The Law of the Iterated Logarithm for Random Dynamical System with Jumps and State-Dependent Jump Intensity

In this paper our considerations are focused on some Markov chain associated with certain piecewise-deterministic Markov process with a state-dependent jump intensity for which the exponential ergodicity was obtained in [4]. Using the results from [3] we show that the law of iterated logarithm holds for such a model.

Non-equilibrium thermodynamics and the free energy principle in biology

According to the free energy principle, life is an “inevitable and emergent property of any (ergodic) random dynamical system at non-equilibrium steady state that possesses a Markov blanket” (Friston in J R Soc Interface 10(86):20130475, 2013). Formulating a principle for the life sciences in terms of concepts from statistical physics, such as random dynamical system, non-equilibrium steady state and ergodicity, places substantial constraints on the theoretical and empirical study of biological systems. Thus far, however, the physics foundations of the free energy principle have received hardly any attention. Here, we start to fill this gap and analyse some of the challenges raised by applications of statistical physics for modelling biological targets. Based on our analysis, we conclude that model-building grounded in the free energy principle exacerbates a trade-off between generality and realism, because of a fundamental mismatch between its physics assumptions and the properties of actual biological targets.

Simple SIR models with Markovian control

We consider a random dynamical system, where the deterministic dynamics are driven by a finite-state space Markov chain. We provide a comprehensive introduction to the required mathematical apparatus and then turn to a special focus on the susceptible-infected-recovered epidemiological model with random steering. Through simulations we visualize the behaviour of the system and the effect of the high-frequency limit of the driving Markov chain. We formulate some questions and conjectures of a purely theoretical nature.

Uniform attractors of stochastic three-component Gray-Scott system with multiplicative noise

<p style='text-indent:20px;'>In a bounded domain, we study the long time behavior of solutions of the stochastic three-component Gray-Scott system with multiplicative noise. We first show that the stochastic three-component Gray-Scott system can generate a non-autonomous random dynamical system. Then we establish some uniform estimates of solutions for stochastic three-component Gray-Scott system with multiplicative noise. Finally, the existence of uniform and cocycle attractors is proved.</p>

Non-autonomous invariant sets and attractors: Random dynamical system

In this paper the concepts of pullback attractor ,pullback absorbing family in (deterministic) dynamical system are defined in (random) dynamical systems. Also some main result such as (existence) of pullback attractors ,upper semi-continuous of pullback attractors and uniform and global attractors are proved in random dynamical system .

Uniform attractors for a class of stochastic evolution equations with multiplicative fractional noise

This paper studies the (random) uniform attractor for a class of non-autonomous stochastic evolution equations driven by a time-periodic forcing and multiplicative fractional noise with Hurst parameter bigger than 1/2. We first establish the existence and uniqueness results for the solution to the considered equation and show that the solution generates a jointly continuous non-autonomous random dynamical system (NRDS). Moreover, we prove the existence of the uniform attractor for this NRDS through stopping time technique. Particularly, a compact uniformly absorbing set is constructed under a smallness condition imposed on the fractional noise.

Stability Assessment of Stochastic Differential-Algebraic Systems via Lyapunov Exponents with an Application to Power Systems

In this paper, we discuss stochastic differential-algebraic equations (SDAEs) and the asymptotic stability assessment for such systems via Lyapunov exponents (LEs). We focus on index-1 SDAEs and their reformulation as ordinary stochastic differential equations (SDEs). Via ergodic theory, it is then feasible to analyze the LEs via the random dynamical system generated by the underlying SDEs. Once the existence of well-defined LEs is guaranteed, we proceed to the use of numerical simulation techniques to determine the LEs numerically. Discrete and continuous QR decomposition-based numerical methods are implemented to compute the fundamental solution matrix and use it in the computation of the LEs. Important computational features of both methods are illustrated via numerical tests. Finally, the methods are applied to two applications from power systems engineering, including the single-machine infinite-bus (SMIB) power system model.

On the existence of a σ-finite acim for a random iteration of intermittent Markov maps with uniformly contractive part

For an annealed type random dynamical system arising from non-uniformly expanding maps which admits uniformly contractive branches, we establish the existence of an absolutely continuous [Formula: see text]-finite invariant measure. We also show when the invariant measure is infinite.

Random uniform exponential attractors for non-autonomous stochastic lattice systems and FitzHugh–Nagumo lattice systems with quasi-periodic forces and multiplicative noise

We first give an existence criterion for a random uniform exponential attractor for a jointly continuous non-autonomous random dynamical system defined on the product space of [Formula: see text]-weighted spaces of infinite sequences. Then, based on this criterion, we prove the existence of random uniform exponential attractors for stochastic lattice systems and stochastic FitzHugh–Nagumo lattice systems that are both with quasi-periodic forces.