Narrow Results

Discrete scale invariance, which corresponds to a partial breaking of the scaling symmetry, is reflected in the existence of a hierarchy of characteristic scales l₀,l₀λ,l₀λ²,…, where λ is a preferred scaling ratio and l₀ a microscopic cut-off. Signatures of discrete scale invariance have recently been found in a variety of systems ranging from rupture, earthquakes, Laplacian growth phenomena, "animals" in percolation to financial market crashes. We believe it to be a quite general, albeit subtle phenomenon. Indeed, the practical problem in uncovering an underlying discrete scale invariance is that standard ensemble averaging procedures destroy it as if it was pure noise. This is due to the fact, that while λ only depends on the underlying physics, l₀ on the contrary is realization-dependent. Here, we adapt and implement a novel so-called "canonical" averaging scheme which re-sets the l₀ of different realizations to approximately the same value. The method is based on the determination of a realization-dependent effective critical point obtained from, e.g., a maximum susceptibility criterion. We demonstrate the method on diffusion limited aggregation and a model of rupture.

We analyze two theoretical approaches to ensemble averaging for integrable systems in quantum chaos, spectral averaging (SA) and parametric averaging (PA). For SA, we introduce a new procedure, namely, rescaled spectral averaging (RSA). Unlike traditional SA, it can describe the correlation function of spectral staircase (CFSS) and produce persistent oscillations of the interval level number variance (IV). PA while not as accurate as RSA for the CFSS and IV, can also produce persistent oscillations of the global level number variance (GV) and better describes saturation level rigidity as a function of the running energy. Overall, it is the most reliable method for a wide range of statistics.

This paper gives a short overview of various applications of stabilization by vibration, along with the exposition of the geometrical mechanism of this phenomenon. More specifically, the following observation is described: a rapidly vibrated holonomic system can be approximated by a certain associated nonholonomic system. It turns out that effective forces in some rapidly vibrated (holonomic) systems are the constraint forces of an associated auxiliary nonholonomic constraint. In particular, we review a simple but remarkable connection between the curvature of the pursuit curve (the tractrix) on the one hand and the effective force on the pendulum with vibrating support. The latter observation is a part of a recently discovered close relationship between two standard classical problems in mechanics: (1) the pendulum whose suspension point executes fast periodic motion along a given curve, and (2) the Chaplygin skate (known also as the Prytz planimeter, or the "bicycle"). The former is holonomic, the latter is nonholonomic. The holonomy of the skate shows up in the effective motion of the pendulum. This relationship between the pendulum with a twirled pivot and the Chaplygin skate has somewhat unexpected physical manifestations, such as the drift of suspended particles in acoustic waves. Finally, a higher-dimensional example of "geodesic motion" on a vibrating surface is described.

Dynamical behavior of a nonsmooth master system which is coupled to a nonsmooth Nonlinear Energy Sink (NES) during free and forced oscillations is studied analytically and numerically. Invariant manifolds of the system and their stable zones at different time scales are revealed and finally application of coupled nonsmooth NES to the passive control process of the main nonsmooth system is highlighted.

We provide the nine topological global phase portraits in the Poincaré disk of the family of the centers of Kukles polynomial differential systems of the form $ẋ=-y,$$ẏ=x+a{x}^{5}y+b{x}^3{y}^3+cx{y}^5,$ where $x,y\in ℝ$ and $a,b,c$ are real parameters satisfying $a^2+b^2+c^2\ne 0.$ Using averaging theory up to sixth order we determine the number of limit cycles which bifurcate from the origin when we perturb this system first inside the class of all homogeneous polynomial differential systems of degree $6,$ and second inside the class of all polynomial differential systems of degree $6.$

The effect of multiplicative stochastic perturbations on planar Hamiltonian systems is investigated. It is assumed that perturbations fade with time and preserve a stable equilibrium of the limiting system. The paper investigates bifurcations associated with changes in the stability of the equilibrium and with the appearance of new stochastically stable states in the perturbed system. It is shown that depending on the structure and the parameters of the decaying perturbations, the equilibrium can remain stable or become unstable. In some intermediate cases, a practical stability of the equilibrium with estimates for the length of the stability interval is justified. The performed stability analysis is based on a combination of the averaging method and the construction of stochastic Lyapunov functions.

We introduce a simple and straightforward averaging procedure, which is a generalization of one which is commonly used in electrodynamics, and show that it possesses all the characteristics we require for linearized averaging in general relativity and cosmology — for weak-field and perturbed FLRW situations. In particular, we demonstrate that it yields quantities which are approximately tensorial in these situations, and that its application to an exact FLRW metric yields another FLRW metric, to first-order in integrals over the local coordinates. Finally, we indicate some important limits of any linearized averaging procedure with respect to cosmological perturbations which are the result of averages over large amplitude small and intermediate scale inhomogeneities, and show our averaging procedure can be approximately implemented by that of Zotov and Stoeger in these cases.

We consider a predator-prey model in a multi-patch environment. We assume the existence of two time scales: the migration process takes place on the behavioural level and is thus much faster than the population dynamics. Each population is subdivided into subpopulations which correspond to the spatial distribution. The model is thus a large system of ordinary differential equations. We assume that the migration rates are fastly oscillating: it is the case for some aquatic populations for example. Indeed, these populations undergo regular vertical movements in the water column every day. In order to study our model, we use a reduction method which allows us to simplify the initial model. It is then possible to bring to light that some properties emerge from the coupling between the fast migration process and the slow population dynamics. We give an explicit example of the emerging property.

