Autonomous Dynamical Systems Research Articles

English

Fuzzy cell mapping is a novel computational technique that combines fuzzy logic and a simple cell mapping method. In a simple cell mapping method, the information about mapping locations of image cells is never incorporated into the method. This limits the usage of a simple cell mapping method. In our fuzzy cell mapping method, we account for the mapping locations of image cells and incorporate the information by employing triangular membership functions. This paper demonstrates the use of fuzzy cell mapping on nonlinear dynamical systems. Our results indicate that fuzzy cell mapping can give good estimates on the global behavior of autonomous dynamical systems that posses limit cycles and strange attractors.   Key words: Fuzzy cell mapping, nonlinear dynamical systems, limit cycle, strange attractor.

On the rational stability of autonomous dynamical systems. Applications to control chained systems

In this paper, several sufficient conditions for rational stability are provided. We applied our conditions to show that all chained systems are rationally stabilized-by Hölderian feedback laws – in partial sense. In addition, we show that the backstepping techniques can be extended to rational stabilizability theory.

Symplectic feedback using Hamiltonian Lie algebra and its applications to an inverted pendulum

From the symplectic representation of an autonomous nonlinear dynamical system with holonomic constraints, i.e., those that can be represented through a symplectic form derived from a Hamiltonian, we present a new proof on the realization of the symplectic feedback action, which has several theoretical advantages in demonstrating the uniqueness and existence of this type of solution. Also, we propose a technique based on the interpretation, construction and characterization of the pull-back differential on the symplectic manifold as a member of a one-parameter Lie group. This allows one to synthesize the control law that governs a certain system to achieve a desired behavior; and the method developed from this is applied to a classical system such as the inverted pendulum.

Fractal entropy of nonautonomous systems

We define formulas of entropy dimension for a nonautonomous dynamical system consisting of a sequence of continuous self-maps of a compact metric space. This study reveals analogues of basic propositions for entropy dimension, such as the power rule, product rule and commutativity, etc. These properties allow us to convert to an equality an inequality found by de Carvalho (1997) concerning the product rule for the autonomous dynamical system. We also prove a subadditivity rule of entropy dimension for one-dimensional dynamics based on our previous work.

Non-autonomous Morse-decomposition and Lyapunov functions for gradient-like processes

We define (time dependent) Morse-decompositions for non-autonomous evolution processes (non-autonomous dynamical systems) and prove that a non-autonomous gradient-like evolution process possesses a Morsedecomposition on the associated pullback attractor. We also prove the existence of an associated Lyapunov function which describes the gradient behavior of the system. Finally, we apply these abstract results to non-autonomous perturbations of autonomous gradient-like evolution processes (semigroups or autonomous dynamical systems).

Dynamical Movement Primitives: Learning Attractor Models for Motor Behaviors

Nonlinear dynamical systems have been used in many disciplines to model complex behaviors, including biological motor control, robotics, perception, economics, traffic prediction, and neuroscience. While often the unexpected emergent behavior of nonlinear systems is the focus of investigations, it is of equal importance to create goal-directed behavior (e.g., stable locomotion from a system of coupled oscillators under perceptual guidance). Modeling goal-directed behavior with nonlinear systems is, however, rather difficult due to the parameter sensitivity of these systems, their complex phase transitions in response to subtle parameter changes, and the difficulty of analyzing and predicting their long-term behavior; intuition and time-consuming parameter tuning play a major role. This letter presents and reviews dynamical movement primitives, a line of research for modeling attractor behaviors of autonomous nonlinear dynamical systems with the help of statistical learning techniques. The essence of our approach is to start with a simple dynamical system, such as a set of linear differential equations, and transform those into a weakly nonlinear system with prescribed attractor dynamics by means of a learnable autonomous forcing term. Both point attractors and limit cycle attractors of almost arbitrary complexity can be generated. We explain the design principle of our approach and evaluate its properties in several example applications in motor control and robotics.

