Spatio-temporal dynamics induced by competing instabilities in two asymmetrically coupled nonlinear evolution equations
- Physikalisch-Technische Bundesanstalt, Abbestrasse 2-12, 10587 Berlin (Germany)
- CNR-Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi - Via Madonna del Piano 10, I-50019 Sesto Fiorentino (Italy)
Pattern formation often occurs in spatially extended physical, biological, and chemical systems due to an instability of the homogeneous steady state. The type of the instability usually prescribes the resulting spatio-temporal patterns and their characteristic length scales. However, patterns resulting from the simultaneous occurrence of instabilities cannot be expected to be simple superposition of the patterns associated with the considered instabilities. To address this issue, we design two simple models composed by two asymmetrically coupled equations of non-conserved (Swift-Hohenberg equations) or conserved (Cahn-Hilliard equations) order parameters with different characteristic wave lengths. The patterns arising in these systems range from coexisting static patterns of different wavelengths to traveling waves. A linear stability analysis allows to derive a two parameter phase diagram for the studied models, in particular, revealing for the Swift-Hohenberg equations, a co-dimension two bifurcation point of Turing and wave instability and a region of coexistence of stationary and traveling patterns. The nonlinear dynamics of the coupled evolution equations is investigated by performing accurate numerical simulations. These reveal more complex patterns, ranging from traveling waves with embedded Turing patterns domains to spatio-temporal chaos, and a wide hysteretic region, where waves or Turing patterns coexist. For the coupled Cahn-Hilliard equations the presence of a weak coupling is sufficient to arrest the coarsening process and to lead to the emergence of purely periodic patterns. The final states are characterized by domains with a characteristic length, which diverges logarithmically with the coupling amplitude.
- OSTI ID:
- 22402520
- Journal Information:
- Chaos (Woodbury, N. Y.), Vol. 24, Issue 4; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); ISSN 1054-1500
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
GENERAL PHYSICS
BIFURCATION
CHAOS THEORY
COMPUTERIZED SIMULATION
COUPLING
EQUATIONS
INSTABILITY
MATHEMATICAL EVOLUTION
NONLINEAR PROBLEMS
ORDER PARAMETERS
PERIODICITY
PHASE DIAGRAMS
SPACE DEPENDENCE
STEADY-STATE CONDITIONS
TIME DEPENDENCE
TRAVELLING WAVES
WAVELENGTHS