On periodic orbits of nonlinear dynamical systems with many degrees of freedom

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.

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

Journal of Applied Mechanics ◽

10.1115/1.1558079 ◽

2003 ◽

Vol 70 (3) ◽

pp. 328-338 ◽

Cited By ~ 11

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.

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Symmetries and itineracy in nonlinear systems with many degrees of freedom

Behavioral and Brain Sciences ◽

10.1017/s0140525x01250092 ◽

2001 ◽

Vol 24 (5) ◽

pp. 813-813 ◽

Cited By ~ 6

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.”

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