The Birman–Williams theorem gives a connection between the collection of unstable periodic orbits (UPOs) contained within chaotic attractor and topology that attractor, for three-dimensional systems. In certain cases, fractal dimension in partial differential equation (PDE) is less than three, even though embedded an infinite-dimensional space. Here, we study Kuramoto–Sivashinsky PDE at onset chaos. We use two different dimensionality-reduction techniques—proper orthogonal decomposition...

Unstable periodic orbits are believed to underpin the dynamics of turbulence, but by their nature hard find computationally. We present a family methods converge such unstable for incompressible Navier–Stokes equations, based on variations an integral objective functional, and using traditional gradient-based optimisation strategies. Different approaches handling incompressibility condition considered. The variational applied specific case periodic, two-dimensional Kolmogorov flow compared...

It has recently been speculated that long-time average quantities of hyperchaotic dissipative systems may be approximated by weighted sums over unstable invariant tori embedded in the attractor, analogous to equivalent periodic orbits, which are inspired rigorous orbit theory and have shown much promise fluid dynamics. Using a new numerical method for converging two-tori chaotic partial differential equation (PDE), exploiting symmetry breaking relative orbits detect those tori, we identify...

A Koopman decomposition of a complex system leads to representation in which nonlinear dynamics appear be linear. The existence linear framework with analyze dynamical systems brings new strategies for prediction and control, while the approach is straightforward apply large datasets using dynamic mode (DMD). However, it can challenging connect output DMD analysis since there are relatively few analytical results available, algorithm itself known struggle situations involving propagation...

We study the dynamical system of a forced stratified mixing layer at finite Reynolds number $Re$, and Prandtl $Pr=1$. consider hyperbolic tangent background velocity profile in two cases uniform buoyancy stratifications. The is such way that these profiles are steady solution governing equations. As well-known, if minimum gradient Richardson flow, $Ri_m$, less than certain critical value $Ri_c$, flow linearly unstable to Kelvin-Helmholtz instability both cases. Using Newton-Krylov iteration,...

Abstract

The eigenspectrum of the Koopman operator enables decomposition nonlinear dynamics into a sum functions state space with purely exponential and sinusoidal time dependence. For limited number dynamical systems, it is possible to find these eigenfunctions exactly analytically. Here, this done for Korteweg–de Vries equation on periodic interval using inverse scattering transform some concepts algebraic geometry. To authors’ knowledge, first complete analysis partial differential equation, which...

One approach to understand the chaotic dynamics of nonlinear dissipative systems is study non-chaotic yet dynamically unstable invariant solutions embedded in system's attractor. The significance zero-dimensional fixed points and one-dimensional periodic orbits capturing time-periodic widely accepted for high-dimensional systems, including fluid turbulence, while higher-dimensional tori representing quasiperiodic have rarely been considered. We demonstrate that 2-tori are generically...

In dynamical systems governed by differential equations, a guarantee that trajectories emanating from given set of initial conditions do not enter another can be obtained constructing barrier function satisfies certain inequalities on the phase space. Often, these amount to nonnegativity polynomials and enforced using sum-of-squares conditions, in which case functions constructed computationally convex optimization over polynomials. To study how well such computations characterize sets...

We formulate, for continuous-time dynamical systems, a sufficient condition to be gradient-like system, i.e. that all bounded trajectories approach stationary points and therefore periodic orbits, chaotic attractors, etc. do not exist. This is based upon the existence of an auxiliary function defined over state space in way analogous Lyapunov stability equilibrium. For polynomial functions can found computationally by using sum-of-squares optimisation. demonstrate this method finding such...

Abstract We formulate, for continuous-time dynamical systems, a sufficient condition to be gradient-like system, i.e. that all bounded trajectories approach stationary points and therefore periodic orbits, chaotic attractors, etc do not exist. This is based upon the existence of an auxiliary function defined over state space in way analogous Lyapunov stability equilibrium. For polynomial functions can found computationally by using sum-of-squares optimisation. demonstrate this method finding...

The Birman-Williams theorem gives a connection between the collection of unstable periodic orbits (UPOs) contained within chaotic attractor and topology that attractor, for three-dimensional systems. In certain cases, fractal dimension in partial differential equation (PDE) is less than three, even though embedded an infinite-dimensional space. Here we study Kuramoto-Sivashinsky PDE at onset chaos. We use two different dimensionality-reduction techniques - proper orthogonal decomposition...

Unstable periodic orbits (UPOs) are believed to be the underlying dynamical structures of spatio-temporal chaos and turbulence. Finding these UPOs is however notoriously difficult. Matrix-free loop convergence algorithms deform entire space-time fields (loops) until they satisfy evolution equations. Initial guesses for robust variational thus in a high-dimensional state space, rendering their generation highly challenging. Usually generated with recurrency methods, which most suited shorter...

Close to a saddle-node bifurcation, when two invariant solutions collide and disappear, the behavior of dynamical system can closely resemble that solution which is no longer present at chosen parameter value. For bifurcating equilibria in low-dimensional ODEs, influence such 'ghosts' on temporal system, namely delayed transitions, has been studied previously. We consider spatio-temporal PDEs characterize phenomenon ghosts by defining representative state-space structures, we term 'ghost...

The breaking of internal gravity waves in the abyssal ocean is thought to be responsible for much mixing necessary close oceanic buoyancy budgets. exact mechanism by which these break down into turbulence remains an active area research and can have significant implications on efficiency. Recent evidence has suggested that both shear instabilities convective play a role wave high Richardson number mean flow. We perform systematic analysis stability configuration superimposed background flow...

It has recently been speculated that statistical properties of chaos may be captured by weighted sums over unstable invariant tori embedded in the chaotic attractor hyperchaotic dissipative systems; analogous to periodic orbits formalized within orbit theory. Using a novel numerical method for converging 2-tori PDE, we identify many quasiperiodic, unstable, 2-torus solutions modified Kuramoto-Sivashinsky equation exhibiting dynamics with two positive Lyapunov exponents. The set covers...