Dynamics of Localized States with Periodic Forcing
Patterns and Simulations
By Punit Gandhi, Cédric Beaume, Edgar Knobloch
We study localized states in the Swift–Hohenberg equation when timeperiodic parametric forcing is introduced:
$$u_t= r u\left(1+\partial_{x}^2\right)^2u+bu^2u^3,\qquad r=r_0+\rho\sin 2\pi t/T,\qquad b=1.8.$$
The presence of a timedependent forcing introduces a new characteristic time which creates a series of resonances with the depinning time of the fronts bounding the localized pattern.
The resulting dynamics partition the parameter space by the net number of wavelengths of the pattern gained or lost during the course of a forcing cycle.
Asymptotic analysis can provide insight into and, in some cases, quantitative predictions about the sweetspot and pinching structure that is generated.
For more details, see
SIAM Journal on Applied Dynamical Systems 2015 14:2, 860–892
Author Institutional Affiliation  University of California, Berkeley; Imperial College, London 
Author Email  
Keywords  Swift–Hohenberg Equation, Parametric Forcing, Localized Structures 
Documents to download

Canard trajectories(.jpg, 68.28 KB)  3228 download(s)
The asymptotic theory described above also predicts the existence of canard trajectories that follow an unstable part of the autonomous solution branch for part of the cycle before quickly jumping back to a stable part. The first panel shows the predictions for the transition between regions with a constant number of wavelengths lost (blue) or gained (red) within a cycle of the forcing on top of results from simulations. The middle panels show numerical simulations of canard trajectories that are located near the transitions and whose parameters are indicated with corresponding colored symbols. The right panels show the position of the front and its velocity for the two ``doubleheaded" canards that are depicted. The value \(\rho=(r_+r_)/2+0.001\) was used here.

An adiabatic theory(.jpg, 272.75 KB)  3211 download(s)
In the adiabatic limit \(T\to\infty\), we can employ a quasistatic approximation for calculating the number of wavelengths lost and gained within a cycle of the forcing. Combining this theory with numerical fits of depinning times when a constant forcing is applied produces quantitative agreement to simulations well outside of the \(T\to \infty\) limit. The right panel shows the partitioning as calculated from numerical simulations for \(\rho=0.1\) and the left panel shows the corresponding predictions of transitions between regions with a constant number of wavelengths lost (blue) or gained (red) within a cycle of the forcing. The dark region is the predicted \(PO\) where the growth exactly balances the decay.

Partitioning of parameter space(.gif, 1.24 MB)  3234 download(s)
Simulations initialized with stable localized solutions at \(r_0\) of the autonomous \((\rho=0)\) system. The left panel indicates number of spatial periods added/lost per cycle for an oscillation amplitude \(\rho=0.1\) in the \((r_0,T)\) parameter plane. The central dark region (\(PO\)) corresponds to periodic orbits where there is no net change in spatial extent. The light blue region to the left corresponds to decay by one wavelength on each side of the localized state per cycle, the next to decay by two wavelengths per cycle etc. The regions to the right of \(PO\) correspond instead to net growth by one wavelength, two wavelengths etc. on each side of the localized state per cycle. The large white region to the left indicates the location of decay to the trivial state within one cycle period as a result of the dominance of an amplitude mode. Transition zones where irregular behavior is observed are shown in gray. The dots indicate the locations in parameter space of the solutions represented in the other panels. The right panels show the dynamics of the localized state in time (bottom plot), the value of the forcing parameter \(r\), and the trajectory in the frontamplitude plane (top plots).

Birth of sweet spots and pinched zones(.gif, 483.07 KB)  3229 download(s)
A carefully constructed asymptotic limit provides an analytical handle on the emergence of the sweet spot and pinching structure observed. The amplitude and average value of the forcing are tuned such that the extrema of the forcing remain within an asymptotically small vicinity of the edges of the pinning interval \((r_{\pm})\): \(\rhop \ll 1\), where \(p=(r_+r_)/2\), and assume \(r_0r_c \ll 1\), where \(r_c=(r_++r_)/2\) is the center of the pinning interval, so that \(\rhop/r_0r_c \sim\mathcal{O}(1)\). Additionally, we choose the period of the forcing cycle such that \(\rhop T\sim \mathcal{O}(1)\) in order to observe a finite number of depinning events. The figure shows the predicted transitions between regions with a constant number of wavelengths lost (blue) or gained (red) within a cycle of the forcing. The left panel shows the case that \(\rho=p\) where the system never leaves the pinning region and the two sets of resonance bands asymptote to \(r_0\to r_c\) as \(T\to\infty\). The right panel shows an asymptotically small sweet spot and pinching structure when \(\rho=p+0.001\), and the center animation shows the transition between the two as \(\rho\) increases.

Splitting and merging of localized states(.gif, 390.87 KB)  3216 download(s)
Near the edge of amplitude collapse, periodic forcing can induce the creation of new fronts within the localized state. These new fronts can also collide and annihilate each other as the pieces of the state merge together once again. This simulation with \(\rho=0.1\), \(r_0=−0.276228387\), and \(T = 1100\) was conducted on a \(640\pi\) domain.

Breathing localized states(.jpg, 175.5 KB)  3246 download(s)
Four simulations representing typical dynamics of localized states when a time periodic forcing is introduced with \(\rho=0.1\), \(r_0=0.28\) and different values of \(T\). The states breathe, or expand for part of the cycle via nucleation of new wavelengths of the pattern and then shrink by wavelength annihilation later in the cycle. We observe growing states, persistent periodic patterns, decaying states, and collapse to the trivial state. For each case, the top is a spacetime plot of the solution while the bottom shows its trajectory in frontamplitude variables. Here \(x=f \equiv 2 \int_0^{\Gamma/2} x u^2 dx / \int_0^{\Gamma/2} u^2 dx\) represents the location of the right edge of the localized state and \(A=\mathrm{max}(u)\) is its amplitude; \(\Gamma\) is the domain size. Steady state localized (blue) and periodic (green) states of the autonomous (\(\rho=0\)) problem are plotted for reference.