How limit cycles and quasi-cycles are related in systems with intrinsic noise

Abstract

Fluctuations and noise may alter the behavior of dynamical systems considerably. For example, oscillations may be sustained by demographic fluctuations in biological systems where a stable fixed point is found in the absence of noise. We here extend the theoretical analysis of such stochastic effects to models which have a limit cycle for some range of the model parameters. We formulate a description of fluctuations about the periodic orbit which allows the relation between the stochastic oscillations in the fixed point phase and the oscillations in the limit cycle phase to be elucidated. In the case of the limit cycle, a suitable transformation into a co-moving frame allow fluctuations transverse and longitudinal with respect to the limit cycle to be effectively decoupled. While longitudinal fluctuations are of a diffusive nature, those in the transverse direction follow a stochastic path more akin to an Ornstein-Uhlenbeck process. Their power spectrum is computed analytically within a van Kampen expansion in the inverse system size. This is carried out in two different ways, and the subsequent comparison with numerical simulations illustrates the effects that can occur due to diffusion in the longitudinal direction.

arXiv comment15 pages, 14 figures

References in corpus (7)

• Predator-prey cycles from resonant amplification of demographic stochasticity
• Coexistence versus extinction in the stochastic cyclic Lotka-Volterra model
• Amplified biochemical oscillations in cellular systems
• How limit cycles and quasi-cycles are related in systems with intrinsic noise
• Van Kampen's expansion approach in an opinion formation model
• Anomalous finite-size effects in the Battle of the Sexes
• Finite Size Fluctuations in Interacting Particle Systems

Cited by in corpus (39)

• Oscillatory Dynamics in Rock-Paper-Scissors Games with Mutations
• Bacterial growth: a statistical physicist's guide
• Intrinsic noise in systems with switching environments
• Stochastic lattice gas model describing the dynamics of an epidemic
• How limit cycles and quasi-cycles are related in systems with intrinsic noise
• Patterns of non-normality in networked systems
• Evolutionary dynamics, intrinsic noise and cycles of co-operation
• Turing patterns in systems with high-order interactions
• Limit cycles, complex Floquet multipliers and intrinsic noise
• Metastability in a stochastic neural network modeled as a velocity jump Markov process
• Intrinsic fluctuations in stochastic delay systems: theoretical description and application to a simple model of gene regulation
• Finite-size and correlation-induced effects in Mean-field Dynamics
• Macroscopic Stochastic Thermodynamics
• Synchronization dynamics in non-normal networks: the trade-off for optimality
• Cycles of cooperation and defection in imperfect learning
• Stochastic effects in a seasonally forced epidemic model
• Gaussian approximations for stochastic systems with delay: chemical Langevin equation and application to a Brusselator system
• Noise-driven oscillations in microbial population dynamics
• Persistence, extinction and spatio-temporal synchronization of SIRS cellular automata models
• Demographic noise and piecewise deterministic Markov processes
• Learning to play public good games
• Desynchronization and pattern formation in a noisy feedforward oscillators network
• Intrinsic noise, Delta-Notch signalling and delayed reactions promote sustained, coherent, synchronised oscillations in the presomitic mesoderm
• Stochastic Description of a Bistable Frustrated Unit
• Long delay times in reaction rates increase intrinsic fluctuations
• Finite Size Effects in Addition and Chipping Processes
• Fixation and escape times in stochastic game learning
• Supercluster states and phase transitions in aggregation-fragmentation processes
• An exactly solvable coarse-grained model for species diversity
• Elasticity, stability, and quasi-oscillations of cell-cell junctions in solid confluent epithelia
• Predicting properties of the stationary probability currents for two-species reaction systems without solving the Fokker-Planck equation
• Noise induced oscillations in non-equilibrium steady state systems
• Pattern reconstruction through generalized eigenvectors on defective networks
• Phase Resetting in an Asymptotically Phaseless System: On the Phase Response of Limit Cycles Verging on a Heteroclinic Orbit
• Relation between the convective field and the stationary probability distribution of chemical reaction networks
• Stabilization of hyperbolic reaction-diffusion systems on directed networks through the generalized Routh-Hurwitz criterion for complex polynomials
• Dynamical Tipping in a Quantum Limit Cycle
• Dissipation and fluctuations of CMOS ring oscillators close to criticality
• Dynamic Space Filling