Introduction
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
.
Although most slow earthquakes have been detected
in subduction zones, there are reports of these in other types of faults .
Observations suggest that this occurs between the seismogenic zone and the frictionally stable zone (Fig. ) surrounding the critical value of ordinary-earthquake nucleation; i.e., on parts of faults where
the behavior is transitional between frictional properties on the rocks and slow, steady
deformation , but
some investigation revealed that slow earthquakes have been observed in shallow
regions .
Figure shows two stability regions and transition zones:
at shallow depth and on the base of the seismogenic layer , the latter of which is the focus of the study presented in
this paper.
determined that at the border of the stability transition there is a
region in which self-sustained oscillatory motion occurs into the
conditionally stable region, below the critical point, where
slow earthquakes are nucleated. These oscillations are observed in the presence or absence
of external forces such as vibrations from neighbor faults but eventually tend to stabilize.
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.
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Slow earthquakes occur in a variety of stick slip . say that observational
studies have provided information for their characterization (variability in duration, recurrence, and propagation velocity), but the mechanism of
slow earthquakes is still unclear. Experimental data show that
the physical and mechanical parameters that control changes in slow slip events
are the rates of convergence, frictional parameters and effective normal
stress, under rate- and state-dependent constitutive
properties of Dieterich–Ruina friction law .
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 . infer that simple friction
laws by themselves do not
provide an explanation for the complex behavior of slow earthquakes. Some investigations have
suggested that fluids play an important role in this mechanism .
Some studies support this idea: conclusions from revised literature establish that failures are lubricated on the shear area regardless
of the composition of
the rocks and the frictional weakening mechanism involved .
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 .
Experimental research on
the role of fluids in the ordinary-earthquake mechanisms has captured specific
features associated with the molecular-layer lubrication on the border between two
surfaces in contact, with different frictional properties
when considering lubricants in volume and dry interfaces .
The initiation slip on the microscopic scale is associated with a shear melting
transition in the lubricant layer so that microscopic
scales contribute to better understanding of the friction mechanism.
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 .
and , used a spring-block model
where complex oscillatory behavior
was found near to a critical value of the earthquakes nucleation.
This behavior is presented as changing when there is a variation in any parameter related
to the critical value.
introduced, in a spring-block model,
an alternative friction law, where the state variable is interpreted in terms of the
shear melting of the lubricant
(molecular layer of lubricant) between solid surfaces in contact. They considered
that by incorporating the microscopic mechanism it could determine other behavior and found
that the transition from steady sliding to stick slip is typically discontinuous and sometimes
hysteretic. This transition is associated with a subcritical Hopf bifurcation (set of seismic parameters
for the critical point of nucleation). , , and observed a sudden and
discontinuous onset in the amplitude of oscillations at the
bifurcation point. Surrounding the transition, complex oscillatory behavior was observed in these cases.
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 .
Some nonlinear dynamical systems show self-sustained oscillation (SSO) motion , one of
them being relative to the earthquake physics mechanism
. The SSO behavior would be explained in the context of the
complex system of faults, such that the modeling of movement in a
single failure would be affected by external forces .
These forces can be generated due to vibrations or stress
transferred from neighboring faults, that make the system go from a limit cycle to another,
leaving different paths of recurrence .
In this paper the main objective is to propose an upper limit for the SSO region
in the frictionally unstable zone (Fig. ).
The upper limit is a function of mathematical and numerical relations in terms of seismic and frictional parameters.
This limit
describes the complexity of the oscillatory movement in the nearest
region to the critical point of nucleation. Another objective is to determine the role of the fluids in this region.
Dieterich–Ruina–Stribeck one-degree-of-freedom oscillator. Here, v0 is a reference velocity, v is the block velocity,
k is the constant of deformation, and β3 is the dynamical viscosity coefficient; τ(t) is an external and perturbing
force with angular frequency ω.
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The oscillatory behavior of the system is studied through analysis of the Madariaga spring-block model
complemented by the Stribeck effect . The Stribeck effect
shows the transition from dry interfaces or lubricated
at the border until separated by a layer of lubricant as a shear melting. This effect
takes into account the microscopic mechanism between
the contacting surfaces during displacement. The Dieterich–Ruina–Stribeck (DR-S) oscillator (Fig. )
depicts the behavior of the kinetic mechanism between tectonic blocks in the Earth's crust undergoing stick-slip
effects from friction. The oscillator is coupled to a driven plate by the rheological properties of rocks, and to a
static plate by frictional properties. The slider is on the rough and lubricated surface. The relative displacement,
rate of displacement, and acceleration are given by u=y-v0t, u˙=v-v0, and u¨=v˙,
respectively, where y is the block displacement and v=y˙.
