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Limit cycles, complex Floquet multipliers, and intrinsic noise
Richard P Boland1, Tobias Galla, Alan J McKane
1Theoretical Physics, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom. richard.o.poland@postgrad.man.ac.uk
Summary
Intrinsic noise can induce sustained oscillations in chemical reaction systems approaching limit cycles. This phenomenon, observed in models like the Brusselator and Willamowski-Rössler system, amplifies oscillations near instability boundaries.
Area of Science:
- Chemical kinetics
- Nonlinear dynamics
- Statistical physics
Background:
- Deterministic chemical systems can exhibit oscillatory behavior approaching a fixed point or limit cycle.
- Previous research indicated that noise can induce sustained oscillations in systems with oscillatory decay to a fixed point.
Purpose of the Study:
- To investigate the impact of intrinsic noise on chemical reaction systems whose deterministic dynamics lead to a limit cycle attractor.
- To determine if noise can induce sustained oscillations in systems already exhibiting limit cycle behavior.
Main Methods:
- Analysis of correlation functions and spectral properties of fluctuations.
- Utilizing comoving Frenet frames for analyzing system dynamics.
- Employing computational simulations for model systems like the driven and coupled Brusselators and the Willamowski-Rössler system.
Main Results:
- Intrinsic noise can indeed induce large-amplitude, sustained oscillations in systems with limit cycle attractors.
- The effect is general and occurs when the Floquet multipliers of the limit cycle are complex.
- Oscillation amplitude increases as the system approaches the instability boundary.
Conclusions:
- Intrinsic noise plays a crucial role in generating sustained oscillations in chemical systems, even those with stable limit cycles.
- The findings highlight a general mechanism for noise-induced oscillations in nonlinear dynamical systems.
- Understanding this noise effect is vital for controlling and predicting the behavior of complex chemical oscillators.
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