Analytical solutions for period-m flows and chaos in nonlinear dynamical systems are presented through the generalized harmonic balance method. The mechanism for a period-m motion jumping to another period-n motion in numerical computation is found. The Hopf bifurcation of periodic motions is equivalent to the period-doubling bifurcation via Poincare mappings of dynamical systems. The stable and unstable period-m motions can be obtained analytically. In addition, the stable and unstable chaotic motions can be achieved analytically. The methodology presented in this paper is independent of small parameters. The nonlinear damping, periodically forced, Duffing oscillator was investigated as an example to demonstrate the analytical solutions of periodic motions and chaos.

Complex networks such as the neuronal ones composed of neurons coupled by chemical synapses are known to exhibit a large variety of forms of activity. Certain neurophysiological experiments have shown that neuronal processes are accompanied by short transient activity of individual elements or small groups of elements. Such a behavior is called the sequential dynamics. In the framework of dynamical systems theory this behavior is related to the existence of a collection of metastable invariant sets joined by transient (often heteroclinic) trajectories in the phase space. The sequential dynamics can be treated as a process of successive switchings among these sets. Such a treatment allows one to explain temporal order in which the elements become activated and to single out parameters of the system responsible for its prediction. In the talk it is supposed to speak about both the situations where metastable sets are just equilibrium points or limit cycles and where they are more complex sets.

It is well known that a sound wave exerts an average “radiation” force (RF) on a particle and that two particles can attract each other (Bjerknes force). The RF is, in general, defined as a period-averaged force exerted on the medium by sound wave. There exists a wide range of literature on acoustic radiation force acting on small particles and bubbles, as well as on their motion in an acoustic field. The early history of such studies is associated with such names as Lord Rayleigh, Bjerknes, Brillouin, Langevin, and other first-class physicists. RF is commonly presented as a unidirectional force acting on the absorbing or reflecting targets in the wave path. A classical example of such an action is sound pressure on a wall or other interface. As regards the distributed RF, one of the most thorough studies has been carried out by Eckart (1947) who consistently derived equations for the average force and the resulting motion in a viscous fluid known as acoustic streaming (“acoustic wind”). The expressions for RF acting on a small particle were obtained by Yosioka and Kawasima (1955) and Gor’kov (1962). By present the acoustic radiation force has found numerous biomedical applications, including: Motion of small particles and bubbles for stirring and mixing small volumes of biochemical fluids; Collecting microparticles, including bacteria, in standing ultrasonic waves, for separation and diagnostics; Generation of shear waves in focused ultrasonic beams allowing diagnostics of lesions and tumors.In this presentation we outline nonlinear theoretical models related to the acoustic RF, as well as some biomedical applications.

Dynamics of the ensemble of two model neurons interacting through electrical synapse is investigated. Both neurons are described by a two-dimensional map. It is shown that in four-dimensional phase space a chaotic attractor of relaxation type exists corresponding to spike-bursting chaotic oscillations. A new characteristic estimating the degree of spike-bursting synchronization is introduced. Dependence of the synchronism degree on the coupling strength is shown for some coupling interval where only incomplete activity synchronization occurs. A probabilistic study provides a dynamical explanation of memory effects of these phenomena using Perron-Frobenius operator formalism.

When we attempt to control a linear system in which some noise has been added, typically we need a control higher or equal to the amount of noise added. When we have a region in phase space where there is a chaotic saddle, all initial conditions will escape from it after a transient with the exception of a set of points of zero Lebesgue measure. The action of an external noise makes all trajectories escape even faster. Attempting to avoid those escapes by applying a control smaller than noise seems to be an impossible task. Here we show, however, that this goal is indeed possible, based on a geometrical property found typically in this situation: the existence of a horseshoe. The horseshoe implies that there exists what we call safe sets, which assures that there is a general strategy that allows one to keep trajectories inside that region with a control smaller than noise. We call this type of control partial control of chaos that allows one to keep the trajectories of a dynamical system close to the saddle even in presence of a noise stronger than the applied control. In this talk recent further progress and new results on this control strategy including the sculpting algorithm to compute safe sets are presented. This is joint work with James A Yorke (USA), Samuel Zambrano and Juan Sabuco (Spain).

From computation point of view, in nonlinear dynamical systems attractors can be regarded as self-exciting and hidden attractors. Self-exciting attractors can be localized numerically by standard computational procedure, in which after transient process a trajectory, started from a point of unstable manifold in a small neighborhood of unstable equilibrium, reaches an attractor and computes it. Hidden attractor attractor, a basin of attraction of which does not contain neighborhoods of equilibria. While classical attractors in well-known dynamical systems of Van der Pol, Beluosov-Zhabotinsky, Lorenz, Chua, and many others are self-exiting attractors and can be obtained numerically by standard computational procedure, for localization of hidden attractors it is necessary to develop special analytical-numerical procedures, in which on the first step initial data is chosen in the basin of attraction and then numerical localization (visualization) of attractor is performed. Simplest examples of hidden attractors are internal nested limit cycles (hidden oscillations) in two-dimensional systems (see, e.g., results on the second part of 16th Hilbert's problem). Other examples of hidden oscillations are counterexamples to Aizerman's and Kalman's conjectures on absolute stability in automatic control theory (where unique stable equilibrium point coexists with stable periodic solution). In 2010, for the first time, a chaotic hidden attractor was computed by the authors in generalized Chua's circuit and then one was discovered in classical Chua's circuit. This survey devoted to analytical-numerical methods for hidden attractors localization and its application to well-known problems and systems.

