Small Random Perturbations Research Articles

Thermalisation for small random perturbations of dynamical systems

We consider an ordinary differential equation with a unique hyperbolic attractor at the origin, to which we add a small random perturbation. It is known that under general conditions, the solution of this stochastic differential equation converges exponentially fast to an equilibrium distribution. We show that the convergence occurs abruptly: in a time window of small size compared to the natural time scale of the process, the distance to equilibrium drops from its maximal possible value to near zero, and only after this time window the convergence is exponentially fast. This is what is known as the cut-off phenomenon in the context of Markov chains of increasing complexity. In addition, we are able to give general conditions to decide whether the distance to equilibrium converges in this time window to a universal function, a fact known as profile cut-off.

Refinement of dynamic equilibrium using small random perturbations

AbstractWe propose a refinement process of dynamic equilibria based on small random perturbations (SRPs) of the backward perfect foresight (bpf) equilibrium map in a class of one‐step, forward‐looking dynamic models. An equilibrium is selected if its stationary measure is the limit of the stationary measures associated with the processes generated by the SRPs of the bpf maps, as the perturbation size approaches 0. We show that, for full measure sets of parameter values of a large class of one‐parameter families of unimodal bpf maps, only determinate cycles or the chaotic sunspot equilibrium defined by Araujo and Maldonado (2000) is selected. Two examples are provided illustrating such refinement process.

Statistical approach for parameter identification by Turing patterns

Prevailing theories in biological pattern formation, such as in morphogenesis or multicellular structures development, have been based on purely chemical processes, with the Turing models as the prime example. Recent studies have challenged the approach, by underlining the role of mechanical forces. A quantitative discrimination of competing theories is difficult, however, due to the elusive character of the processes: different mechanisms may result in similar patterns, while patterns obtained with a fixed model and fixed parameter values, but with small random perturbations of initial values, will significantly differ in shape, while being of the “same” type. In this sense each model parameter value corresponds to a family of patterns, rather than a fixed solution. For this situation we create a likelihood that allows a statistically sound way to distinguish the model parameters that correspond to given patterns. The method allows us to identify model parameters of reaction-diffusion systems by using Turing patterns only, i.e., the steady-state solutions of the respective equations without the use of transient data or initial values. The method is tested with three classical models of pattern formation: the FitzHugh-Nagumo model, Gierer-Meinhardt system and Brusselator reaction-diffusion system. We quantify the accuracy achieved by different amounts of training data by Bayesian sampling methods. We demonstrate how a large enough ensemble of patterns leads to detection of very small but systematic structural changes, practically impossible to distinguish with the naked eye.

Young bilateral supernova remnants evolving into a turbulent interstellar magnetic field

ABSTRACT We employ 3D magnetohydrodynamic simulations to study the morphology and synchrotron emission of young supernova remnants evolving in a turbulent interstellar magnetic field, seeking to shed new light on to the polarization structure of the emission and on the debate concerning the quasi-parallel and quasi-perpendicular acceleration mechanisms. In the simulations, we consider a non-homogeneous interstellar medium magnetic field by introducing small random perturbations in the direction and intensity of the field. In order to analyse the dependence of the radio morphology on the degree of magnetic field perturbation and the observer’s point of view, we compute synthetic maps of the polarized intensity, position-angle, polarization fraction, and the polar-reference angle. By comparing the distribution of this angle to the polarization intensity, we show that it is possible to identify what type of acceleration mechanism is taking place at the main shock front.

Analysis of a stochastic single species model with Allee effect and jump-diffusion

In this paper, we consider the effects of the small and abrupt random perturbations in the environment, and formulate a stochastic single species model with Allee effect and jump-diffusion. We first prove that the model admits a unique solution which is global and positive. Then we study the stochastic permanence and extinction of the model. In addition, we estimate the growth rate of the solution. Our results reveal that the properties of the model have close relationships with the jump-diffusion. Finally, we work out several numerical simulations to validate the theoretical results.

