Early Warnings Of Noise-induced Critical Transitions In Multi-stable Systems

Noise-induced critical transitions appear widespread in practical systems.Their appearance means that the system evolves from one dynamical state to another contrasting one,and may cause some catastrophic effects on human living environment,economy and health.Furthermore,it is often difficult to revert a system to the previous state once a noise-induced critical transition has occurred.Therefore,early warnings of noise-induced critical transitions have been always one of the hottest topics in the investigation of nonlinear stochastic dynamics.The related theories and techniques have been applied in many areas to guide the identification and prediction of catastrophic transitions.At present,detecting early warning signals of critical transitions usually focus on some statistical indicators based on critical slowing down,such as variance and autocorrelation.However,these indicators work well when the critical point of a critical transition is near a bifurcation point of the corresponding deterministic system,namely,perturbations are relatively small.They cannot identify an earlier ctitical transition under strong fluctuations.In addition,these indicators can only provide early warning signals when a critical transition is already awfully close,so that we do not have enough time to adopt the management to hinder it.Then,this thesis is to explore more general early-warning indicators and propose the new concept of parameter dependent basin of the unsafe regime(PDBUR)to quantify the range of the control parameter where noise-induced critical transitions may occur.The contents of this dissertation are presented as:1.More general early-warning indicators of Gaussian white noise-induced critical transitions in multi-stable systems are established.By analyzing the critical transition in bi-stable and tri-stable systems with different Gaussian white noise intensities,we find that the critical transition takes place before the bifurcation point of the corresponding deterministic system,and it occurs much earlier with increasing noise intensity.Then,the largest Lyapunov exponent and Shannon entropy are used to warn an impending critical transition.Our results show that these two measures can accurately predict a Gaussian white noise-induced critical transition in a given bistable system with different intensities.Moreover,both two indicators can be further extended to warn two critical transitions in a tristable system.Compared with the classical early-warning indicators,the variance and the autocorrelation,the robustness of the largest Lyapunov exponent and Shannon entropy is verified.2.The ranges where Gaussian white noise-induced single critical transitions may occur in a bi-stable system are quantified.Based on the relationship between the Gaussian white noise-induced critical transition and the corresponding dynamic transition,the escape probability is introduced into the prediction of critical transition.Then,the region that the current state of the given model is absorbed into the opposite state is measured via the escape probability,which is named as absorbed region.To determine how large the range of the absorbed region is,a Gaussian white noise-induced critical transition occurs,a new concept of PDBUR is introduced.A global analysis of the critical transition with respect to the control parameter and noise intensity is presented.3.Quantification of the ranges where Gaussian white noise-induced double critical transitions may occur in a tri-stable system is realized.A Gaussian white noise-induced tri-stable model with desirable,sub-desirable and undesirable states may undergo two critical transitions from the desirable state to the sub-desirable one(CT1)and from the sub-desirable state to the undesirable one(CT2).By analyzing the trend of the escape probability,the absorbed regions that the current state of the system is absorbed into the opposite state corresponding to CT1 and CT2 are defined.Finally,the PDBUR is extended to quantify the ranges of the control parameter and the noise intensity where CT1 and CT2 may occur.4.The PDBUR of asymmetric Lévy-noise-induced critical transitions between non-adjacent states in a tri-stable system is established.Lévy noise with large jumps can induce a critical transition from the desirable state directly to the undesirable state in a tri-stable system.To warn this catastrophic transition,the difference form of the escape probability that the given system escapes to the the undesired state is given.Then,the absorbed region that the current state of the system is absorbed into the undesired state under Lévy noise is calculated.Finally,the new concept of the PDBUR of Lévy-noise-induced critical transitions is defined,and the ranges of the control parameter and the Lévy noise parameters where asymmetric Lévy-noise-induced critical transitions are quantified.5.Early warning indicators of the critical transition in a periodically driven bistable system with Gaussian white noise are established,and the effects of Gaussian white noise and periodic force on slowing down an imminent critical transition is uncovered.The relationship between the system’s response and the periodic forcing is first observed via linear dynamics approximation.Then,two indicators,the phase lag and amplitude difference of the system’s response,are established.Both indicators can announce a Gaussian white noise-induced critical transition with small intensity,while they fail in the case of large noise intensity.Based on the moment-expanding scheme,a large noise can be reduced to small noises,and then both indicators work well again.Finally,the increase of the residence probability or the mean first passage time,and a decrease of the mean velocity verify that Gaussian white noise and periodic force can slow down an impending critical transition.