Dimensionality hyper-reduction and machine learning for dynamical systems with varying parameters

Thesis (Ph.D.)--University of Washington, 2016-08

Machine learning and data decompositions for complex networked dynamical systems

Thesis (Ph.D.)--University of Washington, 2016-08

José Rodrigues Santos de Sousa Ramos (1948-2007): The diversity of dynamical systems

José Rodrigues Santos de Sousa Ramos, mathematician of great merit, passed away on January 1st, 2007, in Lisbon. He was buried in Sobral da Adiça. His death was a huge loss for the development of mathematics in Portugal. The course of time will increase the dimension of this loss. Therefore, we dedicated this theme issue on Dynamical Systems to recall his memory and underline his work. We never forget you.

Window of Chaotic Delayed Synchronization

We consider a general coupling of two chaotic dynamical systems and we obtain conditions that provide delayed synchronization. We consider four different couplings that satisfy those conditions. We define Window of Delayed Synchronization and we obtain it analytically. We use four different free chaotic dynamics in order to observe numerically the analytically predicted windows for the considered couplings.

Branching processes and computational collapse of discretized unimodal mappings

In computer simulations of smooth dynamical systems, the original phase space is replaced by machine arithmetic, which is a finite set. The resulting spatially discretized dynamical systems do not inherit all functional properties of the original systems, such as surjectivity and existence of absolutely continuous invariant measures. This can lead to computational collapse to fixed points or short cycles. The paper studies loss of such properties in spatial discretizations of dynamical systems induced by unimodal mappings of the unit interval. The problem reduces to studying set-valued negative semitrajectories of the discretized system. As the grid is refined, the asymptotic behavior of the cardinality structure of the semitrajectories follows probabilistic laws corresponding to a branching process. The transition probabilities of this process are explicitly calculated. These results are illustrated by the example of the discretized logistic mapping.

Complete synchronization and delayed synchronization in couplings

We consider a general coupling of two identical chaotic dynamical systems, and we obtain the conditions for synchronization. We consider two types of synchronization: complete synchronization and delayed synchronization. Then, we consider four different couplings having different behaviors regarding their ability to synchronize either completely or with delay: Symmetric Linear Coupled System, Commanded Linear Coupled System, Commanded Coupled System with delay and symmetric coupled system with delay. The values of the coupling strength for which a coupling synchronizes define its Window of synchronization. We obtain analytically the Windows of complete synchronization, and we apply it for the considered couplings that admit complete synchronization. We also obtain analytically the Window of chaotic delayed synchronization for the only considered coupling that admits a chaotic delayed synchronization, the commanded coupled system with delay. At last, we use four different free chaotic dynamics (based in tent map, logistic map, three-piecewise linear map and cubic-like map) in order to observe numerically the analytically predicted windows.

Complete synchronization and delayed synchronization in couplings

We consider a general coupling of two identical chaotic dynamical systems, and we obtain the conditions for synchronization. We consider two types of synchronization: complete synchronization and delayed synchronization. Then, we consider four different couplings having different behaviors regarding their ability to synchronize either completely or with delay: Symmetric Linear Coupled System, Commanded Linear Coupled System, Commanded Coupled System with delay and symmetric coupled system with delay. The values of the coupling strength for which a coupling synchronizes define its Window of synchronization. We obtain analytically the Windows of complete synchronization, and we apply it for the considered couplings that admit complete synchronization. We also obtain analytically the Window of chaotic delayed synchronization for the only considered coupling that admits a chaotic delayed synchronization, the commanded coupled system with delay. At last, we use four different free chaotic dynamics (based in tent map, logistic map, three-piecewise linear map and cubic-like map) in order to observe numerically the analytically predicted windows.

Estimating Attractor Dimension on the Nonlinear Pendulum Time Series

Chaotic dynamical systems exhibit trajectories in their phase space that converges to a strange attractor. The strangeness of the chaotic attractor is associated with its dimension in which instance it is described by a noninteger dimension. This contribution presents an overview of the main definitions of dimension discussing their evaluation from time series employing the correlation and the generalized dimension. The investigation is applied to the nonlinear pendulum where signals are generated by numerical integration of the mathematical model, selecting a single variable of the system as a time series. In order to simulate experimental data sets, a random noise is introduced in the time series. State space reconstruction and the determination of attractor dimensions are carried out regarding periodic and chaotic signals. Results obtained from time series analyses are compared with a reference value obtained from the analysis of mathematical model, estimating noise sensitivity. This procedure allows one to identify the best techniques to be applied in the analysis of experimental data.