Particles in finite and infinite one-dimensional periodic chains

Particle motion in one-dimensional crystal chains is studied with the help of the transfer matrix method. The transition from a finite to an infinite chain is analyzed. In the cases where an analytic solution is impossible, the method allows calculating the energy spectra with reasonable accuracy, based on the known cell potential. It turns out that the structure of allowed and forbidden energy bands arising in an ideal lattice contains some features that are absent in the real world. This means that the model of an ideal lattice should be extended in order to describe reality. We show that accounting for small random perturbations of periodicity may serve as such an extension. Light propagation in a layered medium (including a photonic crystal) is studied using the same method.

Using Parsimony-Guided Tree Proposals to Accelerate Convergence in Bayesian Phylogenetic Inference.

Sampling across tree space is one of the major challenges in Bayesian phylogenetic inference using Markov chain Monte Carlo (MCMC) algorithms. Standard MCMC tree moves consider small random perturbations of the topology, and select from candidate trees at random or based on the distance between the old and new topologies. MCMC algorithms using such moves tend to get trapped in tree space, making them slow in finding the globally most probable trees (known as “convergence”) and in estimating the correct proportions of the different types of them (known as “mixing”). Here, we introduce a new class of moves, which propose trees based on their parsimony scores. The proposal distribution derived from the parsimony scores is a quickly computable albeit rough approximation of the conditional posterior distribution over candidate trees. We demonstrate with simulations that parsimony-guided moves correctly sample the uniform distribution of topologies from the prior. We then evaluate their performance against standard moves using six challenging empirical data sets, for which we were able to obtain accurate reference estimates of the posterior using long MCMC runs, a mix of topology proposals, and Metropolis coupling. On these data sets, ranging in size from 357 to 934 taxa and from 1740 to 5681 sites, we find that single chains using parsimony-guided moves usually converge an order of magnitude faster than chains using standard moves. They also exhibit better mixing, that is, they cover the most probable trees more quickly. Our results show that tree moves based on quick and dirty estimates of the posterior probability can significantly outperform standard moves. Future research will have to show to what extent the performance of such moves can be improved further by finding better ways of approximating the posterior probability, taking the trade-off between accuracy and speed into account. [Bayesian phylogenetic inference; MCMC; parsimony; tree proposal.]

On the excursion area of perturbed Gaussian fields

We investigate Lipschitz-Killing curvatures for excursion sets of random fields on ℝ2under a very specific perturbation, namely a small spatial-invariant random perturbation with zero mean. An expansion formula for mean curvatures is derived when the magnitude of the perturbation vanishes, which recovers the Gaussian Kinematic Formula at the limit by contiguity of the model. We develop an asymptotic study of the perturbed excursion area behaviour that leads to a quantitative non-Gaussian limit theorem, in Wasserstein distance, for fixed small perturbations and growing domain. When letting both the perturbation vanish and the domain grow, a standard Central Limit Theorem follows. Taking advantage of these results, we propose an estimator for the perturbation variance which turns out to be asymptotically normal and unbiased, allowing to make inference through sparse information on the field.

Threshold Models of Collective Behavior II: The Predictability Paradox and Spontaneous Instigation

Collective behavior can be notoriously hard to predict. We revisited a possible explanation suggested by Granovetter’s classic threshold model: collective behavior can unexpectedly fail, despite a group’s strong interest in the outcome, because of the sensitivity of cascades to small random perturbations in group composition and the distribution of thresholds. Paradoxically, we found that a small amount of randomness in individual behavior can make collective behavior less sensitive to these perturbations and therefore more predictable. We also examined conditions in which collective behavior unexpectedly succeeds despite the group’s weak interest in the outcome. In groups with an otherwise intractable start-up problem, individual randomness can lead to spontaneous instigation, making outcomes more sensitive to the strength of collective interests and therefore more predictable. These effects of chance behavior become much more pronounced as group size increases. Although randomness is often assumed to be a theoretically unimportant residual category, our findings point to the need to bring individual idiosyncrasy back into the study of collective behavior.

On the fast convergence of random perturbations of the gradient flow

We consider in this work small random perturbations (of multiplicative noise type) of the gradient flow. We prove that under mild conditions, when the potential function is a Morse function with additional strong saddle condition, the perturbed gradient flow converges to the neighborhood of local minimizers in O ( ln ( ε − 1 ) ) time on the average, where ε is the scale of the random perturbation. Under a change of time scale, this indicates that for the diffusion process that approximates the stochastic gradient method, it takes (up to logarithmic factor) only a linear time of inverse stepsize to evade from all saddle points. This can be regarded as a manifestation of fast convergence of the discrete-time stochastic gradient method, the latter being used heavily in modern statistical machine learning.