Traditionally asymptotic in time properties and time averages over an infinite time interval was studied. It occurred that there are various interesting and important properties of chaotic dynamical systems which manifest themselves in a finite time.

In this talk we propose new dynamical discrete model of the olivo-cerebellar system. The model consists of three layers of interacting elements: inferior olive neurons, Purkinje cells and deep cerebellar nuclei neurons, and also of axonal and synaptic connections between neurons. Each element of the structure is described by a two-dimensional nonlinear map with a set of parameters for each type of neurons. Dynamic properties of different types of neurons have been described, spontaneous and stimulus-induced dynamics of the system has been investigated. Unlike previously proposed models, our model takes into account such processes as axonal interaction of neurons of different layers, as well as the interaction of the inferior olive neurons through the electrical synapses with the property of plasticity. It has been shown that the inclusion of these factors plays a significant role in the formation of spatio-temporal activity, spontaneous and stimulus-induced, in the layer of inferior olive neurons.

Most of the networks and databases humans have deal with contain large albeit finite number of units. Their structure maintaining functional consistency of the components is essentially not random and calls for a precise quantitative description of relations between nodes or data units and all network parts, as having important implications for the network robustness. A network can be seen as a discrete time dynamical system possessing a finite number of states. The behavior of such a dynamical system can be studied by means of a transfer operator which describes the time evolution of distributions in phase space. The transfer operator can be represented by a stochastic matrix determining a discrete time random walk on the graph in which a walker picks at each node between the various available edges with equal probability. The Laplace operator associated to random walks possesses a group generalized inverse that can be used in order to define a probabilistic Riemannian manifold with a random metric on any finite connected undirected graph, or a database. In contrast to classical graph theory paying attention to the shortest paths of least cost, in the developed probabilistic approach all possible paths between a pair of vertices in a connected graph or a pair of units in a database are taken into account, although some paths shall be more probable than others. In such a formulation of graph theory, the distance is nothing else as a "path integral". The probabilistic geometrization of data enables us to attack applied problems which could not even be started otherwise. In particular, we report on the applications of the probabilistic approach for the analysis of urban structures, evolution of languages and musical compositions.

Dynamics of many-body long-range interacting systems are investigated, using the XY-Hamiltonian mean-field model as a case study. We show that regular trajectories, associated with invariant tori of the single-particle dynamics emerge as the number of particles is increased. Moreover, the construction of stationary solutions and studies of the maximal Lyapunov exponent of the systems show the same trend towards integrability. This feature provides a dynamical interpretation of the emergence of long-lasting out-of-equilibrium regimes observed generically in long-range systems. Extension beyond the mean-field system is considered and displays similar features.

Genetic regulatory networks are usually modeled by systems of piecewise affine differential equations. Finite state models, better known as logical networks, are also used. In this talk we present a class of models of regulatory networks which may be situated in the middle of the spectrum; they present both discrete and continuous aspects. They consist of a network of units whose states are quantified by a continuous real variable. The state of each unit in the network evolves according to a contractive transformation chosen from a finite collection of possible transformations. The particular transformation chosen at each time step depends on the state of the neighboring units. In this way we obtain a network of coupled contractions. In this talk we will briefly describe some of the phenomenology deployed by this kind of networks, we will present some results on the relationship between the topological structure of the underlying network and the dynamical behavior of the corresponding regulatory model. We will end by formulating some conjectures concerning the general behavior of this kind of models which we will contrast with some unusual phenomenology encountered in the more general class of piecewise contracting transformations.

Intrinsic neuronal and circuit properties control the responses of large ensembles of neurons by creating spatio-temporal patterns of activity, which are used for sensory processing, memory formation and other cognitive tasks. The modeling of such systems requires single neuron models capable to display both realistic response properties and computational efficiency. This paper discusses a set of reduced models based on difference equations (map-based models) to simulate intrinsic dynamics of biological neurons. Such phenomenological models are designed to capture the main response properties of specific types of neurons to ensure the realistic model behavior within sufficient dynamic range of the inputs. This approach allows fast simulations and efficient parameter space analysis of networks containing hundreds of thousands of neurons of different types using a conventional workstation. Drawing on results obtained with large-scale networks of map-based neurons, we will discuss spatio-temporal cortical network dynamics as a function of parameters of synaptic interconnections and intrinsic states of the neurons. The paper focuses on the modeling and analysis of the common synchronization regimes and the role of oscillations and synchrony in formation of complex oscillatory patterns of neuronal activity. The paper also considers an application of map-based neuron models to replicate the behavior a Central Pattern Generator of undulatory locomotion. The system has been implemented with DSP that provides real-time simulation of locomotion activity and used controls swimming patterns of a biomimetic lamprey-based robot.

