scholarly journals An upper limit for slow-earthquake zones: self-oscillatory behavior through the Hopf bifurcation mechanism from a spring-block model under lubricated surfaces

2017 ◽  
Vol 24 (3) ◽  
pp. 419-433
Author(s):  
Valentina Castellanos-Rodríguez ◽  
Eric Campos-Cantón ◽  
Rafael Barboza-Gudiño ◽  
Ricardo Femat

Abstract. 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 L, the implications of fluids, and the effects of external perturbations in the complex dynamic oscillatory behavior, as well as in the stationary solution, are take into consideration.

2016 ◽  
Author(s):  
Valentina Castellanos-Rodríguez ◽  
Eric Campos-Cantón ◽  
Rafael Barboza-Gudiño ◽  
Ricardo Femat

Abstract. The complex oscillatory behavior of a springblock model is analyzed via the Hopf bifurcation mechanism. The mathematical springblock model is generated by considering the Dieterich–Ruinas's friction law and the 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 L, the implications of fluids, and the effects of external perturbations in the complex dynamic oscillatory behavior as well as in the stationary solution, are take into consideration.


1990 ◽  
Vol 63 (6) ◽  
pp. 1508-1519 ◽  
Author(s):  
L. C. Barrio ◽  
W. Buno

1. The effects of regular and random inhibition at moderate rates on the sensory response evoked by sinusoidal stretches were investigated in slowly and rapidly adapting stretch receptors of crayfish (RM1 and RM2, respectively). 2. Although the RM1 has pacemaker properties and the RM2 is spontaneously silent, inhibitory postsynaptic potential (IPSP) effects were similar in both mechanosensory neurons. The most common consequence was the expected reduction of the sensory response and the increase of the elongation needed to reach firing threshold. With regular IPSPs there were regions where pre- and postsynaptic spikes alternated at fixed integer ratios, usually 1:1, more rarely 1:2 and 1:3. Increases or decreases of the sensory excitation caused sudden postsynaptic accelerations or decelerations when specific length bounds were crossed and where pre- and postsynaptic alternations changed to lower (e.g., from 1:1 to 1:2) or higher ratios (e.g., from 1:2 to 1:1), respectively. 3. Paradoxical effects were also observed because increasing or decreasing the inhibitory rate for a given alternation ratio (e.g., 1:1) accelerated or decelerated the output rate, respectively. Alternations and paradoxical behaviors disappeared with IPSP pattern irregularization. Random IPSPs strongly irregularized the receptor's output. Inhibition, especially if the pattern was irregular, could excite under special conditions. 4. With regular IPSPs, mechanical sensitivity became zero at the lengths at which receptors were silenced, low during alternations, and maximum at transitions between successive alternation ratios. Irregular IPSPs did not have this delinearizing consequence. 5. In conclusion, inhibition introduced important complex modifications in the coding of mechanosensory information. Effects were similar in both receptor types, indicating that self-sustained oscillations are not fundamental. The observed changes were critically dependent on pre- and postsynaptic rate and pattern. They cannot be explained by simple summation of converging sensory and inhibitory inputs and represent another observation of the complex dynamic behavior of periodically driven nonlinear systems.


2016 ◽  
Vol 26 (11) ◽  
pp. 1650181 ◽  
Author(s):  
Junhai Ma ◽  
Wenbo Ren

On the basis of our previous research, we deepen and complete a kind of macroeconomics IS-LM model with fractional-order calculus theory, which is a good reflection on the memory characteristics of economic variables, we also focus on the influence of the variables on the real system, and improve the analysis capabilities of the traditional economic models to suit the actual macroeconomic environment. The conditions of Hopf bifurcation in fractional-order system models are briefly demonstrated, and the fractional order when Hopf bifurcation occurs is calculated, showing the inherent complex dynamic characteristics of the system. With numerical simulation, bifurcation, strange attractor, limit cycle, waveform and other complex dynamic characteristics are given; and the order condition is obtained with respect to time. We find that the system order has an important influence on the running state of the system. The system has a periodic motion when the order meets the conditions of Hopf bifurcation; the fractional-order system gradually stabilizes with the change of the order and parameters while the corresponding integer-order system diverges. This study has certain significance to policy-making about macroeconomic regulation and control.


