scholarly journals Symmetries and itineracy in nonlinear systems with many degrees of freedom

2001 ◽  
Vol 24 (5) ◽  
pp. 813-813 ◽  
Author(s):  
Michael Breakspear ◽  
Karl Friston
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Brain Function ◽  
Degrees Of Freedom ◽  
Nonlinear Dynamical Systems ◽  
Research Direction ◽  
Potential Contribution ◽  
Nonlinear Dynamical

Tsuda examines the potential contribution of nonlinear dynamical systems, with many degrees of freedom, to understanding brain function. We offer suggestions concerning symmetry and transients to strengthen the physiological motivation and theoretical consistency of this novel research direction: Symmetry plays a fundamental role, theoretically and in relation to real brains. We also highlight a distinction between chaotic “transience” and “itineracy.”

HIGH-DIMENSIONAL CHAOS IN DISSIPATIVE AND DRIVEN DYNAMICAL SYSTEMS

2009 ◽  
Vol 19 (09) ◽  
pp. 2823-2869 ◽  
Author(s):  
Z. E. MUSIELAK ◽  
D. E. MUSIELAK
Keyword(s):  
Dynamical Systems ◽  
Degrees Of Freedom ◽  
Nonlinear Dynamical Systems ◽  
High Dimensional ◽  
Van Der Pol ◽  
Nonlinear Dynamical ◽  
Onset Of Chaos ◽  
General Rules ◽  
Van Der Pol Oscillators ◽  
Control And Synchronization

Studies of nonlinear dynamical systems with many degrees of freedom show that the behavior of these systems is significantly different as compared with the behavior of systems with less than two degrees of freedom. These findings motivated us to carry out a survey of research focusing on the behavior of high-dimensional chaos, which include onset of chaos, routes to chaos and the persistence of chaos. This paper reports on various methods of generating and investigating nonlinear, dissipative and driven dynamical systems that exhibit high-dimensional chaos, and reviews recent results in this new field of research. We study high-dimensional Lorenz, Duffing, Rössler and Van der Pol oscillators, modified canonical Chua's circuits, and other dynamical systems and maps, and we formulate general rules of high-dimensional chaos. Basic techniques of chaos control and synchronization developed for high-dimensional dynamical systems are also reviewed.

Carathe´odory Controllability Criterion for Nonlinear Dynamical Systems

1978 ◽  
Vol 100 (3) ◽  
pp. 209-213 ◽  
Author(s):  
G. Langholz ◽  
M. Sokolov
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Control Theory ◽  
Linear Systems ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical ◽  
Modern Control Theory ◽  
Open Question ◽  
Modern Control

The question of whether a system is controllable or not is of prime importance in modern control theory and has been actively researched in recent years. While it is a solved problem for linear systems, it is still an open question when dealing with bilinear and nonlinear systems. In this paper, a controllability criterion is established based on a theorem by Carathe´odory. By associating a given dynamical system with a certain Pfaffian equation, it is argued that the system is controllable (uncontrollable) if its associated Pfaffian form is nonintegrable (integrable).

Multiscale, Multiphenomena Modeling and Simulation at the Nanoscale: On Constructing Reduced-Order Models for Nonlinear Dynamical Systems With Many Degrees-of-Freedom

2003 ◽  
Vol 70 (3) ◽  
pp. 328-338 ◽  
Author(s):  
E. H. Dowell ◽  
D. Tang
Keyword(s):  
Dynamical Systems ◽  
Degrees Of Freedom ◽  
Nonlinear Dynamical Systems ◽  
Total System ◽  
Reduced Order Models ◽  
Efficient Computation ◽  
Linear Dynamical Systems ◽  
Nonlinear Dynamical ◽  
Reduced Order ◽  
Value Decomposition

The large number of degrees-of-freedom of finite difference, finite element, or molecular dynamics models for complex systems is often a significant barrier to both efficient computation and increased understanding of the relevant phenomena. Thus there is a benefit to constructing reduced-order models with many fewer degrees-of-freedom that retain the same accuracy as the original model. Constructing reduced-order models for linear dynamical systems relies substantially on the existence of global modes such as eigenmodes where a relatively small number of these modes may be sufficient to describe the response of the total system. For systems with very many degrees-of-freedom that arise from spatial discretization of partial differential equation models, computing the eigenmodes themselves may be the major challenge. In such cases the use of alternative modal models based upon proper orthogonal decomposition or singular value decomposition have proven very useful. In the present paper another facet of reduced-order modeling is examined, i.e., the effects of “local” nonlinearity at the nanoscale. The focus is on nanoscale devices where it will be shown that a combination of global modal and local discrete coordinates may be most effective in constructing reduced-order models from both a conceptual and computational perspective. Such reduced-order models offer the possibility of reducing computational model size and cost by several orders of magnitude.

scholarly journals Dynamical Movement Primitives: Learning Attractor Models for Motor Behaviors

2013 ◽  
Vol 25 (2) ◽  
pp. 328-373 ◽  
Author(s):  
Auke Jan Ijspeert ◽  
Jun Nakanishi ◽  
Heiko Hoffmann ◽  
Peter Pastor ◽  
Stefan Schaal
Keyword(s):  
Motor Control ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Nonlinear Dynamical Systems ◽  
Emergent Behavior ◽  
Nonlinear Dynamical ◽  
Movement Primitives ◽  
Model Complex ◽  
Linear Differential ◽  
Goal Directed Behavior

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.