Long-time dynamics for a class of Kirchhoff models with memory

This paper is concerned with a class of Kirchhoff models with memory effects \documentclass[12pt]{minimal}\begin{document}$u_{tt}\break + \alpha \Delta ^2 u - \mbox{div}(\vert \nabla u \vert ^{p-2}\nabla u) - \int _{0}^{\infty } \mu (s) \Delta ^{2}u(t-s)ds - \Delta u_t + f(u) = h,$\end{document}utt+αΔ2u−div(|∇u|p−2∇u)−∫0∞μ(s)Δ2u(t−s)ds−Δut+f(u)=h, defined in a bounded domain of \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^N$\end{document}RN. This non-autonomous equation corresponds to a viscoelastic version of Kirchhoff models arising in dynamics of elastoplastic flows and plate vibrations. Under assumptions that the exponent p and the growth of f(u) are up to the critical range, it turns out that the model corresponds to an autonomous dynamical system in a larger phase space, by adding a component which describes the relative displacement history. Then the existence of a global attractor is granted. Furthermore, in the subcritical case, this global attractor has finite Hausdorff and fractal dimensions.

Realization of Parameters Identification in Only Locally Lipschitzian Dynamical Systems with Multiple Types of Time Delays

Most of the existing results for identifying parameters in dynamical systems through adaptive coupling techniques always require a uniform Lipschitz condition on system vector fields. However, it is not this requirement but the only locally Lipschitz condition that is more suitable for many nonlinear systems in real applications. Also time delays sometimes are essential for modeling real systems, so that in addition to the discrete form, time delays in the distributed form could be considered. This paper thus considers the problem of parameters identification in only locally Lipschitzian dynamical systems with multiple types of time delays through adaptive coupling techniques. In particular, the paper proposes some linear independence conditions for realization of parameters identification in both nonautonomous and autonomous systems and draws a comparison between these conditions and the persistent of excitation condition which is used frequently in the literature. Moreover, the paper proposes an index to determine the nonlinear degree of the considered systems, which enables us to distinguish the systems in which the parameters identification can be globally or only locally achieved. The paper also uses some representative and elaborate examples to illustrate the practical usefulness of the obtained theoretical results. During the discussion, the paper takes into full consideration the property of infinite dimensions for nonautonomous or autonomous dynamical systems with time delays.

Chaotic Properties on Time Varying Map and Its Set Valued Extension

Every autonomous dynamical system （X, f）induces a set-valued dynamical system on the space of compact subsets of X. In this paper we have investigated some chaotic relations between a nonautonomous dynamical system and its set valued extension.

Chaotic Dynamics Characteristic Analysis for Matrix Converter

The unstable oscillation of autonomous dynamic system of a matrix converter (MC) is studied based on nonlinear dynamic theory, and its chaotic characteristic is analyzed. Analysis based on the fundamental-harmonic nonlinear state equations shows that the system loses stability via a Hopf bifurcation. The behavior of the system near the critical power is examined through simulation. The trajectories obtained by the fundamental-harmonic nonlinear state equations show some typical chaotic characteristics such as extreme sensitivity to initial values and self-similarity. Power density spectrum and Lyapunov exponents of the MC are obtained from simulation results. Finally, the trajectories drawn based on experimental data show some behaviors very similar to those from simulation results, which implies a possible chaotic status in the electrical drive system fed by MC.

Bi-space pullback attractors for closed processes

In the description of the long-time behavior of solutions to nonautonomous differential equations the notion of a pullback attractor plays a similar role as the global attractor in autonomous dynamical systems. We present the theorem on the existence of a pullback attractor if the evolution process is a family of closed operators. The abstract result is formulated in the context of the smoothing properties of the process and for pullback attractors attracting a given universe, i.e. a chosen class of possibly time-dependent families of sets. We also present an application of the result to reaction-diffusion equations.

Variable Modified Chaplygin Gas in Anisotropic Universe with Kaluza-Klein Metric

In this work, we have considered Kaluza-Klein Cosmology for anisotropic universe where the universe is filled with Variable Modified Chaplygin Gas (VMCG). Here we find normal scalar field ϕ and the self interacting potential V(ϕ) to describe the VMCG Cosmology. We have also graphically analyzed the geometrical parameters named Statefinder Parameters in anisotropic Kaluza-Klein model. Next, we have considered a Kaluza-Klein model of interacting VMCG with dark matter in the Einstein gravity framework. Here we construct the three dimensional autonomous dynamical system of equations for this interacting model with the assumption that the dark energy and the dark matter interacts between themselves and for that we also choose the interaction term. We convert that interaction term to its dimensionless form and perform stability analysis and solve them numerically. We obtain a stable scaling solution of the equations in Kaluza-Klein model and graphically represent solutions.

