A theory of average response to large jump perturbations.

We study the limit behavior of (non)stationary and random chaotic dynamical systems. Several (vector-valued) almost sure invariance principles for (non)stationary dynamical systems and quenched (vector-valued) almost sure invariance principles for random dynamical systems are proved. We also apply our results to stationary chaotic dynamical systems, which admit Young towers, and to (non)uniformly expanding non-stationary and random dynamical systems with intermittencies or uniform spectral gaps. It implies that the systems under study tend to a Brownian motion under various scalings.

In this paper, a plane viscous liquid sheet is considered in contact with two passive media, from which two external perturbations of small intensity, given by local pressure distributions, act on the sheet. Of the two possible ways of action of the sources of perturbation only the symmetric one is pursued here, for which the sheet reacts in the so called varicose mode. The action of the sources is taken either as instantaneous, or as persisting infinitely long. During the persistent action the sources are considered to move with constant velocity along the sheet and the perturbation is changing harmonically with circular frequency ω e . In the paper, the linear response of the viscous sheet to the external perturbation is computed and analyzed. The starting point of the computation is a general formula for the exact linear response of the sheet as an inverse Fourier-Laplace transform, the derivation of which is reproduced in Appendix A of this paper. The formula shows the response to depend on several characteristic numbers, of which one is intrinsic to the unperturbed sheet, whereas the others also imply the properties of the source. The intrinsic number can be taken as in which the sheet thickness h, the density ρ of the liquid, its kinematic viscosity ν, and it surface tension γ with respect to the two ambient media enter. It is related by to the Ohnesorge number Oh. First the concept of the long wave approximation used is discussed in the context: it implies the neglect of all the hard branches ω n (k),n=1,..., of spectral modes of the sheet, for which On the remaining two soft branches ω±(k) of the spectrum, for which lim k →0ω±(k)=0, the asymptotic expansion of ω(k) fork→0 is considered and the whole branches are approximated by the (extrapolation of the) lowest order of the expansion. Finally, the whole Fourier integrand of the signal is substituted by its lowest order asymptotic expression aroundk=0. Within this approximation the response of the sheet to an instantaneous point perturbation is computed in its rest frame of reference. It is a long time asymptotic form of the exact response to the instantaneous point perturbation, the kernel of the general signal evolution of the sheet. The computed asymptotic form of this kernel bears a strong formal resemblance to the evolution kernel of the heat equation. The analogy is, however, not physical: whereas for Γ 2 4 their modes are dispersive and dissipative waves, with their peculiar propagation properties. By adequate application of the superposition principle the responses to instantaneous line perturbation and to harmonically persistent perturbations by moving point and line sources are computed from the asymptotic evolution kernel. Because of the specific form of this kernel, the long time surface deflection responses to the harmonic excitations do not depend separately on the additional characteristic numbers ω e andW=| |/Γ (where ω e and | | are considered in units of 4ν/h2 and 2ν/h, respectively), but only on their ratio ω e /W2. The properties of these signals in dependence on Γ and ω e /W2, as well as their implications for technical processes involving liquid sheets, like courtain coating, are discussed in detail.

A new method for a creation of the information system for sequential identification of states of technological processes or other dynamic systems for their supervision and control is considered. The states of dynamic system can be unknown and can change themselves abruptly or slowly. The method is based on a sequential nonlinear mapping of many-dimensional vectors of pa- rameters (collection of which describes the present state of dynamic systems) into two-dimensional vectors in order to reflect the states and their changes on the PC screen and to observe the situa- tion by means of computer. The mapping error function is chosen and expressions for sequential nonlinear mapping are obtained. The mapping preserves the inner structure of distances among the vectors. Examples are given.