An
external periodic perturbation τ(t) is introduced
in order to show that this system exhibits SSO behavior between the
critical point of nucleation and the region limited by the proposed upper limit.
In Sect. 2 the model is proposed, and the linearized system, stationary
solution,
and criterion for frictional stability are analyzed. In Sect. 3 the
oscillatory behavior is described, the upper limit of SSO region is proposed, and numerical simulations are provided.
Finally, in Sect. 4 a brief discussion of the outcomes
and conclusions are presented.
Nonlinear dynamical system
The model
The faults are lubricated in the shear area as
the most sliding contacts. A model based in a slider block (Fig. )
is essentially a mechanic representation, and the frictional components are velocity-dependent.
explain Stribeck's effect as follows.
The friction force
varies with the sliding speed depending on the
extent to which the interacting contact surfaces are running
under boundary, mixed (fluid in the union between asperities),
or full film lubrication (a layer of fluid as a shear melting). Even dry
contacts show some behavior similar to that in lubricated
contacts in that they have a higher static friction than
dynamic or sliding friction. In lubricated sliding contacts,
the friction decreases with increasing sliding speed until a
mixed or full film situation is obtained, after which the
friction in the contact can be constant, increasing or
decreasing somewhat with increasing sliding speed due to
viscous and thermal effects. This transition is named Stribeck's effect
and has been formulated as follows:
Fs(v)=β1sign(v-v0)+β2e-μ(v)sign(v-v0)+β3v,
where β1 is the Coulomb friction, β2=Fmax-β1 (with Fmax
as the upper limit of static force), β3 represents the dynamical viscosity
coefficient of the fluid, and μ denotes a slip constant or decay parameter for the
mixed lubrication.
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 θ) and velocity ,
one of its formulations is given by two equations :
Fdr(v,θ)=θ+Alnv/v0,θ˙=-v/Lθ+Blnv/v0.
The first equation represents the frictional stress under a stable state (θ˙=0), and the second one corresponds
to evolution of state variable θ that evolves with time, slip, and normal stress history . L is a
characteristic sliding distance over which θ evolves
(required distance to renew the contact population). A and B are material properties,
and we assume B>A>0 for frictional instability.
Now it is possible to derive from Eqs. () and () an alternative friction
law taking into account the macroscopic and microscopic mechanism (frictional properties and dissipation).
The equation of motion derived from Eqs. () and () gives a
first-order differential equation system; a similar expression is given by , but we complemented their system by the term of
Stribeck's effect:
θ˙=-(v/L)[θ+Bln(v/v0)],u˙=v-v0,v˙=-(1/M)[ku+Fdr(v,θ)]-(1/M)F0(v)+τ(t),
where Stribeck's effect from Eq. () is given now by
F0(v)=β1+β2e-μ(v)+β3v.
Note that the slider block velocity is relative to velocity of driver plate. From the second equation of the system Eq. (), we infer that the block will
oscillate with respect to the position of the plate. This equation tells us whether the
oscillator is to the right or to the left with respect to the driver plate. Although the direction of the displacement
of the system is always forward if v-v0>0, i.e., if v>v0, eventually the oscillator will
be more advanced than the plate. Conversely, if v-v0<0, i.e., v<v0 eventually
the oscillator will be to the left of the plate. One of the objectives when introducing the
function sign(v-v0) in the Eq. () is to indicate this effect; assigning value 1 for the first case, -1 for
the second case, and zero otherwise. We eliminate the sign function because this effect
is already considered in the second equation of the system Eq. ().
Defining
new variables θ^, v^, u^, and t^ as θ/A=θ^,
v/v0=v^, u/L=u^, t(v0/L)=t^, the dimensionless system is given by the following
equations:
θ^˙=-v^[θ^+(1+ε)ln(v^)],u^˙=v^-1,v^˙=-γ2[u^+(1/ξ)(θ^+ln(v^))]-αF0(v^)+τ^(t^),
where α=(α1,α2,α3); α1,2=Lβ2v02M, α3=Lβ3v0M.
The external force is τ^(t^)=c^sin(w^t^), where c^=Lv02 and w^=Lwv0. The frictional
parameters α are associated
with frictional coefficients from the Stribeck effect and
αF0(v^)=α1+α2e-μ^v^+α3v^,μ^=v0μ.
The variable x=(θ^,u^,v^) is the dimensionless state variable, and θ^ stands
for the measurement of contact with asperities from the Dieterich–Ruina friction law;
u^ is the dimensionless relative displacement between the
block and the upper plate, and v^>0 is the dimensionless velocity of the block.