Learning to Play Minigolf: A Dynamical System-Based Approach

A current trend in robotics is to define robot motions so that they can be easily adopted to situations beyond those for which the motion was originally designed. In this work, we show how the challenging task of playing minigolf can be efficiently tackled by first learning a basic hitting motion model, and then learning to adapt it to different situations. We model the basic hitting motion with an autonomous dynamical systems, and solve the problem of learning the parameters of the model from a set of demonstrations through a constrained optimization. To hit the ball with the appropriate hitting angle and speed, a nonlinear model of the hitting parameters is estimated based on a set of examples of good hitting parameters. We compare two statistical methods, Gaussian Process Regression and Gaussian Mixture Regression in the context of inferring the hitting parameters for the minigolf task. We demonstrate the generalization ability of the model in various situations. We validate our approach on the 7 Degrees of Freedom (DoF) Barrett WAM arm and 6-DoF Katana arm in both simulated and real environments.

KOLMOGOROV COMPLEXITY AND CHAOTIC PHENOMENON IN COMPUTING THE ENVIRONMENTAL INTERFACE TEMPERATURE

In this paper, we consider the chaotic phenomenon and Kolomogorov complexity in computing the environmental interface temperature. First, the environmental interface is defined in the context of the complex system, in particular for autonomous dynamical systems. Then we consider the following issues in modeling procedure: (i) how to replace given differential equations by appropriate difference equations in modeling of phenomena in the environmental world? (ii) whether a mathematically correct solution to the corresponding differential equation or system of equations is always physically possible and (iii) phenomenon of chaos in autonomous dynamical systems in environmental problems, in particular in solving the energy balance equation to calculate environmental interface temperature. The difference form of this equation for computing the environmental interface temperature is discussed and analyzed depending on parameters of equation, using the Lyapunov exponent and sample entropy. Finally, the Kolmogorov complexity of time series obtained from this difference equation is analyzed.

Stability Boundary Characterization of Nonlinear Autonomous Dynamical Systems in the Presence of Saddle-Node Equilibrium Points

A dynamical characterization of the stability boundary for a fairly largeclass of nonlinear autonomous dynamical systems is developed in this paper. This characterization generalizes the existing results by allowing the existence of saddlenode equilibrium points on the stability boundary. The stability boundary of anasymptotically stable equilibrium point is shown to consist of the stable manifoldsof the hyperbolic equilibrium points on the stability boundary and the stable, stable center and center manifolds of the saddle-node equilibrium points on the stability boundary.

Dynamics of generalized tachyon field

We investigate the dynamics of generalized tachyon field in FRW spacetime. We obtain the autonomous dynamical system for the general case. Because the general autonomous dynamical system cannot be solved analytically, we discuss two cases in detail: β=1 and β=2. We find the critical points and study their stability. At these critical points, we also consider the stability of the generalized tachyon field, which is as important as the stability of critical points. The possible final states of the universe are discussed.

Continuity of Dynamical Structures for Nonautonomous Evolution Equations Under Singular Perturbations

In this paper we study the continuity of invariant sets for nonautonomous infinite-dimensional dynamical systems under singular perturbations. We extend the existing results on lower-semicontinuity of attractors of autonomous and nonautonomous dynamical systems. This is accomplished through a detailed analysis of the structure of the invariant sets and its behavior under perturbation. We prove that a bounded hyperbolic global solutions persists under singular perturbations and that their nonlinear unstable manifold behave continuously. To accomplish this, we need to establish results on roughness of exponential dichotomies under these singular perturbations. Our results imply that, if the limiting pullback attractor of a nonautonomous dynamical system is the closure of a countable union of unstable manifolds of global bounded hyperbolic solutions, then it behaves continuously (upper and lower) under singular perturbations.