An information field theory approach to Bayesian state and parameter estimation in dynamical systems

Basin of attraction of a stable equilibrium point is an effective concept for stability analysis in deterministic systems; however, it does not contain information on the external perturbations that may affect it. Here we introduce the concept of stochastic basin of attraction (SBA) by incorporating a suitable probabilistic notion of basin. We define criteria for the size of the SBA based on the escape probability, which is one of the deterministic quantities that carry dynamical information and can be used to quantify dynamical behavior of the corresponding stochastic basin of attraction. SBA is an efficient tool to describe the metastable phenomena complementing the known exit time, escape probability, or relaxation time. Moreover, the geometric structure of SBA gives additional insight into the system's dynamical behavior, which is important for theoretical and practical reasons. This concept can be used not only in models with small noise intensity but also with noise whose amplitude is proportional or in general is a function of an order parameter. As an application of our main results, we analyze a three potential well system perturbed by two types of noise: Brownian motion and non-Gaussian α-stable Lévy motion. Our main conclusions are that the thermal fluctuations stabilize the metastable system with an asymmetric three-well potential but have the opposite effect for a symmetric one. For Lévy noise with larger jumps and lower jump frequencies ( α=0.5) metastability is enhanced for both symmetric and asymmetric potentials.

We exhibit a fundamental relationship between measures of dynamical and structural stability of linear dynamical systems-e.g. linearized models in the vicinity of equilibria. We show that dynamical stability, quantified via the response to external perturbations (i.e. perturbation of dynamical variables), coincides with the minimal internal perturbation (i.e. perturbations of interactions between variables) able to render the system unstable. First, by reformulating a result of control theory, we explain that harmonic external perturbations reflect the spectral sensitivity of the Jacobian matrix at the equilibrium, with respect to constant changes of its coefficients. However, for this equivalence to hold, imaginary changes of the Jacobian's coefficients have to be allowed. The connection with dynamical stability is thus lost for real dynamical systems. We show that this issue can be avoided, thus recovering the fundamental link between dynamical and structural stability, by considering stochastic noise as external and internal perturbations. More precisely, we demonstrate that a linear system's response to white-noise perturbations directly reflects the intensity of internal white-noise disturbance that it can accommodate before becoming stochastically unstable.

The study of the response of nonlinear systems to external periodic perturbations leads to interesting information1-7. Cool-flame8,9 oscillations occur in a number of combustion reactions, and we discuss an experimental study of the effect of external periodic perturbations on such systems. The application of perturbations to a chemical reaction can reveal important information about the stability, kinetics, and dynamics of the reaction. This technique is well known in the field of relaxation kinetics, in which perturbations are applied to a chemical system at equilibrium. In our work, periodic perturbations are first applied to the input rates of acetaldehyde and oxygen, one at a time, in the combustion of acetaldehyde in a CSTR. We measure periodic responses in five entrainment bands as we vary the frequency and amplitude of the external periodic perturbation. Outside of entrainment bands we find quasi-periodic responses. Nextphase maps10,11 of the experimental results are constructed in real time and used in the observation and interpretation of entrainment and quasi-periodic behavior. Within the fundamental entrainment band, we measure critical slowing down and enhancement of the response amplitude.

When confronting an abrupt external perturbation force during movement, subjects continuously adjust their behaviors to adapt to changes. Such adaptation is of great importance for realizing flexible motor control in varied environments, but the potential cortical neuronal mechanisms behind it have not yet been elucidated. Aiming to reveal potential neural control system compensation for external disturbances, we applied an external orientation perturbation while monkeys performed an orientation reaching task and simultaneously recorded the neural activity in the primary motor cortex (M1). We found that a subpopulation of neurons in the primary motor cortex specially created a time-locked activity in response to a “go” signal in the adaptation phase of the impending orientation perturbation and did not react to a “go” signal under the normal task condition without perturbation. Such neuronal activity was amplified as the alteration was processed and retained in the extinction phase; then, the activity gradually faded out. The increases in activity during the adaptation to the orientation perturbation may prepare the system for the impending response. Our work provides important evidence for understanding how the motor cortex responds to external perturbations and should advance research about the neurophysiological mechanisms underlying motor learning and adaptation.

The classical fluctuation-dissipation theorem predicts the average response of a dynamical system to an external deterministic perturbation via time-lagged statistical correlation functions of the corresponding unperturbed system. In this work we develop a fluctuation-response theory and test a computational framework for the leading order response of statistical averages of a deterministic or stochastic dynamical system to an external stochastic perturbation. In the case of a stochastic unperturbed dynamical system, we compute the leading order fluctuation-response formulas for two different cases: when the existing stochastic term is perturbed, and when a new, statistically independent, stochastic perturbation is introduced. We numerically investigate the effectiveness of the new response formulas for an appropriately rescaled Lorenz 96 system, in both the deterministic and stochastic unperturbed dynamical regimes.