The function f(x) on the right-hand side of Eq. () defines a mapping f:R3→R3. This mapping defines
a vector field on R3. Thus, the system given by Eq. () induces in phase space R3 the flow (φt),
and t∈R such
that each forward trajectory of the initial point x0=x(t=0) is the set {x(t)=φt(x0):t≥0}.
Parameters Π=(ε,ξ,γ) are given as follows:
ε=(B-A)/A,
ξ=kL/A, and γ=(k/M)(L/v0), associated with stress drop during
displacement, deformation, and the oscillation frequency, respectively.
The Eq. () is referred to the system of DR-S
where τ^(t^) is the periodical and deterministic
external force τ^(t^)=c^sin(ω^t^);
here ω^ is the angular frequency. We designated a system as an unperturbed system when τ^(t^)=0
and a perturbed system otherwise. We will denote (θ^,u^, v^):=(θ,u,v), τ^(t^):=τ(t), ω^:=ω,
c^:=c, and μ^:=μ.
Stationary solution at equilibrium point
The stationary solution x=x* of the system Eq. () with τ(t)=0 is given by
x*=(θ*,u*,v*)=(0,η,1),u*=η=(α1+α2e-μ+α3)/γ2,γ>0,
where η corresponds to the relative position of the single slider block. At x* the plate
and the block have the same velocity and the measure of the asperities contact is zero. Note
that θ* and v* do not depend on Π but u* depends on frequency
oscillation γ (consequently on kL) and frictional constants of Stribeck's effect,
both are associated with the energy dissipation.
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 linearized system analysis is very useful to describe some
features of the nonlinear system about a steady-state solution . The local and asymptotic stability of
x* is analyzed with the indirect method of Lyapunov that
consists of the analysis of the eigenvalues of the Jacobian matrix from the linearized system of Eq. () around the stationary
solution .
Let x* be locally asymptotically stable, i. e., every solution of the system φt(x0)=(θ(t),u(t),v(t));
starting near to the stationary solution, it remains at the surrounding of x* all the time, and eventually the solution converges to
x* (convergence to frictional stability).
Let us denote the Jacobian matrix as
Df(x*)=(∂fi(x)/∂xj)|x*,
for i,j=1,2,3, where f(x)=(f1,f2,f3) is the vector field given by the right-hand side of Eq. (), with
τ(t)=0,
(x1,x2,x3)=(θ,u,v), and λi
be the eigenvalues of Df(x*):
Df(x*)=-10-(1+ε)001-γ2/ξ-γ2-γ2/ξ-ϕ,
where ϕ=α2μe-μ-α3.
The characteristic polynomial of Eq. () is given by the following:
P(λ)=a0λ3+a1λ2+a2λ+a3,
whose coefficients are in terms of seismic parameters Π and frictional
coefficients α and μ.
a0=1a1=1+γ2/ξ+ϕa2=γ2(1-ε/ξ)+ϕa3=γ2
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 x*
if a1=-TraceDf(x*)=∑i=13λi<0 that is true under the following condition:
Mv0L+Av0+β2μe-μv0<β3.
Equation () comes directly from a1<0 and the values
of γ, ε, ξ, and ϕ. The Eq. () is the
necessary condition for the system to be subdamped , and oscillations can be observed;
moreover, due to a3=-detDf(x*)<0, x* is hyperbolic. Through
the analysis of the eigenvalues of Df(x*) we will
explain what implies a hyperbolic equilibrium point related to oscillatory behavior.
Oscillatory behavior
Earthquake dynamics are a nonlinear oscillatory phenomenon
,
where the nonlinear complex behavior
is attributable to the friction forces. The analysis of the oscillatory behavior is explored in this section. We use the full nonlinear term in
the numerical simulation in Sect. 3.2 and 3.3.
Analysis of eigenvalues
The equilibrium point x* is locally asymptotically stable if the
real part of all the eigenvalues is negative, i.e., Re(λi)<0,
and it is unstable if at least one eigenvalue of Df(x*) is positive, i.e., Re(λi)≥0.
We are interested in the type of hyperbolic stationary solution x*=(0,η,1):
it has a stable manifold, i.e., Re{λi}<0, Im{λi}=0,
and a unstable manifold that generates oscillations in a plane, i.e., Re{λk}>0, Im{λk}≠0 .
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. () has three eigenvalues with a negative real part are that the coefficients of the characteristic
polynomial holds the following inequalities:
a0,a1,a2,a3>0,a1a2-a0a3>0.
Note that the first inequality holds (a0,a3>0), and a1>0 because ϕ<1 and Π>0; if sign(a2)>0 we
deduce that
ε<ξψandψ=1+ϕ/γ2
as necessary conditions for stability but are not sufficient. The sufficient condition for stability comes from the second inequality of
Eq. () and a2>0 as follows:
ε<ξ1-11+γ2/ξ+ϕ+ϕγ2.