Turing instabilities in a mean field model of electrocortical activity

The mean field model formulated by Liley et al. [1] describes the membrane potential and synaptic interaction of excitatory and inhibitory neuron populations in a two-dimensional slab of cortical tissue. When complemented with periodic boundary conditions, it yields a set of fourteen coupled partial differential equations that encompass both highly nonlinear local interaction and long-range interaction through cable equations. We consider these equations as an autonomous dynamical system and aim to parse it using bifurcation analysis. We focus in particular on Turing-type bifurcations from spatially homogeneous equilibria to time-periodic patterns. Such bifurcations have been considered in simplified models but, to the best of our knowledge, not in a physiologically realistic setting [2]. In this study, we fix the model parameters to values that have previously been shown to lead to physiologically interesting γ-range activity [3]. We locate the primary instability when varying the inhibitory-to-inhibitory connection density, known to strongly affect the power spectrum, and the system size. Although for small system size, spatially homogeneous periodic solutions appear, for system sizes of a few square centimeters we observe centimeter-scale, oscillating patterns, which appear in subcritical Hopf bifurcations and have a strong γ-band component in their power spectrum. An example is shown in Fig. ​Fig.1,1, for a domain of 12.8 by 12.8cm. The black curve denotes stable (solid) and unstable (dashed) equilibria, and the blue curve denotes spatially homogeneous periodic solutions. The latter can be stable to spatially homogeneous perturbations. The primary instability corresponds to a spatial wave number two in both directions. The inlay shows a snapshot of the excitatory membrane potential for the corresponding, mildly unstable, periodic solution near the Hopf bifurcation. Figure 1 Partial bifurcation diagram of the mean-field model. A periodic solution (blue) bifurcates from an equilibrium (black), both spatially homogeneous. The primary instability, however, is a Turing bifurcation to the pattern in the left inlay. The second ... We have implemented a time-stepper in the open-source PETSc environment [4]. It uses a finite difference approximation for the Laplacian and first order implicit time stepping, both for the field equations and for the tangent linear model. The distribution of the grid over CPUs is automated, allowing for efficient parallel computation. This code can be used for the computation of equilibria, traveling waves, periodic orbits, manifolds and other building blocks of dynamical systems analysis, leading to a greater insight in the model's behaviour than could previously be extracted from analysis of spatially homogeneous solutions alone.

Finite-time entropy: A probabilistic approach for measuring nonlinear stretching

Transport and mixing processes in dynamical systems are often difficult to study analytically and therefore a variety of numerical methods have been developed. Finite-time Lyapunov exponents (FTLEs) or related stretching indicators are frequently used as a means to estimate transport barriers. Alternatively, eigenvectors, singular vectors, or Oseledets vectors of numerical transfer operators find almost-invariant sets, finite-time coherent sets, or time-asymptotic coherent sets, respectively, which are minimally dispersed under the dynamics. While these families of approaches (geometric FTLEs and the probabilistic transfer operator) often give compatible results, a formal link is still missing; here we present a small step towards providing a mathematical link.We propose a new entropy-based methodology for estimating finite-time expansive behaviour along trajectories in autonomous and nonautonomous dynamical systems. We introduce the finite-time entropy (FTE) field as a simple and flexible way to capture nonlinear stretching directly from the entropy growth experienced by a small localised density evolved by the transfer operator. The FTE construction elucidates in a straightforward way the connection between the evolution of probability densities and the local stretching experienced.We develop an extremely simple and numerically efficient method of constructing an estimate of the FTE field. The FTE field is instantaneously calculable from a numerical transfer operator — a transition matrix of conditional probabilities that describes a discretised version of the dynamical system; once one has such a transition matrix, the FTE field may be computed “for free”. We also show (i) how to avoid long time integrations in autonomous and time-periodic systems, (ii) how to perform backward time computations by a fast matrix manipulation rather than backward time integration, and (iii) how to easily employ adaptive methods to focus on high-value FTE regions.

Characterization of Stability Region for General Autonomous Nonlinear Dynamical Systems

The existing characterization of stability regions was developed under the assumption that limit sets on the stability boundary are exclusively composed of hyperbolic equilibrium points and closed orbits. The characterizations derived in this technical note are a generalization of existing results in the theory of stability regions. A characterization of the stability boundary of general autonomous nonlinear dynamical systems is developed under the assumption that limit sets on the stability boundary are composed of a countable number of disjoint and indecomposable components, which can be equilibrium points, closed orbits, quasi-periodic solutions and even chaotic invariant sets.