A Stochastic Pitchfork Bifurcation in Most Probable Phase Portraits

We study stochastic bifurcation for a system under multiplicative stable Levy noise (an important class of non-Gaussian noise), by examining the qualitative changes of equilibrium states in its most probable phase portraits. We have found some peculiar bifurcation phenomena in contrast to the deterministic counterpart: (i) When the non-Gaussianity parameter in Levy noise varies, there is either one, two or none backward pitchfork type bifurcations; (ii) When a parameter in the vector field varies, there are two or three forward pitchfork bifurcations; (iii) The non-Gaussian Levy noise clearly leads to fundamentally more complex bifurcation scenarios, since in the special case of Gaussian noise, there is only one pitchfork bifurcation which is reminiscent of the deterministic situation.Comment: 9 pages, 11 figures To appear in International Journal of Bifurcation and Chao

Geometric Methods for Stochastic Dynamical Systems

Noisy fluctuations are ubiquitous in complex systems. They play a crucial or delicate role in the dynamical evolution of gene regulation, signal transduction, biochemical reactions, among other systems. Therefore, it is essential to consider the effects of noise on dynamical systems. It has been a challenging topic to have better understanding of the impact of the noise on the dynamical behaviors of complex systems.Comment: 8 page

State transitions in the Morris-Lecar model under stable L\'evy noise

This paper considers the state transition of the stochastic Morris-Lecar neuronal model driven by symmetric $\alpha$-stable L\'evy noise. The considered system is bistable: a stable fixed point (resting state) and a stable limit cycle (oscillating state), and there is an unstable limit cycle (borderline state) between them. Small disturbances may cause a transition between the two stable states, thus a deterministic quantity, namely the maximal likely trajectory, is used to analyze the transition phenomena in non-Gaussian stochastic environment. According to the numerical experiment, we find that smaller jumps of the L\'evy motion and smaller noise intensity can promote such transition from the sustained oscillating state to the resting state. It also can be seen that larger jumps of the L\'evy motion and higher noise intensity are conducive for the transition from the borderline state to the sustained oscillating state. As a comparison, Brownian motion is also taken into account. The results show that whether it is the oscillating state or the borderline state, the system disturbed by Brownian motion will be transferred to the resting state under the selected noise intensity

Maximal Likely Phase Lines for a Reduced Ice Growth Model

We study the impact of Brownian noise on transitions between metastable equilibrium states in a stochastic ice sheet model. Two methods to accomplish different objectives are employed. The maximal likely trajectory by maximizing the probability density function and numerically solving the Fokker-Planck equation shows how the system will evolve over time. We have especially studied the maximal likely trajectories starting near the ice-free metastable state, and examined whether they evolve to or near the ice-covered metastable state for certain parameters, in order to gain insights into how the ice sheet formed. Furthermore, for the transition from ice-covered metastable state to the ice-free metastable state, we study the most probable path for various noise parameters via the Onsager-Machlup least action principle. This enables us to predict and visualize the melting process of the ice sheet if such a rare event ever does take place.Comment: 15 pages, 13 figure

Most probable dynamics of a genetic regulatory network under stable L\'evy noise

Numerous studies have demonstrated the important role of noise in the dynamical behaviour of a complex system. The most probable trajectories of nonlinear systems under the influence of Gaussian noise have recently been studied already. However, there has been only a few works that examine how most probable trajectories in the two-dimensional system (MeKS network) are influenced under non-Gaussian stable L\'evy noise. Therefore, we discuss the most probable trajectories of a two-dimensional model depicting the competence behaviour in B. subtilis under the influence of stable L\'evy noise. On the basis of the Fokker-Planck equation, we describe the noise-induced most probable trajectories of the MeKS network from the low ComK protein concentration (vegetative state) to the high ComK protein concentration (competence state) under stable L\'evy noise. We demonstrate choices of the non-Gaussianity index $\alpha$ and the noise intensity $\epsilon$ which generate the ComK protein escape from the low concentration to the high concentration. We also reveal the optimal combination of both parameters $\alpha$ and $\epsilon$ making the tipping time shortest. Moreover, we find that different initial concentrations around the low ComK protein concentration evolve to a metastable state, and provide the optimal $\alpha$ and $\epsilon$ such that the distance between the deterministic competence state and the metastable state is smallest.Comment: 21 page

The maximum likelihood climate change for global warming under the influence of greenhouse effect and L\'evy noise

An abrupt climatic transition could be triggered by a single extreme event, an $\alpha$-stable non-Gaussian L\'evy noise is regarded as a type of noise to generate such extreme events. In contrast with the classic Gaussian noise, a comprehensive approach of the most probable transition path for systems under $\alpha$-stable L\'evy noise is still lacking. We develop here a probabilistic framework, based on the nonlocal Fokker-Planck equation, to investigate the maximum likelihood climate change for an energy balance system under the influence of greenhouse effect and L\'evy fluctuations. We find that a period of the cold climate state can be interrupted by a sharp shift to the warmer one due to larger noise jumps, and the climate change for warming $1.5\rm ^oC$ under an enhanced greenhouse effect generates a step-like growth process. These results provide important insights into the underlying mechanisms of abrupt climate transitions triggered by a L\'evy process

Stochastic Bifurcation in Single-Species Model Induced by {\alpha}-Stable Levy Noise

Bifurcation analysis has many applications in different scientific fields, such as electronics, biology, ecology, and economics. In population biology, deterministic methods of bifurcation are commonly used. In contrast, stochastic bifurcation techniques are infrequently employed. Here we establish stochastic P-bifurcation behavior of (i) a growth model with state-dependent birth rate and constant death rate, and (ii) a logistic growth model with state-dependent carrying capacity, both of which are driven by multiplicative symmetric stable Levy noise. Transcritical bifurcation occurs in the deterministic counterpart of the first model, while saddle-node bifurcation takes place in the logistic growth model. We focus on the impact of the variations of the growth rate, the per capita daily adult mortality rate, the stability index, and the noise intensity on the stationary probability density functions of the associated non-local Fokker-Planck equation. In the first model, the bifurcation parameter is the ratio of the population birth rate to the population death rate. In the second model, the bifurcation parameter corresponds to the sensitivity of carrying capacity to change in the size of the population near equilibrium. In each case, we show that as the value of the bifurcation parameter increases, the shape of the steady-state probability density function changes and that both stochastic models exhibit stochastic P-bifurcation. The unimodal density functions become more peaked around deterministic equilibrium points as the stability index increases. While an increase in any one of the other parameters has an effect on the stationary probability density function. That means the geometry of the density function changes from unimodal to flat, and its peak appears in the middle of the domain, which means a transition occurs.Comment: 17 pages, 19 figure