Nonlinear Dynamical Systems, Their Stability, and Chaos

2014 ◽  
Vol 66 (2) ◽  
Author(s):  
Amol Marathe ◽  
Rama Govindarajan
Keyword(s):  
Fluid Flow ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Fluid Mechanics ◽  
Fixed Points ◽  
Chaotic Systems ◽  
Nonlinear Dynamical Systems ◽  
Not Given ◽  
Nonlinear Dynamical ◽  
Detailed Explanation

This introduction to nonlinear systems is written for students of fluid mechanics, so connections are made throughout the text to familiar fluid flow systems. The aim is to present how nonlinear systems are qualitatively different from linear and to outline some simple procedures by which an understanding of nonlinear systems may be attempted. Considerable attention is paid to linear systems in the vicinity of fixed points, and it is discussed why this is relevant for nonlinear systems. A detailed explanation of chaos is not given, but a flavor of chaotic systems is presented. The focus is on physical understanding and not on mathematical rigor.

Periodic Flows to Chaos Based on Discrete Implicit Mappings of Continuous Nonlinear Systems

2015 ◽  
Vol 25 (03) ◽  
pp. 1550044 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Time Delay ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Nonlinear Dynamical Systems ◽  
Duffing Oscillator ◽  
Single Step ◽  
Nonlinear Dynamical ◽  
Periodic Flows ◽  
Mapping Structures ◽  
Stability And Bifurcations

This paper presents a semi-analytical method for periodic flows in continuous nonlinear dynamical systems. For the semi-analytical approach, differential equations of nonlinear dynamical systems are discretized to obtain implicit maps, and a mapping structure based on the implicit maps is employed for a periodic flow. From mapping structures, periodic flows in nonlinear dynamical systems are predicted analytically and the corresponding stability and bifurcations of the periodic flows are determined through the eigenvalue analysis. The periodic flows predicted by the single-step implicit maps are discussed first, and the periodic flows predicted by the multistep implicit maps are also presented. Periodic flows in time-delay nonlinear dynamical systems are discussed by the single-step and multistep implicit maps. The time-delay nodes in discretization of time-delay nonlinear systems were treated by both an interpolation and a direct integration. Based on the discrete nodes of periodic flows in nonlinear dynamical systems with/without time-delay, the discrete Fourier series responses of periodic flows are presented. To demonstrate the methodology, the bifurcation tree of period-1 motion to chaos in a Duffing oscillator is presented as a sampled problem. The method presented in this paper can be applied to nonlinear dynamical systems, which cannot be solved directly by analytical methods.

ALGORITHMIC COMPLEXITY OF SOME PHYSICAL PROCESSES

1994 ◽  
Vol 05 (02) ◽  
pp. 421-424
Author(s):  
B.S. SAGINTAEV
Keyword(s):  
Time Series ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Temporal Variation ◽  
Nonlinear Dynamical Systems ◽  
Algorithmic Complexity ◽  
Physical Processes ◽  
Nonlinear Dynamical ◽  
The Relationship

Measure of algorithmic complexity c(n) which has been suggested by Lempel and Ziv is one of important quantities for characterizing properties of nonlinear dynamical systems. The temporal variation of c(n) is investigated for time series generated by some physical processes. The relationship between algorithmic complexity and other characteristics of nonlinear systems is discussed.

Using Strange Attractors to Control Sound Processing in Live Electroacoustic Composition

2015 ◽  
Vol 39 (3) ◽  
pp. 25-45
Author(s):  
Miroslav Spasov
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Nonlinear Dynamical Systems ◽  
Chaotic Attractors ◽  
Live Electronics ◽  
Tenor Saxophone ◽  
Sound Processing ◽  
Nonlinear Dynamical ◽  
Interactive Composition ◽  
Mathematical Equations

This article explores the possibility of using chaotic attractors to control sound processing with software instruments in live electroacoustic composition. The practice-led investigation involves the Attractors Library, a collection of Max/MSP externals based on iterative mathematical equations representing nonlinear dynamical systems; Attractors Player, a Max/MSP patch that controls the attractors' performance and live processing; and the two compositions based on the software: Strange Attractions for flute, clarinet, horn, and live electronics, and Sabda Vidya No. 2 for flute, tenor saxophone, and live electronics. In the article I discuss some specific attractors' characteristics and their use in interactive composition, relying on the experience from the performances of these two compositions. The idea is to highlight the experience with these nonlinear systems and to encourage other composers to use them in their own works.

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