Theory for relaxation at a subcritical pitchfork bifurcation

Journal of Statistical Physics, 1995

Previous results on first-passage-time statistics for systems driven by dichotomous noise are extended in order to cover the escape from regions including fixed points of the stochastic flow. For such regions a treatment splitting the escape through one or the other boundary is required. The obtained escape probabilities and mean exit times are relevant for the complete characterization of stochastic systems undergoing bifurcations.

Discrete and Continuous Dynamical Systems - Series B, 2004

The bifurcations of strange nonchaotic attractors in quasi-periodically forced systems are poorly understood. A simple two-parameter example is introduced which unifies previous observations of the non-smooth pitchfork bifurcation. There are two types of generalized pitchfork bifurcation which occur in this example, and the corresponding bifurcation curves can be calculated analytically. The example shows how these bifurcations are organized around a codimension two point in parameter space.

Physical Review E, 2010

A Langevin equation whose deterministic part undergoes a saddle-node bifurcation is investigated theoretically. It is found that statistical properties of relaxation trajectories in this system exhibit divergent behaviors near a saddle-node bifurcation point in the weak-noise limit, while the final value of the deterministic solution changes discontinuously at the point. A systematic formulation for analyzing a path probability measure is constructed on the basis of a singular perturbation method. In this formulation, the critical nature turns out to originate from the neutrality of exiting time from a saddle-point. The theoretical calculation explains results of numerical simulations.

We present the results of an experimental and numerical investigation of the effects of noise on pitchfork and Hopf bifurcations. Good quantitative agreement is found between calculations and experiment. In the case of the pitchfork we find that natural imperfections override the effects of the noise. However, novel noise amplification effects have been uncovered in the study of the Hopf bifurcation. These destroy the critical event found in the noise-free case and could be of considerable practical importance in systems containing dynamic bifurcations. † Present address: Schuster Laboratory, University of Manchester, Manchester M13 9PL, UK.

Physica D: Nonlinear Phenomena, 2000

This paper discusses the mathematical analysis of a codimension two bifurcation determined by the coincidence of a subcritical Hopf bifurcation with a homoclinic orbit of the Hopf equilibrium. Our work is motivated by our previous analysis of a Hodgkin-Huxley neuron model which possesses a subcritical Hopf bifurcation . In this model, the Hopf bifurcation has the additional feature that trajectories beginning near the unstable manifold of the equilibrium point return to pass through a small neighborhood of the equilibrium, that is, the Hopf bifurcation appears to be close to a homoclinic bifurcation as well. This model of the lateral pyloric (LP) cell of the lobster stomatogastric ganglion was analyzed for its ability to explain the phenomenon of spike-frequency adaptation , in which the time intervals between successive spikes grow longer until the cell eventually becomes quiescent. The presence of a subcritical Hopf bifurcation in this model was one identified mechanism for oscillatory trajectories to increase their period and finally collapse to a non-oscillatory solution. The analysis presented here explains the apparent proximity of homoclinic and Hopf bifurcations. We also develop an asymptotic theory for the scaling properties of the interspike intervals in a singularly perturbed system undergoing subcritical Hopf bifurcation that may be close to a codimension two subcritical Hopf-homclinic bifurcation.

1998

Volume 2: Modeling, Simulation and Control; Bio-Inspired Smart Materials and Systems; Energy Harvesting, 2016

Accurately predicting the onset of large behavioral deviations associated with saddlenode bifurcations is imperative in a broad range of sciences and for a wide variety of purposes, including ecological assessment, signal amplification, and microscale mass sensing. In many such practices, noise and non-stationarity are unavoidable and everpresent influences. As a result, it is critical to simultaneously account for these two factors toward the estimation of parameters that may induce sudden bifurcations. Here, a new analytical formulation is presented to accurately determine the probable time at which a system undergoes an escape event as governing parameters are swept toward a saddle-node bifurcation point in the presence of noise. The double-well Duffing oscillator serves as the archetype system of interest since it possesses a dynamic saddle-node bifurcation. The stochastic normal form of the saddle-node bifurcation is derived from the governing equation of this oscillator to formulate the probability distribution of escape events. Non-stationarity is accounted for using a time-dependent bifurcation parameter in the stochastic normal form. Then, the mean escape time is approximated from the probability density function (PDF) to yield a straightforward means to estimate the point of bifurcation. Experiments conducted using a double-well Duffing analog circuit verifies that the analytical approximations provide faithful estimation of the critical parameters that lead to the non-stationary and noise-activated saddle-node bifurcation.

Physical Review A, 1990

The influence of additive white noise on the delay of a bifurcation point in the presence of a swept control parameter has been studied by analog experiment and digital simulation. Measurements of the time taken for the second moment (x'(r) ) to reach a given threshold are in excellent agreement with the calculations of Zeghlache, Mandel, and Van den Broeck [Phys. Rev. A 40, 286 (1989)]. It is demonstrated, however, that the mean first-passage time (MFPT) for x'(t) to attain the same threshold, corresponding to the quantity that is usually determined in laser experiments, can be markedly different. It may be either larger or smaller, depending on the conditions under which the measurements are made. A calculation of the MFPT is presented and shown to be in excellent agreement with the experimental measurements and to reduce, in the relevant limit, to the theoretical results previously published by Torrent and San Miguel [Phys. Rev. A 38, 245 (1988)].

arXiv (Cornell University), 2023

Bifurcations in dynamical systems are often studied experimentally and numerically using a slow parameter sweep. Focusing on the cases of period-doubling and pitchfork bifurcations in maps, we show that the adiabatic approximation always breaks down sufficiently close to the bifurcation, so that the upsweep and downsweep dynamics diverge from one another, disobeying standard bifurcation theory. Nevertheless, we demonstrate universal upsweep and downsweep trajectories for sufficiently slow sweep rates, revealing that the slow trajectories depend essentially on a structural asymmetry parameter, whose effect is negligible for the stationary dynamics. We obtain explicit asymptotic expressions for the universal trajectories, and use them to calculate the area of the hysteresis loop enclosed between the upsweep and downsweep trajectories as a function of the asymmetry parameter and the sweep rate.

American Journal of Physics, 2004

Critical slowing down near a bifurcation or limit point leads to a dynamical hysteresis that cannot be avoided by sweeping a control parameter slowly through the critical point. This paper analytically illustrates, with the help of a simple model, the bifurcation shift. We describe an inexpensive experiment using a semiconductor laser where this phenomenon occurs near the threshold of a semiconductor laser.