arXiv subjects
Publications and source records attributed to Jeroen S. W. Lamb.
We propose a notion of random horseshoe and prove density of random horseshoes and existence of a random Young tower with annealed exponential tails for nonuniformly expanding random dynamical systems with diffusive noise.
We study the asymptotic behaviour of stationary densities of one-dimensional random diffeomorphisms, at the boundaries of their support, which correspond to deterministic fixed points of extremal diffeomorphisms. In particular, we show how this stationary density at a boundary depends on the underlying noise distribution, as well as the linearisation of the extremal diffeomorphism at the boundary point (in case the corresponding fixed point is hyperbolic), or the leading nonlinear term of the extremal diffeomorphism (in case the corresponding fixed point is not hyperbolic).
We develop an early-warning signal for bifurcations of one-dimensional random difference equations with additive bounded noise, based on the asymptotic behaviour of the stationary density near a boundary of its support. We demonstrate the practical use in numerical examples.
We numerically study bifurcations of attractors of the Hénon map with additive bounded noise with spherical reach. The bifurcations are analysed using a finite-dimensional boundary map. We distinguish between two types of bifurcations: topological bifurcations and boundary bifurcations. Topological bifurcations describe discontinuous changes of attractors and boundary bifurcations occur when singularities of an attractor's boundary are created or destroyed. We identify correspondences between topological and boundary bifurcations of attractors and local and global bifurcations of the boundary map.
We establish the existence of intermittent two-point dynamics and infinite stationary measures for a class of random circle endomorphisms with zero Lyapunov exponent, as a dynamical characterisation of the transition from synchronisation (negative Lyapunov exponent) to chaos (positive Lyapunov exponent).
We propose a notion of conditioned stochastic stability of invariant measures on repellers: we consider whether quasi-ergodic measures of absorbing Markov processes, generated by random perturbations of the deterministic dynamics and conditioned upon survival in a neighbourhood of a repeller, converge to an invariant measure in the zero-noise limit. Under suitable choices of the random perturbation, we find that equilibrium states on uniformly expanding repellers are conditioned stochastically stable. In the process, we establish a rigorous foundation for the existence of ``natural measures'', which were proposed by Kantz and Grassberger in 1984 to aid the understanding of chaotic transients.
We consider the product of i.i.d. random matrices sampled according to a probability measure $μ$ supported on a strongly irreducible and proximal subset of a compact set $S\subset GL(d,\mathbb{R})$. We establish the local analyticity of the Lyapunov moment function and the unique stationary measure on the projective space with respect to $μ$ in the total variation topology. As a consequence, we obtain the analyticity of the asymptotic variance and all higher-order Lyapunov moments.
We give an example of a planar set $E\subset \mathbb{R}^2$ for which the boundary $\partial E_\varepsilon$ of its $\varepsilon$-neighbourhood $E_\varepsilon = \{x \in \mathbb{R}^2 \, : \, \textrm{dist}(x, E) \leq \varepsilon \}$ is nowhere $C^1$-smooth, in the sense that the set of singularities on the boundary is countably dense (where we note that the latter set cannot be uncountable). Furthermore, we give an example of a planar set $E$ for which $\partial E_\varepsilon$ has the same properties as above, but in addition contains an uncountable subset, with non-integer Hausdorff dimension, where curvature is not defined. Both constructions make use of a characterisation of those star-shaped sets that are an $\varepsilon$-neighbourhood of one of their subsets.
We establish the existence of a full spectrum of Lyapunov exponents for memoryless random dynamical systems with absorption. To this end, we crucially embed the process conditioned to never being absorbed, the $Q$-process, into the framework of random dynamical systems, allowing us to study multiplicative ergodic properties. We show that the finite-time Lyapunov exponents converge in conditioned probability and apply our results to iterated function systems and stochastic differential equations.
We develop a methodology to learn finitely generated random iterated function systems from time-series of partial observations using delay embeddings. We obtain a minimal model representation for the observed dynamics, using a hidden variable representation, that is diffeomorphic to the original system.
We develop a powerful and general method to provide rigorous and accurate upper and lower bounds for Lyapunov exponents of stochastic flows. Our approach is based on computer-assisted tools, the adjoint method and established results on the ergodicity of diffusion processes. We do not require any structural assumptions on the stochastic system and work under mild hypoellipticity conditions and outside of perturbative regimes. Therefore, our method allows for the treatment of systems that were so far out of reach from existing mathematical tools. We demonstrate our method to exhibit the chaotic nature of four different systems. Finally, we show the robustness of our approach by combining it with continuation methods to produce bounds on Lyapunov exponents over large parameter regions.
