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Register a new userPattern formation with pde2path -- a tutorial
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Authors
Hannes Uecker
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We explain some pde2path setups for pattern formation in 1D, 2D and 3D. A focus is on new pde2path functions for branch switching at steady bifurcation points of higher multiplicity, typically due to discrete symmetries, but we also review general concepts of pattern formation and their handling in pde2path, including localized patterns and homoclinic snaking, again in 1D, 2D and 3D, based on the demo sh (Swift-Hohenberg equation). Next, the demos schnakpat (a Schnakenberg reaction-diffusion system) and chemtax (a quasilinear RD system with cross-diffusion from chemotaxis) simplify and unify previous results in a simple and concise way, CH (Cahn-Hilliard) deals with mass constraints, hexex deals with (multiple) branch points of higher degeneracy in a scalar problem on a hexagonal domain, and shgc illustrates some global coupling. The demos acS, actor, schnakS and schnaktor (the Allen-Cahn and Schnakenberg models on spheres and tori) consider pattern formation on curved surfaces, cpol considers a problem of cell polarization described by bulk-surface coupling, and bruosc (Brusselator) explains how to augment autonomous systems by a time periodic forcing. Along the way we also comment on the choice of meshes and mesh adaptation, on time integration, and we give some examples of branch point continuation and Hopf point continuation to approximate stability boundaries.
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Pattern formation in systems with a conserved quantity is considered by studying the appropriate amplitude equations. The conservation law leads to a large-scale neutral mode that must be included in the asymptotic analysis for pattern formation near onset. Near a stationary bifurcation, the usual Ginzburg--Landau equation for the amplitude of the pattern is then coupled to an equation for the large-scale mode. These amplitude equations show that for certain parameters all roll-type solutions are unstable. This new instability differs from the Eckhaus instability in that it is amplitude-driven and is supercritical. Beyond the stability boundary, there exist stable stationary solutions in the form of strongly modulated patterns. The envelope of these modulations is calculated in terms of Jacobi elliptic functions and, away from the onset of modulation, is closely approximated by a sech profile. Numerical simulations indicate that as the modulation becomes more pronounced, the envelope broadens. A number of applications are considered, including convection with fixed-flux boundaries and convection in a magnetic field, resulting in new instabilities for these systems.
Self-organization, the ability of a system of microscopically interacting entities to shape macroscopically ordered structures, is ubiquitous in Nature. Spatio-temporal patterns are abundantly observed in a large plethora of applications, encompassing different fields and scales. Examples of emerging patterns are the spots and stripes on the coat or skin of animals, the spatial distribution of vegetation in arid areas, the organization of the colonies of insects in host-parasitoid systems and the architecture of large complex ecosystems. Spatial self-organization can be described following the visionary intuition of Alan Turing, who showed how non-linear interactions between slow diffusing activators and fast diffusing inhibitors could induce patterns. The Turing instability, as the mechanism described is universally referred to, was raised to paradigm status in those realms of investigations where microscopic entities are subject to diffusion, from small biological systems to large ecosystems. Requiring a significant ratio of the assigned diffusion constants however is a stringent constraint, which limited the applicability of the theory. Building on the observation that spatial interactions are usually direction biased, and often strongly asymmetric, we here propose a novel framework for the generation of short wavelength patterns which overcomes the limitation inherent in the Turing formulation. In particular, we will prove that patterns can always set in when the system is composed by sufficiently many cells - the units of spatial patchiness - and for virtually any ratio of the diffusivities involved. Macroscopic patterns that follow the onset of the instability are robust and show oscillatory or steady-state behavior.
The computation of coefficients of amplitude systems for Turing bifurcations is a straightforward but sometimes elaborate task, in particular for 2D or 3D wave vector lattices. The Matlab tool ampsys automates such computations for two classes of problems, namely scalar equations of Swift-Hohenberg type and generalizations, and reaction-diffusion systems with an arbitrary number of components. The tool is designed to require minimal user input, and for a number of cases can also deal with symbolic computations. After a brief review of the setup of amplitude systems we explain the tool by a number of 1D, 2D and 3D examples over various wave vector lattices.
Driven many-body systems have been shown to exhibit discrete time crystal phases characterized by broken discrete time-translational symmetry. This has been achieved generally through a subharmonic response, in which the system undergoes one oscillation every other driving period. Here, we demonstrate that classical time crystals do not need to resonate in a subharmonic fashion but instead can also exhibit a continuously tunable anharmonic response to driving, which we show can emerge through a coresonance between modes in different branches of the dispersion relation in a parametrically driven medium. This response, characterized by a typically incommensurate ratio between the resonant frequencies and the driving frequency, is demonstrated by introducing a time crystal model consisting of an array of coupled pendula with alternating lengths. Importantly, the coresonance mechanism is the result of a bifurcation involving a fixed point and an invariant torus, with no intermediate limit cycles. This bifurcation thus gives rise to a many-body symmetry-breaking phenomenon directly connecting the symmetry-unbroken phase with a previously uncharacterized phase of matter, which we call an anharmonic time crystal phase. The mechanism is shown to generalize to driven media with any number of coupled fields and is expected to give rise to anharmonic responses in a range of weakly damped pattern-forming systems, with potential applications to the study of nonequilibrium phases, frequency conversion, and acoustic cloaking.
Morphogenetic patterns are highly sophisticated dissipative structures. Are they governed by the same general mechanisms as chemical and hydrodynamic patterns? Turings symmetry breaking and Wolperts signalling provide alternative mechanisms. The current evidence points out that the latter is more relevant but reality is still far more complicated.
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