Stability region for homogeneous system (τ(t)=0), as a
function of (ε,ξ,γ) for fixed values of ξ. (a) ξ=0.4,
(b) ξ=0.8, and (c) ξ=1. The necessary and sufficient conditions are satisfied.
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For
the region with the sufficient condition Eq. (), all eigenvalues of the Jacobian
matrix Eq. () have negative real part and the equilibrium point x*=(0,η,1) is a sink .
The relationship between the parameters Π associated with the necessary
condition Eq. () and sufficient condition Eq. () are
described in Fig. a, b, and c, for fixed ξ=0.4,0.8,1, respectively.
By means of numerical simulation the stability was computed and found for values of γ>0.5.
In order to determine how many eigenvalues of Df(x*) are real or complex conjugates, Descartes' rule
of signs to analyze the roots of the characteristic polynomial is used. Under the condition Eq. (),
signs of coefficients Eq. () are
(+,+,sign(a2),+).
If sign(a2)>0 there are two possibilities: all eigenvalues are
negative real, i. e., Re{λi}<0, Im{λi}=0, or one eigenvalue
is negative real and the other two are complex conjugates; the last statement corresponds
to the oscillatory behavior.
On the other hand, if sign(a2)<0, there are possibly two positive real
eigenvalues, i.e., Re{λk}>0, Im{λk}=0 and one
negative real; for this case there are no complex conjugate eigenvalues, and hence
nonoscillatory behavior is observed. For all cases there is one negative
real eigenvalue, the other two could be complex conjugates or positive real.
Figure shows the locus of the real part of two eigenvalues
corresponding to the oscillatory and the nonoscillatory behavior; it
describes the relationship between parameters Π. The graph for the
negative real eigenvalue was omitted because we focused on complex
eigenvalues.
Loci of eigenvalues for different values of ξ and γ. The Figure
shows the real part of eigenvalues as a function of parameters ε, ξ and γ.
For (a) and (b) fixed γ=0.8 and 10, respectively; (c) and (d) shows eigenvalues
for fixed ξ=0.8 and 1, respectively.
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The oscillatory behavior is located before the branching, after which the system ceases
to oscillate. The locus of the eigenvalues for different values of ξ=0.4,0.8,1 are shown in Fig. a, b
for γ=0.8 and γ=10, respectively. The range of ε decreases
with the increasing of γ and the decreasing of ξ.
Figure c and d describe
the same behavior as explained above, and it was observed that for values of γ>10 the
range of ε for oscillatory behavior is almost equal, although after the branching
point with the increasing of γ, one of the eigenvalues increases rapidly (i.e., kL
increases, making the system stiffness the biggest or making L increase) and the other one tends to zero, more quickly.
A set of parameters was found that satisfies the necessary condition Eq. () for stability,
within an unstable and oscillatory region which is around the Hopf bifurcation (set of seismic parameters
that satisfies the critical value of nucleation; Fig. ). The region with these
features is proposed for the region of self-sustained oscillations and consequently
for the slow-earthquake zone. The SSO region is in the unstable region
and will be explored in the follow sections.
Unperturbed system τ(t)=0. Projection of attractor onto the plane (u,v). Panels (a), (b), and (c) have the position
of equilibrium point u*=η near zero for γ=10, and ξ decreases as it is the range of u. Panels (d), (e), and (f),
for γ=0.8, have both u* and the range of values for u higher than (a)–(c).
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A Hopf bifurcation, oscillatory range (OR) and the self-sustained oscillation (SSO) region
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. ) if the real part of the complex eigenvalues cross
the imaginary axis. Periodic orbits and limit cycles are either created or destroyed for the nearest values of
the Hopf bifurcation . If all neighboring
trajectories approach the limit cycle then it can be said that the limit
cycle is stable or an attractor. Stable limit cycles are important because they model systems that exhibit SSO behavior ,
therefore starting at Hopf bifurcation, the oscillatory behavior around it will be analyzed.
There are three necessary conditions in order for a Hopf bifurcation to occur:
(i) The existence of an equilibrium point, f(x*)=0; (ii) the Jacobian
matrix Df(x*) has a couple of eigenvalues on the imaginary axis, i. e., Re{λ1,2}=0 and Im{λ1,2}≠0; and (iii) the
cross-velocity of eigenvalues through imaginary axis must be different to zero.
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 a1, a2, and a3 from Eq. () .