Abstract. A dynamical version of the Bourgain-Fremlin-Talagrand dichotomy shows that the enveloping semigroup of a dynamical system is either very large and contains a topological copy of βN, or it is a “tame ” topological space whose topology is determined by the convergence of sequences. In the latter case we say that the dynamical system is tame. We show that (i) a metric distal minimal system is tame iff it is equicontinuous (ii) for an abelian acting group a tame metric mini-mal system is PI (hence a weakly mixing minimal system is never tame), and (iii) a tame minimal cascade has zero topological entropy. We also show that for minimal distal-but-not-equicontinuous systems the canonical map from the enveloping op-erator semigroup onto the Ellis semigroup is never an isomorphism. This answers a long standing open question. We give a complete characterization of minimal systems whose enveloping semigroup is metrizable. In particular it follows that for abelian acting group such a system is equicontinuous.

Bistability and stochastic jumps in an airfoil system with viscoelastic material property and random fluctuations

Modern power systems are incorporated with distributed energy sources to be environmental-friendly and cost-effective. However, due to the uncertainties of the system integrated with renewable energy sources, effective strategies need to be adopted to stabilize the entire power systems. Hence, the system operators need accurate prediction tools to forecast the dynamic system states effectively. In this paper, we propose a Bayesian deep learning approach to predict the dynamic system state in a general power system. First, the input system dataset with multiple system features requires the data pre-processing stage. Second, we obtain the dynamic state matrix of a general power system through the Newton-Raphson power flow model. Third, by incorporating the state matrix with the system features, we propose a Bayesian long short-term memory (BLSTM) network to predict the dynamic system state variables accurately. Simulation results show that the accurate prediction can be achieved at different scales of power systems through the proposed Bayesian deep learning approach.

The statistics of systems with good chaotic properties obey a formal fluctuation-response relation which gives the average linear response of a dynamical system to an external perturbation in terms of two-time correlation functions. Unfortunately, except for particularly simple cases, the appropriate form of correlation function is unknown because an analytic expression for the invariant density is lacking. The simplest situation is that in which the probability distribution is Gaussian. In that case, the fluctuation-response relation is a linear relation between the response matrix and the two-time two-point correlation matrix. Some numerical computations have been carried out in low-dimensional models of hydrodynamic systems. The results show that fluctuation-response relation for Gaussian distributions is not a useful approximation. Nevertheless, these calculations show that, even for non-Gaussian statistics, the response function and the two-time correlations can have similar qualitative features, which may be attributed to the existence of the more general fluctuation-response relation.

Responsive systems have recently gained much interest in the scientific community in attempts to mimic dynamic functions in biological systems. One of the fascinating potential applications of responsive systems lies in catalysis. Inspired by nature, novel responsive catalytic systems have been built that show analogy with allosteric regulation of enzymes. The design of responsive catalytic systems allows control of catalytic activity and selectivity. In this Review, advances in the field over the last four decades are discussed and a comparison is made amongst the dynamic responsive systems based on the principles underlying their catalytic mechanisms. The catalyst systems are sorted according to the triggers used to achieve control of the catalytic activity and the distinct catalytic reactions illustrated.

This paper deals with a general methodology to assess the influence of the clearance size and the friction coefficient on the dynamic response of planar rigid multi-body systems including revolute joints with clearance. When there is a clearance in a revolute joint, impacts between the journal and the bearing can occur, and consequently, local deformations take place. The impact is internal and the response of the system is performed using a continuous contact force model. The friction effect because of the contact between joint elements is also included. The dynamic response of the systems is obtained numerically by solving the constraint equations and the contact-impact forces produced in the clearance joint, simultaneously with the differential equations of motion and a set of initial conditions. Numerical results for two simple mechanisms with revolute clearance joints are presented and discussed. In the present work, the clearance size and friction effects are analysed separately. Through the use of Poincaré maps, both periodic and chaotic responses of the systems are observed. The results predict the existence of the periodic or regular motion at certain clearance sizes and friction coefficients and chaotic or non-linear in other cases.