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Invariant manifolds of unstable periodic orbits organize the dynamics of chaotic orbits in phase space. They provide insight into the mechanisms of transport and chaotic advection and have important applications in physical situations…
We investigate how the following properties are related to each other: i)-A manifold is "transversally" exponentially stable; ii)-The "transverse" linearization along any solution in the manifold is exponentially stable; iii)-There exists a…
Stochastic processes of evolving shapes are used in applications including evolutionary biology, where morphology changes stochastically as a function of evolutionary processes. Due to the non-linear and often infinite-dimensional nature of…
In this paper, we study the long-time stability behavior of a class of linear stochastic evolution equations in a Hilbert space with multiplicative noise. Explicit sufficient conditions for $p$-th moment and almost sure exponential…
Invariant manifolds are important sets arising in the stability theory of dynamical systems. In this article, we take a brief review of invariant sets. We provide some results regarding the existence of invariant lines and parabolas in…
We study the stochastic evolution of four species in cyclic competition in a well mixed environment. In systems composed of a finite number $N$ of particles these simple interaction rules result in a rich variety of extinction scenarios,…
We provide an alternative, simpler proof of the existence of thick triangulations for noncompact $\mathcal{C}^1$ manifolds. Moreover, this proof is simpler than the original one given in \cite{pe}, since it mainly uses tools of elementary…
The exponential ordering is exploited in the context of non-auto\-no\-mous delay systems, inducing monotone skew-product semiflows under less restrictive conditions than usual. Some dynamical concepts linked to the order, such as…
The recent interest in human dynamics has led researchers to investigate the stochastic processes that explain human behaviour in different contexts. Here we propose a generative model to capture the essential dynamics of survival analysis,…
In Hamiltonian systems subjected to periodic perturbations the stable and unstable manifolds of the unstable periodic orbits provide the dynamical "skeleton" that drives the mixing process and bounds the chaotic regions of the phase space.…
Laboratory experiments with bacterial colonies, under well-controlled conditions often lead to evolutionary diversification, where at least two ecotypes emerge from an initially monomorphic population. Empirical evidence suggests that such…
In this paper, we consider a class of evolution equations driven by finite-dimensional $\gamma$-H\"{o}lder rough paths, where $\gamma\in(1/3,1/2]$. We prove the global-in-time solutions of rough evolution equations(REEs) in a sutiable…
We identify an issue in recent approaches to learning-based control that reformulate systems with uncertain dynamics using a stochastic differential equation. Specifically, we discuss the approximation that replaces a model with fixed but…
This paper presents a survey of maximal inequalities for stochastic convolutions in $2$-smooth Banach spaces and their applications to stochastic evolution equations.
Living materials such as membranes, cytoskeletal assemblies, cell collectives and tissues can often be described as active solids -- materials that are energized from within, with elastic response about a well defined reference…
Population structure can have a significant effect on evolution. For some systems with sufficient symmetry, analytic results can be derived within the mathematical framework of evolutionary graph theory which relate to the outcome of the…
We formulate and prove a {\it Local Stable Manifold Theorem\/} for stochastic differential equations (sde's) that are driven by spatial Kunita-type semimartingales with stationary ergodic increments. Both Stratonovich and It\^o-type…
In this paper, we consider the dynamics of solutions to complex-valued evolutionary partial differential equations (PDEs) and show existence of heteroclinic orbits from nontrivial equilibria to zero via computer-assisted proofs. We also…
The paper is concerned with a class of stochastic evolution equations in Hilbert space with random coefficients driven by Teugel's martingales and an independent multi-dimensional Brownian motion and its optimal control problem. Here…
A method for constructing evolution equations admitting a master symmetry is proposed. Several examples illustrating the method are presented. It is also noted that for certain evolution equations master symmetries can be useful for…