The cross velocity is given by the derivative
r′(Π0)=ddγ(Re{λ(Π)})|γ=γ0≠0, where
Π0=(ε0,ξ0,γ0) is a set of fixed values and λ(Π)
is an eigenvalue of Df(x*) at Π. The derivative is with respect to
the bifurcation parameter, γ (oscillation frequency). The cross velocity is
r′(Π0)=a3′(Π0)-b2a1′(Π0)+λ0a2′(Π0)2(λ02+b2),a1(Π0)=-λ0,anda2(Π0)=b2.
The polynomial from Eq. () has a couple of imaginary roots λ1,2
if there is a Π such that the following two relations are satisfied:
a3=a1a2,a2>0.
If Π=Π0 satisfies the Eq. (), then
the complex roots of Eq. () with real part zero are determined by
λ1,2=±ib0,b0=a2,λ3=-a3a2.
From Eqs. () and (), the eigenvalues of Eq. () for the Hopf bifurcation are given by
λ1,2(Π0)=±iγ02-γ02εHBξ0+ϕ,λ3(Π0)=-γ02γ02-γ02εHBξ0+ϕ.
The Jacobian matrix Eq. () has two complex conjugate
eigenvalues with a positive real part for values of ε which
depend on the fixed values γ0 and ξ0. These eigenvalues
correspond to oscillatory regions in the unstable regime, and they are observed from
the Hopf bifurcation to the beginning of bifurcation as it is depicted in the
Fig. ;
this interval is called the oscillatory range OR. We want to find
the limits of OR and the SSO in terms of seismic parameters.
From Eqs. () and () it is determined εHB,
i.e., the ε value for the Hopf bifurcation
εHB=ξ01-11+γ02ξ0+ϕ+ϕγ02,
where εHB is the lower limit of OR. The
upper limit is determined by the discriminant of the
third-order polynomial a0λ3+a1λ2+a2λ+a3:
D=3a2-a1293+9a1a2-27a3-2a13542.
If D>0 there are two complex conjugate roots
and one is real; if D=0 all are real roots, and at
least two are equal; and if D<0 all roots are real and unequal.
We are interested in D=0, which implies that the oscillatory behavior
finishes (there are not complex roots). From Eqs. (), (),
and ()
D=3γ02-γ02εD=0ξ0+ϕ-1+γ02ξ0+ϕ293+91+γ02ξ0+ϕγ02-γ02εD=0ξ0+ϕ-27γ02-21+γ02ξ0+ϕ3542=0,
and we can resolve by εD=0, according to
Fig. . The OR for ε is in the interval
OR∈(εHB,εD=0).
In the OR, the necessary condition Eq. () for stability is satisfied by
a set of parameters Π around the Hopf bifurcation; the region with these features
is proposed for the SSO. The proposed interval for SSO is
SSO∈(εHB,ξ0ψ0),ψ0=1+ϕγ02.
Some numerical results are summarized in Table .
Relationship between parameters Π on the oscillatory
range for Fig. a and b εHB is the value of the Hopf bifurcation; (±λ1,2)HB is the eigenvalues on
the imaginary axis; εD=0 is the value when the discriminant of the
characteristic polynomial is equal to zero, i.e., the upper limit of OR; SSO is the interval of self-sustained oscillations; and r′(Π0) is the cross-velocity of eigenvalues.
γ
ξ
εHB
±(λ1,2)HB
εD=0
SSO∈(εHB,ξ0ψ0)
rε′(Π0)
0.8
0.4
0.1981
0.5030i
2.081779482130000
(0.1981,0.3562)
0.3042
0.8
0.2499
0.6083i
3.704570372000000
(0.2499,0.7123)
0.2058
1.0
0.2534
0.6385i
4.507300000000000
(0.2534,0.8904)
0.1749
10
0.4
0.3981
0.6313i
1.668570861615000
(0.3981,0.3997)
0.4981
0.8
0.7931
0.8911i
2.601243068598171
(0.7931,0.7994)
0.4963
1.0
0.9894
0.9954i
3.018387963893687
(0.9894,0.9993)
0.4953
Our interest is for the case that ε=(B-A)/A>0, or B-A>0, which means that the stress drop is negative and consequently an
unstable regime is observed; under this assumption, ξψ it is a positive amount implying that ψ>0.
Equation ()
is a necessary condition for stability, which is maintained within a set of values of parameters where the system is unstable (Table ).
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 .
The system under forcing conditions
This section aims to numerically describe the oscillatory behavior within and
outside the range proposed for the SSO region Eq. (), under
forcing and nonforced conditions. We want to prove numerically that the
proposed upper limit determines the changes in oscillatory behavior, below
and above this. For more theoretical background into the theory of periodically forced systems near a point of Hopf bifurcation,
see and references therein.
We are interested in the oscillatory behavior when the values of parameters Π are nearest to the Hopf bifurcation.
