Lévy on-off intermittency

Abstract

We present a new form of intermittency, Lévy on-off intermittency, which arises from multiplicative $\alpha$-stable white noise close to an instability threshold. We study this problem in the linear and nonlinear regimes, both theoretically and numerically, for the case of a pitchfork bifurcation with fluctuating growth rate. We compute the stationary distribution analytically and numerically from the associated fractional Fokker-Planck equation in the Stratonovich interpretation. We characterize the system in the parameter space $(\alpha ,\beta )$ of the noise, with stability parameter $\alpha \in (0,2)$ and skewness parameter $\beta \in [-1,1]$. Five regimes are identified in this parameter space, in addition to the well-studied Gaussian case $\alpha =2$. Three regimes are located at $1<\alpha <2$, where the noise has finite mean but infinite variance. They are differentiated by $\beta$ and all display a critical transition at the deterministic instability threshold, with on-off intermittency close to onset. Critical exponents are computed from the stationary distribution. Each regime is characterized by a specific form of the density and specific critical exponents, which differ starkly from the Gaussian case. A finite or infinite number of integer-order moments may converge, depending on parameters. Two more regimes are found at $0<\alpha \le 1$. There, the mean of the noise diverges, and no critical transition occurs. In one case the origin is always unstable, independently of the distance $\mu$ from the deterministic threshold. In the other case, the origin is conversely always stable, independently of $\mu$. We thus demonstrate that an instability subject to non-equilibrium, power-law-distributed fluctuations can display substantially different properties than for Gaussian thermal fluctuations, in terms of statistics and critical behavior.

References in corpus (54)

• Understanding individual human mobility patterns
• Lévy walks
• The fundamental solution of the space-time fractional diffusion equation
• Universal emission intermittency in quantum dots, nanorods, and nanowires
• Levy Flights in External Force Fields: Langevin and Fractional Fokker-Planck Equations, and their Solutions
• Levy Flight Superdiffusion: An Introduction
• Non-diffusive transport in plasma turbulence: a fractional diffusion approach
• Non-ergodic Intensity Correlation Functions for Blinking Nano Crystals
• Anomalous jumping in a double-well potential
• Leapover lengths and first passage time statistics for Lévy flights
• Lévy flights in a steep potential well
• Lineshape Theory and Photon Counting Statistics for Spectral Fluctuations in Quantum Dots: a Lévy Walk Process
• Switching times in long-overlap Josephson junctions subject to thermal fluctuations and non-Gaussian noise sources
• Anomalous transport effects on switching currents of graphene-based Josephson junctions
• Effects of Lévy noise on the dynamics of sine-Gordon solitons in long Josephson junctions
• Escape driven by $\alpha$-stable white noises
• Verhulst model with Levy white noise excitation
• Spatio-temporal propagation of COVID-19 pandemics
• Lévy Noise-Induced Stochastic Resonance in a Bistable System
• Generalized Fokker-Planck equation: Derivation and exact solutions
• Josephson-based threshold detector for Lévy distributed fluctuations
• Critical Transitions in Thin Layer Turbulence
• On-off Intermittency in Stochastically Driven Electrohydrodynamic Convection in Nematics
• Stationary states in Langevin dynamics under asymmetric Lévy noises
• Voltage drop across Josephson junctions for Lévy noise detection
• Resonant activation in the presence of non-equilibrated baths
• Levy stable noise induced transitions: stochastic resonance, resonant activation and dynamic hysteresis
• Analogies between Scaling in Turbulence, Field Theory and Critical Phenomena
• Harmonic oscillator under Levy noise: Unexpected properties in the phase space
• Turbulence as a problem in non-equilibrium statistical mechanics
• Transport in a Levy ratchet: Group velocity and distribution spread
• Fractional Fokker-Planck equation for Lévy flights in nonhomogeneous environments
• Low frequency noise controls on-off intermittency of bifurcating systems
• Multiplicative Lévy processes: Itô versus Stratonovich interpretation
• Steady-State Lévy Flights in a Confined Domain
• Condensates in thin-layer turbulence
• Stationary states in single-well potentials under symmetric Levy noises
• Stability analysis of a noise-induced Hopf bifurcation
• Nonlinear stochastic equations with multiplicative Lévy noise
• Effects of the low frequencies of noise on On-Off intermittency
• Lévy walk dynamics in an external harmonic potential
• Noise-induced bifurcations, Multiscaling and On-Off intermittency
• Intermittency in spherical Couette dynamos
• Low-dimensional dynamics of phase oscillators driven by Cauchy noise
• Onset of three-dimensionality in rapidly rotating turbulent flows
• Multiplicative Levy noise in bistable systems
• Lévy noise-driven escape from arctan potential wells
• Can Lèvy noise induce coherence and stochastic resonances in a birhythmic van der Pol system?
• Lévy noise induced transitions and enhanced stability in a birhythmic van der Pol system
• Anomalous exponents at the onset of an instability
• On-off intermittency over an extended range of control parameter
• Intermittency of three-dimensional perturbations in a point-vortex model
• Critical exponents in zero dimensions
• Growth rate distribution and intermittency in kinematic turbulent dynamos : which moment predicts the dynamo onset?

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