Reliable Smart Road Signs

In this paper, we propose a game theoretical adversarial intervention detection mechanism for reliable smart road signs. A future trend in intelligent transportation systems is “smart road signs” that incorporate smart codes (e.g., visible at infrared) on their surface to provide more detailed information to smart vehicles. Such smart codes make road sign classification problem aligned with communication settings more than conventional classification. This enables us to integrate well-established results in communication theory, e.g., error-correction methods, into road sign classification problem. Recently, vision-based road sign classification algorithms have been shown to be vulnerable against (even) small scale adversarial interventions that are imperceptible for humans. On the other hand, smart codes constructed via error-correction methods can lead to robustness against small scale intelligent or random perturbations on them. In the recognition of smart road signs, however, humans are out of the loop since they cannot see or interpret them. Therefore, there is no equivalent concept of imperceptible perturbations in order to achieve a comparable performance with humans. Robustness against small scale perturbations would not be sufficient since the attacker can attack more aggressively without such a constraint. Under a game theoretical solution concept, we seek to ensure certain measure of guarantees against even the worst case (intelligent) attackers that can perturb the signal even at large scale. We provide a randomized detection strategy based on the distance between the decoder output and the received input, i.e., error rate. Finally, we examine the performance of the proposed scheme over various scenarios.

Improving Selfish Routing for Risk-Averse Players

We investigate how and to which extent one can exploit risk-aversion and modify the perceived cost of the players in selfish routing so that the Price of Anarchy (PoA) wrt. the total latency is improved. The starting point is to introduce some small random perturbations to the edge latencies so that the expected latency does not change, but the perceived cost of the players increases, due to risk-aversion. We adopt the simple model of γ-modifiable routing games, a variant of selfish routing games with restricted tolls. We prove that computing the best γ-enforceable flow is NP-hard for parallel-link networks with affine latencies and two classes of heterogeneous risk-averse players. On the positive side, we show that for parallel-link networks with heterogeneous players and for series-parallel networks with homogeneous players, there exists a nicely structured γ-enforceable flow whose PoA improves fast as γ increases. We show that the complexity of computing such a γ-enforceable flow is determined by the complexity of computing a Nash flow for the original game. Moreover, we prove that the PoA of this flow is best possible in the worst-case, in the sense that there are instances where (i) the best γ-enforceable flow has the same PoA, and (ii) considering more flexible modifications does not lead to any improvement on the PoA.

Estimates of Initial Scales for Layers with Small Random Negative-Definite Perturbations

In this work, we consider the Schrödinger operator in a multi-dimensional layer with small random perturbations. Perturbations are distributed in periodicity cells of an arbitrarily chosen periodic lattice. To each cell, we put in correspondence a random variable; these random variables are independent and have the same distributions. Perturbations are described by the same abstract symmetric operator depending on the random variable multiplied by a global small parameter. We consider the case where the perturbations shift the bottom part of the spectrum of the unperturbed operator to the left by a quantity of order of the square of the small parameter. Under these conditions, we establish the main result, which is the estimate of initial scales. We also present particular examples that demonstrate the main result.

Diffusion and Persistence of Rotor/Stator Synchronous Full Annular Rub Response Under Weak Random Perturbations

In this paper, the diffusion and persistence of synchronous full annular rub in a rotor/stator system under weak random disturbances are studied. First, the model of the rotor/stator rubbing system and the global response characteristics of the corresponding deterministic system are introduced. Then the diffusion features of the synchronous full annular rub response under small random perturbation are quantified and visualized using the stochastic sensitivity function for the discrete-time systems. The shape of the density distribution of the rotor orbit under different noise intensities and rotation speeds are compared and discussed. Finally, using the confidence ellipsoid of the response, the mechanism of noise-induced escape from the synchronous full annular rub to a large amplitude response of dry friction backward whirl is revealed, and the persistence of full annular rub response responses under different conditions is discussed and analyzed in detail. The results may be helpful in studying complex phenomena in random-disturbed rotor/stator rubbing systems.

