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Full-Text Articles in Physical Sciences and Mathematics
Swarming Patterns In A Two-Dimensional Kinematic Model For Biological Groups, Chad M. Topaz, Andrea L. Bertozzi
Swarming Patterns In A Two-Dimensional Kinematic Model For Biological Groups, Chad M. Topaz, Andrea L. Bertozzi
Chad M. Topaz
We construct a continuum model for the motion of biological organisms experiencing social interactions and study its pattern-forming behavior. The model takes the form of a conservation law in two spatial dimensions. The social interactions are modeled in the velocity term, which is nonlocal in the population density and includes a parameter that controls the interaction length scale. The dynamics of the resulting partial integrodifferential equation may be uniquely decomposed into incompressible motion and potential motion. For the purely incompressible case, the model resembles one for fluid dynamical vortex patches. There exist solutions which have constant population density and compact …
Multifrequency Control Of Faraday Wave Patterns, Chad M. Topaz, Jeff Porter, Mary Silber
Multifrequency Control Of Faraday Wave Patterns, Chad M. Topaz, Jeff Porter, Mary Silber
Chad M. Topaz
We show how pattern formation in Faraday waves may be manipulated by varying the harmonic content of the periodic forcing function. Our approach relies on the crucial influence of resonant triad interactions coupling pairs of critical standing wave modes with damped, spatiotemporally resonant modes. Under the assumption of weak damping and forcing, we perform a symmetry-based analysis that reveals the damped modes most relevant for pattern selection, and how the strength of the corresponding triad interactions depends on the forcing frequencies, amplitudes, and phases. In many cases, the further assumption of Hamiltonian structure in the inviscid limit determines whether the …
Pattern Control Via Multi-Frequency Parametric Forcing, Jeff Porter, Chad M. Topaz, Mary Silber
Pattern Control Via Multi-Frequency Parametric Forcing, Jeff Porter, Chad M. Topaz, Mary Silber
Chad M. Topaz
We use symmetry considerations to investigate control of a class of resonant three-wave interactions relevant to pattern formation in weakly damped, parametrically forced systems near onset. We classify and tabulate the most important damped, resonant modes and determine how the corresponding resonant triad interactions depend on the forcing parameters. The relative phase of the forcing terms may be used to enhance or suppress the nonlinear interactions. We compare our symmetry-based predictions with numerical and experimental results for Faraday waves. Our results suggest how to design multifrequency forcing functions that favor chosen patterns in the lab.