We establish global-in-time averaging for the $L^2$-critical dispersion-managed nonlinear Schrödinger equation in the fast dispersion management regime. In particular, in the case of nonzero average dispersion, we establish averaging with any subcritical data, while in the case of a strictly positive dispersion map, we obtain averaging for data in $L^2$.

In systems which combine slow and fast motions the averaging principle says that a good approximation of the slow motion can be obtained by averaging its parameters in fast variables. This setup arises, for instance, in perturbations of Hamiltonian systems where motions on constant energy manifolds are fast and across them are slow. When these perturbations are deterministic Anosov's theorem says that the averaging principle works except for a small in measure set of initial conditions while Neistadt's theorem gives error estimates in the case of perturbations of integrable Hamiltonian systems. These results are extended here to the case of fast and slow motions given by stochastic differential equations.

Recently an averaging principle for diffusion processes with a null-recurrent fast component without a drift term was obtained in [4]. In this note this result is widened to allow a drift in the fast component. As a corollary a new result on the homogenization for parabolic PDE's is obtained.

Assuming that the fast motion in averaging is sufficiently well mixing we show that the slow motion can be approximated in the L²-sense by a diffusion solving Hasselmann's nonlinear stochastic differential equation and which provides a much better approximation than the one suggested by the averaging principle. Previously, only weak limit theorems in averaging were known which cannot justify, in principle, a nonlinear diffusion approximation of the slow motion.

Quasi-linear perturbations of a two-dimensional flow with a first integral and the corresponding parabolic PDEs with a small parameter at the second-order derivatives are considered in this paper.

The characteristic equation for a linear delay differential equation (DDE) has countably infinite roots on the complex plane. We deal with linear DDEs that are on the verge of instability, i.e. a pair of roots of the characteristic equation (eigenvalues) lie on the imaginary axis of the complex plane, and all other roots have negative real parts. We show that, when the system is perturbed by small noise, under an appropriate change of time scale, the law of the amplitude of projection onto the critical eigenspace is close to the law of a certain one-dimensional stochastic differential equation (SDE) without delay. Further, we show that the projection onto the stable eigenspace is small. These results allow us to give an approximate description of the delay-system using an SDE (without delay) of just one dimension. The proof is based on the martingale problem technique.

We extend the Erdős–Rényi law of large numbers to the averaging setup both in discrete and continuous time cases. We consider both stochastic processes and dynamical systems as fast motions whenever they are fast mixing and satisfy large deviations estimates. In the continuous time case we consider flows with large deviations estimates which allow a suspension representation and it turns out that fast mixing of corresponding base transformations suffices for our results.

In this paper, we consider a fully-coupled slow–fast system of McKean–Vlasov stochastic differential equations with full dependence on the slow and fast component and on the law of the slow component and derive convergence rates to its homogenized limit. We do not make periodicity assumptions, but we impose conditions on the fast motion to guarantee ergodicity. In the course of the proof we obtain related ergodic theorems and we gain results on the regularity of Poisson type of equations and of the associated Cauchy problem on the Wasserstein space that are of independent interest.

In this paper, the effectiveness of three different operating strategies applied to the Fuzzy ARTMAP (FAM) neural network in pattern classification tasks is analyzed and compared. Three types of FAM, namely average FAM, voting FAM, and ordered FAM, are formed for experimentation. In average FAM, a pool of the FAM networks is trained using random sequences of input patterns, and the performance metrics from multiple networks are averaged. In voting FAM, predictions from a number of FAM networks are combined using the majority-voting scheme to reach a final output. In ordered FAM, a pre-processing procedure known as the ordering algorithm is employed to identify a fixed sequence of input patterns for training the FAM network. Three medical data sets are employed to evaluate the performances of these three types of FAM. The results are analyzed and compared with those from other learning systems. Bootstrapping has also been used to analyze and quantify the results statistically.

We study nonlinear ODE problems in the complex plane, with the right hand side f(u, t) being complex analytic function of the space variable u with the coefficients being nonconstant periodic functions in the time variable t and having non-zero Fourier coefficients for non-negative indices only. Assuming first that , where h is holomorphic in the two variables, we drop out the local a priori estimate for the existence of T-periodic solutions of previous works. We prove that there exists r_o > 0, not depending of T, such that for every r < r₀ the equation has periodic solution. Next we show that the same still holds for the mentioned above class of nonlinear problems. This improves previous author results, also joint with A. Borisovich.

We study semi-linear evolution inclusion in a Banach space E with uniformly convex dual E *. Averaging and partial averaging results on a finite interval is proved, when A is linear m-dissipative and F, G are one sided Lipschitz. When F(t, ·) is one sided Lipschitz with negative constant we prove such results on an infinite interval.

Has applied mathematics disappeared behind the computer screen? High level computer languages and high capacity computers for simulation followed by data mining give powerful tools for studying complex systems. The two examples presented here are a system of difference equations from population genetics and a system of Volterra integral equations from demography, both perturbed by random noise. We analyze these systems using averaging methods and then compare the derived solution with computer simulations of the system. In the first case, mathematical analysis guides mining of the simulated solution data base. In the second, mathematical analysis reveals deterministic chaotic behavior in the averaged system that confounds the simulation/data-mining approach.