We numerically explored the effects of an external, deterministic and periodic force τ(t) acting on the system Eq. (),
for Π=(0.25,0.8,0.8), α=(1.6,0.2,0.1), and μ=3. Such effects are illustrated by varying the angular frequency
ω from τ(t)=sinωt. This numerical analysis is helpful for visualizing patterns in the dynamic of the system,
especially those related to oscillatory behavior, such as limit cycles and periodic orbits.
Projection of the attractor onto the plane (u,v) and time series of the perturbed system . Panels (a), (b), and (c) show
the projection onto the plane for ω=0.1,1.2, and 2, respectively.
Panels (d), (e), and (f) display their respective time series for u.
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Figure shows numerical results of typical oscillations projected onto the plane (u,v) and time series generated by Eq. ().
The system Eq. () presents different behaviors when the parameter ω takes values in the interval [0.1,2].
For example when ω∈[0.1,0.8), a type of complex behavior for low frequencies is observed (Fig. a).
Figure b shows that for values of ω∈[0.8,1.3), periodic orbits of period two are found. For angular
frequencies ω>1.3 the flow of the system converges to a limit cycle, as shown in Fig. c. Conversely,
Fig. d to f describe the time series of u when ω is 0.1, 1.2 and 2, respectively. The motion is periodical,
this behavior emphasizes the periodic motion of the DR-S. The time series for low frequencies (ω<0.8) are more complex than the other cases.
Bifurcation diagram for unperturbed system
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 τ(t)=sin(ωt)=0, i.e., ω=0. In order to show that the behavior of the unperturbed system displays SSO,
the control parameter γ=(k/M)(L/v0) is varied, which in turn is related to frequency of oscillation of the
slider block and to the characteristic length of displacement L. Numerical results are for fixed ξ=0.8, and the value
for ε=0.25 that holds the necessary condition Eq. ().
Bifurcation diagram for an unperturbed system τ(t)=0, bifurcation parameter γ versus local maximum
of time series ulmax(t), and Π=(0.25,0.8,γ).
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An enlarged view of Fig. .
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Under the necessary condition Eq. (),
the system without external perturbations oscillates, and
multi-periodic orbits are created over an approximate range of values γ∈[0.6,1].
Eventually, the flow of the system converges to a limit cycle as is depicted
in Figs. and .
This oscillatory behavior is observed when Π is nearest to the Hopf bifurcation in absence of external forces.
The range of ulmax(t) decreases when γ increases.
Bifurcation diagram for perturbed system
According to previous outcomes, we considered the forced system τ(t)=sinωt centered at ω={0.1,1.2},
for fixed ξ=0.8, and two values of ε. Two values for ε={0.25,1} are explored. The first value
holds the necessary condition and the second one does not. The bifurcation parameter is γ. We named case one of the
analysis of bifurcation with fixed ε=0.25, and the second case for ε=1; for both cases ξ=0.8 is fixed.
Bifurcation diagram γ versus local maximum from the time series ulmax(t) Π=(0.25,0.8,γ), μ=3,
α=(1.6,0.2,0.1), sub-damped system. In panel (a) ω=0.1, and in (b) ω=1.2.
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Bifurcation diagram γ versus local maximum from time series ulmax(t), ω=1.2, Π=(1,0.8,γ) μ=3,
α=(1.6,0.2,0.1), sub-damped system. (a) There are three types of behavior: Type I, II, and III. (b) Enlarged view of Type I.
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The bifurcation diagram for case one is displayed in Fig. , and it shows the qualitative behavior of system relative to
the variable of position u and oscillation frequency γ. Figure displays γ versus its local maximum
of time series ulmax(t). Figure a for ω=0.1 shows that the most orbits are weakly attractive until approximate
values of γ>0.9 when limit cycles are observed.
Conversely, Fig. b for ω=1.2 shows limit cycles, which are observed approximately when
γ∈[0.55,0.66), γ>1.17, and γ=.765; whereas there are two points of bifurcation from orbits
of period one to orbits of period two at γ=0.66 and γ=0.765. The orbits of period two have the form of
Fig. b, and the limit cycles take the form as in Fig. c. The orbits are strongly attractive,
which suggests that they could be stable at least during a short period. According to the necessary condition for stability,
the behavior of ulmax(t) relative to γ decreases for both ω=0.1 and ω=1.2 because after γ
reaching the values 0.9 and 1.17, respectively, the system falls into the limit cycle, in such a way that when γ
increases, the range of values for ulmax(t) decrease. The type of behavior displayed in Fig. is expected when ε<ξψ.
The region of SSO could be numerically explained by this analysis; if the system is perturbed slightly by external
forces τ(t), it always returns to the standard cycle.