Modality analysis of patterns in reaction-diffusion systems with random perturbations

In this paper, a distributed Brusselator model with diffusion is investigated. It is well known that this model undergoes both Andronov-Hopf and Turing bifurcations. It is shown that in the parametric zone of diffusion instability the model generates a variety of stable spatially nonhomogeneous structures (patterns). This system exhibits a phenomenon of the multistability with the diversity of stable spatial structures. At the same time, each pattern has its unique parametric range, on which it may be observed. The focus is on analysis of stochastic phenomena of pattern formation and transitions induced by small random perturbations. Stochastic effects are studied by the spatial modality analysis. It is shown that the structures possess different degrees of stochastic sensitivity.

Large deviation principle for dynamical systems coupled with diffusion–transmutation processes

In this paper, we introduce a mathematical apparatus that is relevant for understanding a dynamical system with small random perturbations and coupled with the so-called transmutation process — where the latter jumps from one mode to another, and thus modifying the dynamics of the system. In particular, we study the exit problem, i.e., an asymptotic estimate for the exit probabilities with which the corresponding processes exit from a given bounded open domain, and then formally prove a large deviation principle for the exit position joint with the type occupation times as the random perturbation vanishes. Moreover, under certain conditions, we also determine the exit place and the type of distribution at the exit time and, as a consequence of this, such information also give the limit of the Dirichlet problem for the corresponding partial differential equation systems with a vanishing small parameter.

On the $$L^\infty $$ L ∞ -Uniqueness of Dynamical Systems with Small Random Perturbation

This article is devoted to study the \(L^\infty \)-uniqueness (in the sense of essential self-adjointness) for the generator of a dynamical system with small random perturbation. As consequence is obtained the \(L^1\)-uniqueness of the weak solution of the Cauchy problem for the associated Fokker–Planck–Kolmogorov equation.

DYNAMIC COMPLEXITY OF A PREDATOR-PREY MODEL FOR IPM WITH NONLINEAR IMPULSIVE CONTROL INCORPORATING A REGULATORY FACTOR FOR PREDATOR RELEASES

The success of integrated pest management (IPM) depends on spraying the correct amount of pesticides at an appropriate time and releases of natural enemies or pathogens of the pest in appropriate proportions at critical times, with little cost and minimal effects on the environment. Therefore, control decisions require information on instantaneous killing rates of pesticides and numbers of natural enemies to be released, variables that should depend on the densities of both pest and natural enemy population densities in the field. To describe such a control strategy we have proposed a mathematical model of IPM involving releases of natural enemies in relation to a regulatory factor. The threshold condition for the existence and stability of the pest free periodic solution is provided using a cobweb model, the comparison principle and Floquet theory, which reveals the effects of nonlinear control actions on pest outbreaks. Bifurcation analyses show that the dynamics of the proposed model can be very complex, including multiple attractors and switch-like transition patterns following small random perturbations. Moreover, the random perturbations and nonlinear impulsive control measures could generate complex switching patterns, which show that the pest population could have outbreaks in complex ways due to environmental noise.

Random perturbations of hyperbolic dynamics

A sequence of large invertible matrices given by a small random perturbation around a fixed diagonal and positive matrix induces a random dynamics on a high-dimensional sphere. For a certain class of rotationally invariant random perturbations it is shown that the dynamics approaches the stable fixed points of the unperturbed matrix up to errors even if the strength of the perturbation is large compared to the relative increase of nearby diagonal entries of the unperturbed matrix specifying the local hyperbolicity.

The survival analysis of a stochastic Lotka-Volterra competition model with a coexistence equilibrium.

In this paper, we propose and analyze a two-species Lotka-Volterra competition model with random perturbations that relate to the inter-specific competition rates and the coexistence equilibrium of the corresponding deterministic system. The stochasticity in inter-specific competition (between species) is more important than that in intra-specific competition (within species). We pose two assumptions and then obtain su cient conditions for coexistence and for competitive exclusion respectively, and find that small random perturbations will not destroy the dynamic behaviors of the corresponding deterministic system. Moreover, if one species goes extinct, the convergence rate to zero is obtained by investigating the Lyapunov exponent. Finally, we provide several numerical examples to illustrate our mathematical results.