Motivated by important geophysical applications we consider a dynamic model of the magma-plug system previously derived by Iverson et al. (2006) under the influence of stochastic forcing. Due to strong nonlinearity of the friction force for a solid plug along its margins, the initial deterministic system exhibits impulsive oscillations. Two types of dynamic behavior of the system under the influence of the parametric stochastic forcing have been found: random trajectories are scattered on both sides of the deterministic cycle or grouped on its internal side only. It is shown that dispersions are highly inhomogeneous along cycles in the presence of noises. The effects of noise-induced shifts, pressure stabilization and localization of random trajectories have been revealed by increasing the noise intensity. The plug velocity, pressure and displacement are highly dependent of noise intensity as well. These new stochastic phenomena are related to the nonlinear peculiarities of the deterministic phase portrait. It is demonstrated that the repetitive stick–slip motions of the magma-plug system in the case of stochastic forcing can be connected with drumbeat earthquakes.

It is well-known that the behavior of volcanic systems is enormously complex
so that a lot of nonlinear feedbacks lead to multiple states even during
a single eruption

Many uncertainties in physical parameters of volcanic dynamics

It is well-known, that an interplay between nonlinearity and noise can
generate various probabilistic phenomena such as noise-induced transitions

Some of the silicic volcanoes analyzed in detail over the last few decades
represent complex periodic systems

A scheme of the plug dynamics.

Time series of the cycle shown in Fig. 2:

In order to explain interactions between solid-state extrusion and persistent
drumbeat earthquakes at MSH, Iverson et al. (2006) developed a model based on
recurrent stick–slip motions of the solid plug along its margins with the
friction force

The following system of reduced governing equations based on the laws of
conservation of the solid plug linear momentum, solid plug mass and conduit
fluid mass was derived and discussed in detail by

In present paper we focus on the autonomous case, when

In Fig. 2,

Essential details of the phase portrait are shown in Fig. 2b by an enlarged
fragment of Fig. 2a. As one can see, there exists a pseudo-separatrix (dashed
red line) which divides two types of dynamics. If the initial state lies to
the left of this red curve, then the trajectory quickly verges towards the
vertical part of the cycle (arrows pointing to the left). If the initial
state lies to the right of this pseudo-separatrix, then the phase trajectory
goes away from the cycle (arrows pointing to the right), and only after
a long excursion, the trajectory approaches to the vertical part of the
cycle. Physically it means that small deviations in

Note, that this cycle is not a limit cycle in the classical mathematical
sense. Indeed, for different values of

Stochastic cycles for

In order to study possible deviations of the friction force from expression
(Eq.

At first we analyze the following

If the noise intensity is small enough, such bundle has a small dispersion
and is localized near the deterministic cycle (green lines in Fig. 5a). As
the noise intensity increases, along with the natural increase of dispersion,
the following unexpected phenomenon is observed: the bundle's right side of
stochastic trajectories is shifted inside the deterministic cycle (blue and
red lines in Fig. 5a). Some details of the corresponding probabilistic
distributions are presented in Figs. 5a, b. The probability density functions
of

This stochastic phenomenon can be explained by the phase portrait
peculiarities of an initial deterministic system (see Fig. 2b) near the upper
part of vertical fragment of the cycle. In the deterministic case, the phase
trajectory slowly moves along the vertical part of the cycle up to point

Under the further increase of noise intensity, random states of the system
(Eqs.

The dynamics of plug displacement is shown in Fig. 7. If the noise intensity is large enough so that the system leaves its cycle, the plug displacement increases with noise. If the system is within its cycle, the displacement is also within the corresponding deterministic stepwise curve (black line in Fig. 7).

Random trajectories

Displacement

Stochastic cycles

Influence of

In order to study an influence of possible changes in magma influx, let us
consider a role of

As one can see, the volcanic model under consideration demonstrates quite
different qualitative and quantitative responses to the random perturbations
of different parameters. The system is extremely sensitive to

The phase portrait of a deterministic system contains a point of unstable
equilibrium and a pseudo-separatrix, which subdivides the system into
different dynamic areas (point

In order to analyze the role of variations of two main parameters of the plug
motion (friction force

An important point is that an increase in dispersion occurs in the vicinity
of the pseudo-separatrix under the influence of

It is known that the eruption of Mount St. Helens was accompanied by rather
regular repetitive long-period (or drumbeat) earthquakes over a long time.
Moreover, such drumbeat events were more random from time to time. In
addition, subevents in the form of randomly occurring series of smaller
seismic events (produced by a separate random process) have been imposed upon
these long-period events

This work was supported by the Ministry of Education and Science of the Russian Federation under the project no. 315. Edited by: R. Gloaguen Reviewed by: C. Michaut and G. Wake