The oscillatory patterns of behaviour exhibited by the simple kinetic scheme A + B → 2B; rate = k 1 ab , B → C; rate = k 2 b /(1 + rb ) are examined in detail. For systems with slowly decaying catalysts, such that k 2 ≪ k 1 a 2 0 a reduced asymptotic form of the governing equations allows a full analytical treatment. Oscillations begin as the residence time is increased through a point of Hopf bifurcation ז * res . The bifurcation is always supercritical, with the amplitudes of the concentration variations increasing from infinitesimally small values. The amplitudes grow initially as ( ז res – ז * res ) ½ , and tend to finite limiting magnitudes at very long residence times. At this limit, the oscillations in A have a ‘saw-tooth’ waveform, with B varying in a pulse-like manner. Numerical solutions of the full kinetic equations for a non-zero inflow of the catalyst B reveal how the system approaches a second Hopf bifurcation as the residence time is increased. The oscillatory amplitude now reaches a maximum and then decays back to zero. The applicability of this model to real chemical systems is discussed with particular reference to the gas phase oxidation of carbon monoxide.


1999 ◽  
Vol 09 (04) ◽  
pp. 629-644 ◽  
Author(s):  
ALEXANDER POGROMSKY ◽  
TORKEL GLAD ◽  
HENK NIJMEIJER

The paper deals with the problem of destabilization of diffusively coupled identical systems. Following a question of Smale [1976], it is shown that globally asymptotically stable systems being diffusively coupled, may exhibit oscillatory behavior. It is shown that if the diffusive medium consists of hyperbolically nonminimum phase systems and the diffusive factors exceed some threshold value, the origin of the overall system undergoes a Poincaré–Andronov–Hopf bifurcation resulting in oscillatory behavior.


2006 ◽  
Vol 47 (4) ◽  
pp. 527-540
Author(s):  
Claire E. Trenham ◽  
Larry K. Forbes

AbstractThis paper analyses a model for combustion of a self-heating chemical (such as pool chlorine), stored in drums within a shipping container. The system is described by three coupled nonlinear differential equations for the concentration of the chemical, its temperature and the temperature within the shipping container. Self-sustained oscillations are found to occur, as a result of Hopf bifurcation. Temperature and concentration profiles are presented and compared with the predictions of a simpler two-variable approximation for the system. We study the period of oscillation and its variation with respect to the ambient temperature and the reaction parameter. Nonlinear resonances are found to exist, as the solution jumps between branches having different periods.


1998 ◽  
Vol 08 (06) ◽  
pp. 1347-1354 ◽  
Author(s):  
Andrew V. Shobukhov

Oscillatory and chaotic regimes arising from Andronov-Hopf bifurcation are obtained in a model of fiber-optic interferometric strain gauge with delayed nonlinear feedback as a result of analytical and numerical investigation. Complicated behavior of solutions in the case of infinite number of equilibrium points is reported.


2000 ◽  
Vol 10 (12) ◽  
pp. 2701-2716 ◽  
Author(s):  
IRENE M. MOROZ ◽  
RAYMOND HIDE

This paper forms the second part of a two-part study into the effects of azimuthal eddy currents in the Faraday disk self-exciting homopolar dynamo, connected in series with the coil when the applied couple driving the disk is steady. The Lorentz couple driving the armature of the motor is a general quadratic function I(1 - ε + εSI) of the current I(t), where 0 ≤ ε ≤ 1. Here we investigate how cases with 0 < ε < 1 relate to the two special cases of ε = 0 and ε = 1, considered in Part I of our study [Hide & Moroz, 1999]. One key difference is that the lack of reflectional symmetry in the general ε problem means that the linear stability curves for the onset of both steady and oscillatory behavior for both the trivial and the nontrivial equilibrium solutions no longer coincide. This results in distinct Takens–Bogdanov double-zero bifurcations for these states, as well as multiple branches to the Hopf bifurcation curves, associated with bifurcations from the nontrivial equilibrium states. Multiple bifurcations involving simultaneous steady and nondegenerate oscillatory solutions are also possible.


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