Coupled maps and pattern formation on the Sierpinski gasket.

Area of Science:

• Complex Systems
• Fractals
• Nonlinear Dynamics

Background:

• Coupled map lattices (CMLs) are used to model complex spatio-temporal phenomena.
• The Sierpinski gasket is a well-known fractal structure with unique topological and geometric properties.
• Understanding synchronization and pattern formation in fractal systems is crucial for various scientific fields.

Purpose of the Study:

• To investigate the bifurcation structure of coupled maps defined on a Sierpinski gasket lattice.
• To analyze the impact of fractal geometry on the stability of synchronized states and the emergence of complex spatial patterns.
• To explore bistability and domain growth in coupled cubic map lattices on this fractal structure.

Main Methods:

• Numerical simulations of coupled quadratic and cubic maps on a Sierpinski gasket lattice.
• Analysis of bifurcation diagrams to identify stability boundaries.
• Characterization of spatially inhomogeneous states and their complex structures.
• Study of domain growth and bistability phenomena in coupled cubic maps.

Main Results:

• The fractal nature of the Sierpinski gasket leads to unusual stability boundaries for periodic synchronized states.
• Complex, spatially inhomogeneous states with intricate structures emerge.
• Bistability and domain growth processes are observed and studied in coupled cubic map lattices.
• The study reveals unique dynamic behaviors not typically seen in Euclidean lattices.

Conclusions:

• The Sierpinski gasket lattice significantly influences the dynamics of coupled maps, creating novel spatio-temporal patterns.
• Fractal geometry plays a critical role in determining the stability and complexity of synchronized states.
• This research provides insights into the behavior of complex systems on fractal substrates, with implications for physics and mathematics.