We study the geometry, topological properties and smoothness of the boundaries of closed $\varepsilon$-neighbourhoods $E_\varepsilon = \{x \in \mathbb{R}^2 \, : \, \textrm{dist}(x, E) \leq \varepsilon \}$ of compact planar sets $E \subset \mathbb{R}^2$. We develop a novel technique for analysing the boundary, and use this to obtain a classification of singularities (i.e.~non-smooth points) on $\partial E_\varepsilon$ into eight categories. We show that the set of singularities is either countable or the disjoint union of a countable set and a closed, totally disconnected, nowhere dense set. Furthermore, we characterise, in terms of local geometry, those $\varepsilon$-neighbourhoods whose complement $\overline{\mathbb{R}^2 \setminus E_\varepsilon}$ is a set with positive reach. It is known that for all bounded $E \subset \mathbb{R}^d$ and all $\varepsilon > 0$, the boundary $\partial E_\varepsilon$ is $(d-1)$-rectifiable. Improving on this, we identify a sufficient condition for the boundary to be uniformly rectifiable, and provide an example of a planar $\varepsilon$-neighbourhood that is not Ahlfors regular. In terms of the topological structure, we show that for a compact set $E$ and $\varepsilon > 0$ the boundary $\partial E_\varepsilon$ can be expressed as a disjoint union of an at most countably infinite union of Jordan curves and a possibly uncountable, totally disconnected set of singularities. Finally, we show that curvature is defined almost everywhere on the Jordan curve subsets of the boundary.
We study the global topological structure and smoothness of the boundaries of $\varepsilon$-neighbourhoods $E_\varepsilon = \{x \in \mathbb{R}^2 \, : \, \textrm{dist}(x, E) \leq \varepsilon \}$ of planar sets $E \subset \mathbb{R}^2$. We show that for a compact set $E$ and $\varepsilon > 0$ the boundary $\partial E_\varepsilon$ can be expressed as a disjoint union of an at most countably infinite union of Jordan curves and a possibly uncountable, totally disconnected set of singularities. We also show that curvature is defined almost everywhere on the Jordan curve subsets of the boundary.
In a class of heterogeneous random networks, where each node dynamics is a random dynamical system, interacting with neighbor nodes via a random coupling function, we characterize the hub behavior as the mean-field, subject to statistically controlled fluctuations. In particular, we prove that the fluctuations are small over exponentially long time scales and obtain Berry-Esseen estimates for the fluctuation statistics at any fixed time. Our results provide a mathematical explanation for several numerical observations, including the scaling relation between system size and frequency of large fluctuations, as well as system size induced desynchronization. To our best knowledge, these are the first characterizations of mean-field fluctuations on networks with a degree distribution that follows a power-law, a common feature for many realistic systems.
This work investigates a three-dimensional slow-fast stochastic system with quadratic nonlinearity and additive noise, inspired by fluid dynamics. The deterministic counterpart exhibits a periodic orbit and a slow manifold. We demonstrate that, under specific parameter regimes, this deterministic slow manifold can serve as an approximate parameterization of the fast variable by the slow variables within the stochastic system. Building upon this parameterization, we derive a two-dimensional reduced model, a stochastic Hopf normal form, that captures the essential dynamics of the original system. Both the original and the reduced systems possess ergodic invariant measures, characterizing their long-term behavior. We quantify the discrepancy between the original system and its slow approximation by deriving error estimates involving the Wasserstein distance between the marginals of these invariant measures along the radial component. These error bounds are shown to be controlled by a parameterization defect, which measures the quality of the fast-slow variable parameterization. A key technical innovation lies in the application of Girsanov's theorem to obtain these error estimates in the presence of oscillatory instabilities. Furthermore, we extend our analysis to regimes exhibiting an "inverted" timescale separation, where the variable to be parameterized evolves on a slower timescale than the resolved variables. To address these more challenging scenarios, we introduce path-dependent coefficients in the parameterizing manifold, enabling the derivation of robust error bounds for the corresponding reduced model. Numerical simulations complement our theoretical findings, providing insights into the model's behavior and exploring parameter regimes beyond the scope of our analytical results.
We propose a notion of random horseshoe for one-dimensional random dynamical systems. We prove the abundance of random horseshoes for a class of circle endomorphisms subject to additive noise, large enough to make the Lyapunov exponent positive. In particular, we provide conditions which guarantee that given any pair of disjoint intervals, for almost every noise realization, there exists a positive density sequence of return times to these intervals such that the induced dynamics are the full shift on two symbols.
In this paper, we consider absorbing Markov chains $X_n$ admitting a quasi-stationary measure $μ$ on $M$ where the transition kernel $\mathcal P$ admits an eigenfunction $0\leq η\in L^1(M,μ)$. We find conditions on the transition densities of $\mathcal P$ with respect to $μ$ which ensure that $η(x) μ(\mathrm d x)$ is a quasi-ergodic measure for $X_n$ and that the Yaglom limit converges to the quasi-stationary measure $μ$-almost surely. We apply this result to the random logistic map $X_{n+1} = ω_n X_n (1-X_n)$ absorbed at $\mathbb R \setminus [0,1],$ where $ω_n$ is an i.i.d sequence of random variables uniformly distributed in $[a,b],$ for $1\leq a <4$ and $b>4.$
We establish the existence and uniqueness of quasi-stationary and quasi-ergodic measures for almost surely absorbed discrete-time Markov chains under weak conditions. We obtain our results by exploiting Banach lattice properties of transition functions under natural regularity assumptions.