The bifurcation diagram for case two is displayed in the Fig. a for ω=1.2. The necessary condition
Eq. () fails; and the dynamic of the system shows three types of behavior: behavior Type I (Fig. b)
is observed approximately at γ∈(0.6,1.55). Type I shows periodic orbits of period one and two that appear to be
alternating, and bifurcations from orbits of period one to period two occur, then the system reaches a limit cycle at γ=1.55
but now with Type II behavior for approximate values of γ∈(1.55,3.5). In Type II, ulmax(t) increases while
maintaining the limit cycle until behavior Type III is observed approximately for γ∈(3.5,5.5), which displays periodic
orbits with different period. The behavior displayed in Fig. corresponds to an unstable and oscillatory region outside of the SSO.
Type of Hopf bifurcation
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 . A subcritical Hopf bifurcation occurs when a small
perturbation can lead to either decaying oscillations due to a stable equilibrium or a jump to large sustained
oscillations in the system due to an unstable limit cycle. For the analysis of the bifurcation type, the main
challenge is the numerical stiffness, due to the nonlinear logarithmic term.
The set of parameters Π does not cross the Hopf bifurcation if γ>γHB. Small disturbances decay
after ringing for a while and a stable spiral is observed. The block and the driver plate are moving at constant rate v=1,
and the relative position is η. Conversely, for γ<γHB, the parameter values cross the Hopf bifurcation.
The equilibrium state loses stability and an unstable spiral is observed. This type of bifurcation is expected for smooth, noncatastrophic changes (see Supplement).
The slow earthquakes are almost imperceptible because the displacement rate is very low compared to ordinary earthquakes and they are generated
for parameter values around the critical value of nucleation. Hopf bifurcation is supercritical within the proposed limits for unforced system.
To find chaotic behavior or strange attractors with the nonforced system it is necessary to vary epsilon very far from the
value of the Hopf bifurcation that we are analyzing.
However, have found chaotic behavior for small values of Π by introducing a time delay in the friction term. They have
found two types of Hopf bifurcation depending on the variation of the time delay.
Similarly, by introducing the external force τ(t) a subcritical Hopf bifurcation could be given for some critical τ(t) and slight
variation of the ε and ξ parameters. Disturbances do not allow the system to remain at an equilibrium point, resulting in
continuous oscillations or chaos. For the case when the set of parameters Π crosses the Hopf bifurcation, continuous oscillations
were found in both displacement and velocity only by varying the bifurcation parameter (see Supplement). Determining critical values of Π and τ(t)
requires more concrete study.
Discussion and conclusions
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 necessary condition is maintained even for a set of parameters
within the frictionally unstable region. This behavior was
studied by bifurcation diagrams through the Hopf bifurcation mechanism.
The complex oscillatory behavior discussed in this paper
is determined by the variation of parameter γ,
fixed values ε and ξ, and the necessary condition for
stability Eq. (). The necessary condition depends on the parameter γ, and this depends on
the characteristic length L, suggesting that the complex
oscillatory behavior should be observed for a range of values for the
parameter γ. The unperturbed and perturbed cases have shown how the system describes behavior as it is found in systems
with SSO; this is in the case when the necessary condition
Eq. ()
is maintained (Figs. and ). There are self-oscillations that create multi-periodic orbits but eventually
converge to a limit cycle and the range of ulmax(t) decreases when γ increases.
The γ parameter seems to be relevant, and consequently the characteristic length L.
, , and derived
some conclusions concerned with L from laboratory experiments and numerical simulations. They
associated L with the size of small earthquakes, with frequency, and with SSO. If L decreases
then the block is frictionally less stable, i.e., there are more frequent displacements; moreover, they found that under a range of
parameter values the oscillations can change from periodic to aperiodic
and vice versa, but the border of the transition was not defined.
Under the assumption B>A>0→ε>0 and Eq. (),
Hopf bifurcation occurs for any γ≥0.5 within the standard
0.4<ξ0<1;
moreover, εHB→ξ0 when γ increases in such way
that for any γ≥10 the values of εHB are equal. For these
values, ε is bounded in 0<εHB<1.
This would have some implications related to the slow-earthquake nucleation. For the SSO region,
for 0.5<γ≤10, the oscillation frequency and fluid are approximately determinant
for the systems that have unstable oscillations. For small values of γ there are more
transitions in the dynamical oscillatory behavior, when the necessary
condition Eq. () holds and does not it as it is shown
in Figs. , , and .
On the other hand, for γ≥10 there are less transitions, and neither
the fluid nor frictional coefficients of Stribeck in the medium affect
the oscillatory behavior (ϕ/γ2→0→ψ→1).