Computer-Assisted Proofs and Symbolic Computations

We discuss some main points of computer-assisted proofs based on reliable numerical computations. Such so-called self-validating numerical methods in combination with exact symbolic manipulations result in very powerful mathematical software tools. These tools allow proving mathematical statements (existence of a fixed point, of a solution of an ODE, of a zero of a continuous function, of a global minimum within a given range, etc.) using a digital computer. To validate the assertions of the underlying theorems fast finite precision arithmetic is used. The results are absolutely rigorous. To demonstrate the power of reliable symbolic-numeric computations we investigate in some details the verification of very long periodic orbits of chaotic dynamical systems. The verification is done directly in Maple, e.g. using the Maple Power Tool intpakX or, more efficiently, using the C++ class library C-XSC.

Local dimension and finite time prediction in spatiotemporal chaotic systems

Predictability is related to the uncertainty in the outcome of future events during the evolution of the state of a system. The cluster weighted modeling (CWM) is interpreted as a tool to detect such an uncertainty and used it in spatially distributed systems. As such, the simple prediction algorithm in conjunction with the CWM forms a powerful set of methods to relate predictability and dimension.

On the Chaotic Suppression of Both Ideal and Non-ideal Duffing Based Vibrating Systems, Using a Magnetorheological Damper

In this paper the dynamics of the ideal and non-ideal Duffing oscillator with chaotic behavior is considered. In order to suppress the chaotic behavior and to control the system, a control signal is introduced in the system dynamics. The control strategy involves the application of two control signals, a nonlinear feedforward control to maintain the controlled system in a periodic orbit, obtained by the harmonic balance method, and a state feedback control, obtained by the state dependent Riccati equation, to bring the system trajectory into the desired periodic orbit. Additionally, the control strategy includes an active magnetorheological damper to actuate on the system. The control force of the damper is a function of the electric current applied in the coil of the damper, that is based on the force given by the controller and on the velocity of the damper piston displacement. Numerical simulations demonstrate the effectiveness of the control strategy in leading the system from any initial condition to a desired orbit, and considering the mathematical model of the damper (MR), it was possible to control the force of the shock absorber (MR), by controlling the applied electric current in the coils of the damper. © 2012 Foundation for Scientific Research and Technological Innovation.

Simultaneous Control of Chaotic Systems Using RBF Networks

Chaos control is a concept that recently acquiring more attention among the research community, concerning the fields of engineering, physics, chemistry, biology and mathematic. This paper presents a method to simultaneous control of deterministic chaos in several nonlinear dynamical systems. A radial basis function networks (RBFNs) has been used to control chaotic trajectories in the equilibrium points. Such neural network improves results, avoiding those problems that appear in other control methods, being also efficient dealing with a relatively small random dynamical noise.

Symbolic dynamics generated by a hybrid chaotic systems

We consider piecewise defined differential dynamical systems which can be analysed through symbolic dynamics and transition matrices. We have a continuous regime, where the time flow is characterized by an ordinary differential equation (ODE) which has explicit solutions, and the singular regime, where the time flow is characterized by an appropriate transformation. The symbolic codification is given through the association of a symbol for each distinct regular system and singular system. The transition matrices are then determined as linear approximations to the symbolic dynamics. We analyse the dependence on initial conditions, parameter variation and the occurrence of global strange attractors.

SYNCHRONIZATION OF A CLASS OF SECOND-ORDER NONLINEAR SYSTEMS

The asymptotic behavior of a class of coupled second-order nonlinear dynamical systems is studied in this paper. Using very mild assumptions on the vector-field, conditions on the coupling parameters that guarantee synchronization are provided. The proposed result does not require solutions to be ultimately bounded in order to prove synchronization, therefore it can be used to study coupled systems that do not globally synchronize, including synchronization of unbounded solutions. In this case, estimates of the synchronization region are obtained. Synchronization of two-coupled nonlinear pendulums and two-coupled Duffing systems are studied to illustrate the application of the proposed theory.

Measuring and Controlling the Chaotic Motion of Profits

The study of economic systems has generated deep interest in exploring the complexity of chaotic motions in economy. Due to important developments in nonlinear dynamics, the last two decades have witnessed strong revival of interest in nonlinear endogenous business chaotic models. The inability to predict the behavior of dynamical systems in the presence of chaos suggests the application of chaos control methods, when we are more interested in obtaining regular behavior. In the present article, we study a specific economic model from the literature. More precisely, a system of three ordinary differential equations gather the variables of profits, reinvestments and financial flow of borrowings in the structure of a firm. Firstly, using results of symbolic dynamics, we characterize the topological entropy and the parameter space ordering of kneading sequences, associated with one-dimensional maps that reproduce significant aspects of the model dynamics. The analysis of the variation of this numerical invariant, in some realistic system parameter region, allows us to quantify and to distinguish different chaotic regimes. Finally, we show that complicated behavior arising from the chaotic firm model can be controlled without changing its original properties and the dynamics can be turned into the desired attracting time periodic motion (a stable steady state or into a regular cycle). The orbit stabilization is illustrated by the application of a feedback control technique initially developed by Romeiras et al. [1992]. This work provides another illustration of how our understanding of economic models can be enhanced by the theoretical and numerical investigation of nonlinear dynamical systems modeled by ordinary differential equations.