concerned with presence of fluids in
the system. The importance of the characteristic length scale

In the last decade, the study of slow earthquakes (tremors, low and very low frequencies events, and
slow slip events) has become of great relevance because of its possible relationship with the occurrence of large earthquakes.
The stress redistribution of slow earthquakes, and the strain in the lowest limit of the seismogenic layer caused by them,
could be helpful for a better understanding of the nucleation process of ordinary earthquakes

Observations suggest that this occurs between the seismogenic zone and the frictionally stable zone (Fig.

Transition zone related to slow-earthquake nucleation. The first dashed red line indicates the lowest limit of the shallow frictionally stable region. The second one shows the deeper limit of the seismogenic layer (upper limit of the deeper frictionally stable region). Self-oscillatory behavior is observed in the second transition.

Slow earthquakes occur in a variety of stick slip

The Dieterich–Ruina friction law has been successfully used to reproduce the stick-slip behavior in the models of earthquakes dynamics and slow earthquakes.
Spring-block models have reproduced these events when coupled with rate-
and state-dependent friction laws, and by contrast, models which have been
used with laws velocity-weakening friction and constant friction have not been
successful

An important issue of the rate- and state-dependent friction law is that it is totally
macroscopic, i.e., it describes the frictional properties of the system
rather than the microscopic mechanism which is responsible for the dissipation

A path to study slow earthquakes is through the spring-block model for ordinary
earthquakes, because both the slow and ordinary earthquakes are related by the
critical value of nucleation. Some spring-block models display complex oscillatory
behavior associated with the transition zone

In terms of dynamical systems based on the spring-block model, the presence of
oscillations and self-oscillations can be explained by the Hopf bifurcation
mechanism

In this paper the main objective is to propose an upper limit for the SSO region
in the frictionally unstable zone (Fig.

Dieterich–Ruina–Stribeck one-degree-of-freedom oscillator. Here,

The oscillatory behavior of the system is studied through analysis of the Madariaga spring-block model

An
external periodic perturbation

The faults are lubricated in the shear area

On the other hand, the well known phenomenological friction law of Dieterich–Ruina is
introduced in rock mechanics to capture experimental
observations of steady state and transient friction that depends on the
displacement history effects (state variable

Now it is possible to derive from Eqs. (

Defining
new variables

Parameters

The stationary solution

In the spring-block model, the logarithmic term in the Dieterich–Ruina friction law has introduced greater difficulties in solving the problem. Due to the
nonlinear term, analytic integration has not been possible, and even numerical solutions present challenges because of the logarithmic term

The dynamical system of earthquakes is a naturally dissipative phenomenon, and due to this feature
the dissipativity condition of the stationary solution is required. Thus, locally the system is dissipative at

Earthquake dynamics are a nonlinear oscillatory phenomenon

The equilibrium point

A sufficient condition for local and asymptotic stability comes from the Routh–Hurwitz criterion, i.e., the sufficient conditions to
ensure that Jacobian matrix Eq. (

Stability region for homogeneous system (

For
the region with the sufficient condition Eq. (

The relationship between the parameters

In order to determine how many eigenvalues of

On the other hand, if sign

Loci of eigenvalues for different values of

The oscillatory behavior is located before the branching, after which the system ceases
to oscillate. The locus of the eigenvalues for different values of

A set of parameters was found that satisfies the necessary condition Eq. (

Unperturbed system

The presence of oscillations in physical systems can be explained through
the mechanism of Hopf bifurcation. When three eigenvalues exist, and two of them
are complex conjugates and the other is a nonzero real, a Hopf bifurcation
occurs (Fig.

There are three necessary conditions in order for a Hopf bifurcation to occur:
(i) The existence of an equilibrium point,

For any system with three variables, the conditions (ii) and (iii) for
obtaining a Hopf bifurcation are given in terms of the characteristic polynomial
coefficients

From Eqs. (

Relationship between parameters

Our interest is for the case that

In the DR-S model any small perturbation in the system can change the dynamical behavior. If the DR-S system is subject to perturbations
from neighboring faults, the seismic fault enters in a limit cycle, but it does not remain long there due to intervening stress perturbations

This section aims to numerically describe the oscillatory behavior within and
outside the range proposed for the SSO region Eq. (

We are interested in the oscillatory behavior when the values of parameters

Projection of the attractor onto the plane

Figure

Qualitative changes in the dynamic of the system are better understood through bifurcation analysis, such as the case when a control
parameter is varied and the bifurcations show the transitions or instabilities of the
system. The unperturbed system is considered
when

Bifurcation diagram for an unperturbed system

An enlarged view of Fig.

Under the necessary condition Eq. (

According to previous outcomes, we considered the forced system

Bifurcation diagram

Bifurcation diagram

The bifurcation diagram for case one is displayed in Fig.

Conversely, Fig.

The region of SSO could be numerically explained by this analysis; if the system is perturbed slightly by external
forces

The bifurcation diagram for case two is displayed in the Fig.

In terms of the flow in phase space, a supercritical Hopf bifurcation occurs when a stable spiral changes into an unstable
spiral surrounded by nearly elliptical limit cycle

The set of parameters

However,

We have analyzed the DR-S model, which
describes the kinetic mechanism during an earthquake. The system displays richness
in their oscillatory dynamic
behavior: from attracting cycles of one and two periods to
a limit cycle and multi-periodic orbits,
depending on the parameter values in the necessary condition for
stability Eq. (

The complex oscillatory behavior discussed in this paper
is determined by the variation of parameter

The

Under the assumption

This would have some implications related to the slow-earthquake nucleation. For the SSO region,
for

This general behavior for small

The relevance of

Complementarily, we found if Stribeck's effect is added to the Madariaga model then the stationary
solution

The relative position

The stability analysis of

The relation between the SSO Eqs. (

The upper limit of SSO behavior is a function of seismic parameters and frictional coefficients concerned with fluids, although
this was established for the base of seismogenic layer, it is likely that it could be applied to the shallow transition zone
in addition to the parameters and constants related to the slip-hardening

Although this investigation is more related to the proposal of a formal pattern in the study of slow slip earthquakes (SSEs), and with a first approximation of the upper limit of the transition zone, this is considered as a preliminary study in order to be applied to the real seismogenic regions. However, the parameters considered for slow earthquakes are still being studied through observations, experiments, and by means of simulations, but there is still not something precise.

The study of SSEs in Cascadia

The proposed upper limit for the SSE zone includes the fluids and oscillation frequency (and consequently

The data set was obtained through simulations with the parameters indicated within the article. We use the fourth-order Runge–Kutta method.

The authors declare that they have no conflict of interests.

This study was supported by CONACyT (support 44731), the Departments of Applied Mathematics and Applied Geosciences at the Instituto Potosino de Investigación Científica y Tecnológica (IPICYT) and the Instituto de Geología, Universidad Autónoma de San luis Potosí, México. Eric Campos Cantón acknowledges the CONACYT financial support for a sabbatical at the Department of Mathematics, University of Houston. He would also like to thank the University of Houston for his sabbatical support and to Matthew Nicol for allowing him to work together closely and his valuable discussions on dynamical systems. The authors also acknowledge technical support from Irwin A. Díaz-Díaz. Edited by: William I. Newman Reviewed by: two anonymous referees