Self-induced topological transition in phononic crystals by nonlinearity management

A new design paradigm of topology has recently emerged to manipulate the flow of phonons. At its heart lies a topological transition to a nontrivial state with exotic properties. This framework has been limited to linear lattice dynamics so far. Here we show a topological transition in a nonlinear regime and its implication in emerging nonlinear solutions. We employ nonlinearity management such that the system consists of masses connected with two types of nonlinear springs, "stiffening" and "softening" types, alternating along the length. We show, analytically and numerically, that the lattice makes a topological transition simply by changing the excitation amplitude and invoking nonlinear dynamics. Consequently, we witness the emergence of a new family of finite-frequency edge modes, not observed in linear phononic systems. We also report the existence of kink solitons at the topological transition point. These correspond to heteroclinic orbits that form a closed curve in the phase portrait separating the two topologically-distinct regimes. These findings suggest that nonlinearity can be used as a strategic tuning knob to alter topological characteristics of phononic crystals. These also provide fresh perspectives towards understanding a new family of nonlinear solutions in light of topology.Comment: 14 pages, 8 figure

Relative Periodic Solutions of the Complex Ginzburg-Landau Equation

A method of finding relative periodic orbits for differential equations with continuous symmetries is described and its utility demonstrated by computing relative periodic solutions for the one-dimensional complex Ginzburg-Landau equation (CGLE) with periodic boundary conditions. A relative periodic solution is a solution that is periodic in time, up to a transformation by an element of the equation's symmetry group. With the method used, relative periodic solutions are represented by a space-time Fourier series modified to include the symmetry group element and are sought as solutions to a system of nonlinear algebraic equations for the Fourier coefficients, group element, and time period. The 77 relative periodic solutions found for the CGLE exhibit a wide variety of temporal dynamics, with the sum of their positive Lyapunov exponents varying from 5.19 to 60.35 and their unstable dimensions from 3 to 8. Preliminary work indicates that weighted averages over the collection of relative periodic solutions accurately approximate the value of several functionals on typical trajectories.Comment: 32 pages, 12 figure

Coupled complex Ginzburg-Landau systems with saturable nonlinearity and asymmetric cross-phase modulation

We formulate and study dynamics from a complex Ginzburg-Landau system with saturable nonlinearity, including asymmetric cross-phase modulation (XPM) parameters. Such equations can model phenomena described by complex Ginzburg-Landau systems under the added assumption of saturable media. When the saturation parameter is set to zero, we recover a general complex cubic Ginzburg-Landau system with XPM. We first derive conditions for the existence of bounded dynamics, approximating the absorbing set for solutions. We use this to then determine conditions for amplitude death of a single wavefunction. We also construct exact plane wave solutions, and determine conditions for their modulational instability. In a degenerate limit where dispersion and nonlinearity balance, we reduce our system to a saturable nonlinear Schr\"odinger system with XPM parameters, and we demonstrate the existence and behavior of spatially heterogeneous stationary solutions in this limit. Using numerical simulations we verify the aforementioned analytical results, while also demonstrating other interesting emergent features of the dynamics, such as spatiotemporal chaos in the presence of modulational instability. In other regimes, coherent patterns including uniform states or banded structures arise, corresponding to certain stable stationary states. For sufficiently large yet equal XPM parameters, we observe a segregation of wavefunctions into different regions of the spatial domain, while when XPM parameters are large and take different values, one wavefunction may decay to zero in finite time over the spatial domain (in agreement with the amplitude death predicted analytically). While saturation will often regularize the dynamics, such transient dynamics can still be observed - and in some cases even prolonged - as the saturability of the media is increased, as the saturation may act to slow the timescale.Comment: 36 page

Revealing the state space of turbulent pipe flow by symmetry reduction

Symmetry reduction by the method of slices is applied to pipe flow in order to quotient the stream-wise translation and azimuthal rotation symmetries of turbulent flow states. Within the symmetry-reduced state space, all travelling wave solutions reduce to equilibria, and all relative periodic orbits reduce to periodic orbits. Projections of these solutions and their unstable manifolds from their $\infty$-dimensional symmetry-reduced state space onto suitably chosen 2- or 3-dimensional subspaces reveal their interrelations and the role they play in organising turbulence in wall-bounded shear flows. Visualisations of the flow within the slice and its linearisation at equilibria enable us to trace out the unstable manifolds, determine close recurrences, identify connections between different travelling wave solutions, and find, for the first time for pipe flows, relative periodic orbits that are embedded within the chaotic attractor, which capture turbulent dynamics at transitional Reynolds numbers.Comment: 24 pages, 12 figure