We consider the dynamics of a two-dimensional ordinary differential equation exhibiting Hopf bifurcation subject to additive white noise and identify three dynamical phases: (I) random attractor with uniform synchronisation trajectories, (II) non-uniform trajectories (III) without trajectories. The attractors in phases are equilibrium points negative Lyapunov exponents while phase there is so-called strange positive exponent. analyse occurrence different as function linear stability origin...

We study metastable behaviour in systems of weakly interacting Brownian particles with localised, attractive potentials which are smooth and globally bounded. In this particular setting, numerical evidence suggests that the converge on a short time scale to "droplet state" is $metastable$, i.e. persists much longer than convergence, before eventually diffusing $0$. article, we provide rigorous quantitative characterisation separation scales. Working at level empirical measure, show (after...

Abstract. Atmospheric blocking exerts a major influence on mid-latitude atmospheric circulation and is known to be associated with extreme weather events. Previous work has highlighted the importance of origin air parcels that define region, especially respect non-adiabatic processes such as latent heating. So far, an objective method clustering individual Lagrangian trajectories passing through into larger and, more importantly, spatially coherent streams not been established. This focus...

We confirm a long-standing conjecture concerning shear-induced chaos in stochastically perturbed systems exhibiting Hopf bifurcation. The method of showing the main chaotic property, positive Lyapunov exponent, is computer-assisted proof. Using recently developed theory conditioned exponents on bounded domains and modified Furstenberg–Khasminskii formula, problem boils down to rigorous computation eigenfunctions Kolmogorov operators describing distributions underlying stochastic process.

We introduce the notion of Lyapunov exponents for random dynamical systems, conditioned to trajectories that stay within a bounded domain asymptotically long times. This is motivated by desire characterize local properties in presence unbounded noise (when almost all are unbounded). illustrate its use analysis bifurcations this context. The theory stochastic differential equations builds on quasi-stationary distributions killed processes and associated quasi-ergodic distributions. show...

We study a transcritical singularity in fast-slow system given by the explicit Euler discretization of corresponding continuous-time normal form. The analysis uses blow-up method and direct trajectory-based estimates. prove that qualitative behaviour is preserved time-discretization with sufficiently small step size. This size fully quantified relative to time scale separation. Our proof also yields results as special case provides more detailed calculations classical (or scaling) chart.

Abstract We prove the positivity of Lyapunov exponents for normal form a Hopf bifurcation, perturbed by additive white noise, under sufficiently strong shear strength. This completes series related results simplified situations which we can exploit studying suitable limits and noise parameters. The crucial technical ingredient making this approach rigorous is result on continuity via Furstenberg–Khasminskii formulas.

Atmospheric blockings -- also known as Quasi-Stationary States (QSAS) exert a major influence on mid-latitude atmospheric circulation and are to be associated with extreme weather events. Previous work has highlighted the importance of origin air parcels that define blocking region, especially respect non-adiabatic processes such moisture transport latent heating. So far, an objective method for clustering individual Lagrangian trajectories passing through QSAS into larger more importantly...

Abstract We study a singularly perturbed fast-slow system of two partial differential equations (PDEs) reaction-diffusion type on bounded domain via Galerkin discretisation. assume that the reaction kinetics in fast variable realise generic fold singularity, whereas slow takes role dynamic bifurcation parameter, thus extending classical analysis fold. Our approach combines spectral discretisation with techniques from geometric singular perturbation theory which are applied to resulting...

In this paper we study coupled fast-slow ordinary differential equations (ODEs) with small time scale separation parameter $\epsilon$ such that, for every fixed value of the slow variable, fast dynamics are sufficiently chaotic ergodic invariant measure. Convergence process to solution a homogenized stochastic equation (SDE) in limit zero, explicit formulas drift and diffusion coefficients, has so far only been obtained case that evolve independently. give sufficient conditions convergence...

In this paper, we study slow manifolds for infinite-dimensional evolution equations. We compare two approaches: an abstract equation framework and a finite-dimensional spectral Galerkin approximation. prove that the constructed within each approach are asymptotically close under suitable conditions. The proof is based upon Lyapunov-Perron methods comparison of local graphs in scales Banach spaces. summary, our main result allows us to change between different characterizations invariant...

We study fast-slow maps obtained by discretization of planar systems in continuous time. focus on describing the so-called delayed loss stability induced slow passage through a singularity systems. This can be related to presence canard solutions. Here we consider three types singularities: transcritical, pitchfork, and fold. First, show that under an explicit Runge--Kutta delay stability, due transcritical or pitchfork singularity, arbitrarily long. In contrast, prove Kahan--Hirota--Kimura...

Abstract For an attracting periodic orbit (limit cycle) of a deterministic dynamical system, one defines the isochron for each point as cross-section with fixed return time under flow. Equivalently, isochrons can be characterized stable manifolds foliating neighborhoods limit cycle or level sets map. In recent years, there has been lively discussion in mathematical physics community on how to define stochastic oscillations, i.e. cycles heteroclinic exposed noise. The main concerned approach...

We consider a fast-slow partial differential equation (PDE) with reaction-diffusion dynamics in the fast variable and slow driven by operator on bounded domain. Assuming transcritical normal form for reaction term viewing as dynamic bifurcation parameter, we analyze passage through subsystem point. In particular, employ spectral Galerkin approximation characterize invariant manifolds finite-dimensional each finite truncation using geometric desingularization via blow-up analysis. addition to...

Using various techniques from dynamical systems theory, we rigorously study an experimentally validated model by [Barkley et al., Nature, 526:550-553, 2015], which describes the rise of turbulent pipe flow via a PDE system reduced complexity. The fast evolution turbulence is governed reaction-diffusion dynamics coupled to centerline velocity, evolves with advection Burgers' type and slow relaminarization term. Applying this spatial ansatz geometric singular perturbation prove existence...

Motivated by the normal form of a fast–slow ordinary differential equation exhibiting pitchfork singularity we consider discrete-time dynamical system that is obtained an application explicit Euler method. Tracking trajectories in vicinity show, how slow manifold extends beyond singular point and give estimate on contraction rate transition mapping. The proof relies blow-up method suitably adapted to discrete setting where precise estimates for cubic map central rescaling chart make key...

Abstract We study the problem of preservation maximal canards for time discretized fast–slow systems with canard fold points. In order to ensure such preservation, certain favorable structure-preserving properties discretization scheme are required. Conventional schemes do not possess properties. perform a detailed analysis an unconventional due Kahan. The uses blow-up method deal loss normal hyperbolicity at point. show that Kahan quadratic vector fields imply similar result as in...

We establish the existence of a full spectrum Lyapunov exponents for memoryless random dynamical systems with absorption. To this end, we crucially embed process conditioned to never being absorbed, Q-process, into framework systems, allowing us study multiplicative ergodic properties. show that finite-time converge in probability and apply our results iterated function stochastic differential equations.

We study numerically a coagulation-fragmentation model derived by Niwa [17] and further elaborated Degond et al. [5]. In [5] unique equilibrium distribution of group sizes is shown to exist in both cases continuous discrete size distributions. provide numerical investigation these equilibria using three different methods approximate the equilibrium: recursive algorithm based on work Ma et. [12], Newton method resolution time-dependent problem. All schemes are validated showing that they...