This general behavior for small γ seems to be independent of the angular
frequency, ω, of the external force because even the unperturbed
system displays this behavior. The function of the ω variation is the generation of the
dominant type of orbits (one, two, or more periods) and its frequency of
transition,
which is bigger when the necessary condition fails. The complex oscillatory behavior
is dependent on small γ values and of the necessary condition
for stability Eq. () that involves fluids from Stribeck's effect.
The relevance of γ seems to be associated with the
presence of fluid. determined that the stationary
solution x*=(0,0,1) is independent of seismic parameters, and the relative
position of the block and the plate will not depend on the frictional coefficients
or seismic parameters; the system would be under the limit of the
unstable regime where the big earthquakes are nucleated. At this solution the block and the
plate have the same velocity and there is no contact with asperities, and hence
there is no displacement between the plate and the slider block.
Complementarily, we found if Stribeck's effect is added to the Madariaga model then the stationary
solution x*=(0,η,1) has the displacement or relative position as a function of
seismic parameter γ. The parameter γ is associated with the oscillation frequency of the block as well as
the frictional constants related to fluid through ψ in the necessary
condition Eq. ().
The relative position u is not necessarily zero, although
the relative displacement is. The role of frequency oscillations and fluid is
decisive for dissipation and consequently it is also for the block position.
If γ is large enough then x*→(0,0,1), i.e.,
if γ or kL increases, then u*→0. The last statement could be
interpreted as k increases making the system stiffer or L increases making
the stress drop slower under some conditions.
The stability analysis of x* was through eigenvalues of Eq. (). If
the real part is negative for all eigenvalues then the stationary solution is a sink and
the system is in the stable regime, the medium properties break any nucleation or propagation
of earthquakes. determined a critical value for frictional stability or instability
under an elastic medium, with an oscillator coupled to the Dieterich friction
law,
σ‾c=-kLa-b,
that depends on the rocks properties, point nucleation, and
frictional parameters a-b, k, yL. The critical value σ‾c corresponds
to the normal effective stress. When any σ‾ holds σ‾≥σ‾c,
then there are changes in the frictional properties, such changes unchained earthquakes. This phenomena is named
frictional instability. reported that the SSOs are in the stable regime, below the critical
value of nucleation Eq. () (Fig. ). In a complementary way this
investigation reveals that the critical value Eq. () of have got
to the upper limit of SSO in the unstable region, and it is related to the upper limit of SSO in
the frictionally unstable region regardless of the Stribeck effect.
The relation between the SSO Eqs. () and () comes
from the definition of Π=(ε,ξ,γ) and the
statement (A-B)=σ(a-b) as follows. From
Eq. ()
σ‾c(b-a)=kL→(B-A)=kL→(B-A)/A=kL/A→ε=ξ; conversely
ε=ξψ→σ‾c(b-a)/A<kLψ/A→σ‾c=-kLψ/(a-b).
A corrected value for SSO is as follows:
1<σ‾c*≡-kLψ(a-b)≡σ‾cψ,
which combines frictional parameters of Dieterich–Ruina friction law and Stribeck's effect.
Equations () and () are equivalent, if γ>10, ψ→1 and σ‾c*→σ‾c.
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 . The fluid presence involves the
frequency of oscillation of the block as a very important element to dissipation and consequently with the stationary solution
(equilibrium point of the system) as well as in the upper limit proposed for slow-earthquake zones. The characteristic length L
has primary relevance for the results of this research.
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 indicates a possible link between the observational and experimental data, with
the parameters involved in the most of models of earthquake's physic coupled to the Dieterich–Ruina's friction law. The slip
amount of SSEs is on centimeter scale but the average slip amount of smaller events are unknown. The effective normal stress in the
range of 3–9 MPa produces a fault slip consistent with some observed SSEs, and B-A is in the range (0.0015 to 0.003) of the slow
slip section. At the top of the slow slip section B-A is 0.003 and 0.001 at the base, A≈0.02, L is in the range 1–50 µm
(real L is unknown), and the rate of convergence (10–50 mm yr-1) represents the range of convergence rates of subduction zones where SSEs
are observed with GPS. These parameters could vary depending on the region in which SSEs occur.
Further, the critical value Kc=(A-B)σ/L depends on L; viscosity = 0.1 (nondimensional) has been used in earthquake
models , but the estimation of the real viscosity depends on the region.
The proposed upper limit for the SSE zone includes the fluids and oscillation frequency (and consequently L), through ψ.
They might be introduced into the simulations and experiments in order to see what the implications are for the recurrence times,
duration, and velocity of SSEs in real seismogenic regions. A final step would be using scaling laws for SSEs to determine whether the real
values of parameters included either experimental and/or simulation data, such as the stiffness Kc and viscosity, take into
account the specific characteristics of the fault.