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Spontaneous traveling waves in oscillatory systems with cross diffusion
1Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, UK.
Summary
Spatially uniform oscillations in active systems can become unstable, leading to the formation of traveling waves. This pattern emerges spontaneously, even without specific boundary conditions, establishing a stable wavelength range.
Area of Science:
- Complex Systems
- Nonlinear Dynamics
- Pattern Formation
Background:
- Active systems exhibit complex behaviors like pattern formation.
- Understanding pattern emergence in distributed systems is crucial.
Purpose of the Study:
- To identify and characterize a novel pattern formation mechanism in one-dimensional active systems.
- To investigate the transition from uniform oscillations to traveling waves.
Main Methods:
- Simulations of one-dimensional, two-component systems.
- Incorporation of predator-prey interactions and pursuit-evasion dynamics.
- Analysis of system stability and perturbation effects.
Main Results:
- Spatially uniform oscillations are unstable in large domains.
- Instability leads to transient focal sources and stable periodic traveling waves.
- Traveling wave formation is independent of boundary conditions.
- A bounded, stable wavelength range emerges, distinct from instability bands.
Conclusions:
- A new mechanism for spontaneous traveling wave formation in active systems is identified.
- The study elucidates the role of local interactions and taxis in emergent patterns.
- Findings contribute to the understanding of self-organization in complex systems.
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