Fractional Calculus (FC) started in 1695 when L'Hôpital wrote a letter to Leibniz asking for the meaning of Dny for n = 1/2. Starting with the ideas of Leibniz many important mathematicians developed the theoretical concepts. During the thirties A. Gemant and O. Heaviside applied FC in the areas of mechanical and electrical engineering, respectively. Nevertheless, these important contributions were somehow forgotten and only during the eighties FC emerged associated with phenomena such as fractal and chaos and, consequently, in the modeling of dynamical systems. This lecture introduces the FC fundamental mathematical concepts, and reviews the main approaches for implementing fractional operators. Based on the FC concepts, are presented several applications in the areas of modeling and control, namely fractional PID, electromagnetism, fractional electrical impedances, evolutionary algorithms, nonlinear system control, and finance.

Polyrhytmicity belongs to important attributes of large assemblies of oscillators; for example, recorded extracellular oscillations of human neurons demonstrate alternating epochs of fast and slow oscillations. To take account of this phenomenon, most of the existing models include interaction of different units whose intrinsic timescales strongly differ. Here, we discuss a possible mechanism which ensures birhytmicity in simple models of ensembles in which all elements share the intrinsic timescale. Although the considered dynamical systems are non-generic, they involve typical properties of many existing models in technics and neuroscience. We consider networks built of oscillatory units with the same eigenfrequency; coupling terms in the governing equations are proportional to velocities of the elements. No restrictions are put either on the symmetry of the coupling or on its pattern (mean field, next neighbors, pairwise or triple interaction etc.).

In the parameter space of the ensemble, destabilization of the equilibrium occurs by means of the Hopf bifurcation. On the large part of the stability boundary, the spectrum of the linearized flow contains not one (as usually) but two pairs of purely imaginary eigenvalues. Within this context the so-called ``double Hopf'' bifurcation becomes a codimension-one phenomenon. Of the two resulting frequencies, one is typically much lower than the individual frequency of an element, whereas the other one is distinctly higher. Accordingly, in the nonlinear regime the ensembles are potentially capable of performing both slow and fast modes of oscillations. We illustrate this general phenomenon by numerical data obtained from ensembles of oscillators with different coupling patterns and demonstrate that after the transition the system can possess two stable limit cycles (respectively, one ``fast'' and one ``slow'') or a quasiperiodic state.

The suspension of a rotating shaft assembly in a magnetic field thereby avoiding mechanical contact and lubrication is an idea that is very attractive and has been the subject of some research in recent years. Apart from the obvious advantage of reducing wear, the magnetic bearings are unique in that the bearing characteristics can be changed in an active control loop just by changing the controller parameters. This makes it possible - at least theoretically - not only to reduce rotor vibration in general, but also to control the rotor in case of instabilities and other such potentially catastrophic situations.

The magnetic bearing forces arise from the electromagnetic field and are strongly nonlinear functions of the rotor displacement and the control current. This talk will focus on several recent results of our research on accurate modeling and nonlinear dynamic analysis of rotors supported on magnetic bearings. In particular, the talk will focus on nonlinear models, bifurcation analysis, effect of support excitation and evidence of horseshoe chaos due to time-varying stiffness. Also, the relevance of these results to engineering design will be presented.

This work presents some techniques for order reduction of parametrically excited nonlinear systems subjected to external and control inputs. This important class of problems arises in the analysis and control of structures with rotating components and periodic in-plane loads or systems described by nonlinear differential equations representing dynamics about a periodic motion. The techniques presented are based on construction of time-varying invariant manifolds. The first problem deals with systems with external periodic excitation where an n dimensional time-periodic nonlinear system is approximated by a system of differential equations of a smaller dimension m << n. Order reduction for all three cases (viz., fundamental, sub and super harmonic) are considered.

Next, an order reduction of a parametrically excited nonlinear closed-loop system with a scalar state feedback is considered. In both cases we apply Lyapunov-Floquét (LF) transformation and separate the dominant and the non-dominant (slave) states. Then the dominant dynamics represented by the reduced order model, can be decoupled from the non-dominant dynamics by constructing an invariant manifold relating the non-dominant states as nonlinear periodic functions of the dominant states. The control problem involves design of a linear as well as a nonlinear controller where the linear controller is designed using a symbolic approach that can place the Floquét multipliers in the desired locations. Examples are included to demonstrate the effectiveness of the method.