For other initial conditions, the equation of motion is given by the exponential of a matrix : for an initial point x 0 ,.

### 1. Background

From the eigenvalues and the eigenvectors of A it is possible to determine if an initial point will converge or diverge to the equilibrium point at the origin. Linear systems display sensitive dependence on initial conditions in the case of divergence. For nonlinear systems this is one of the necessary but not sufficient conditions for chaotic behavior. A discrete-time , affine dynamical system has the form of a matrix difference equation :.

The solutions for the map are no longer curves, but points that hop in the phase space. The orbits are organized in curves, or fibers, which are collections of points that map into themselves under the action of the map. As in the continuous case, the eigenvalues and eigenvectors of A determine the structure of phase space. Points in this straight line run into the fixed point.

There are also many other discrete dynamical systems. It is in the neighborhood of singular points and periodic orbits that the structure of a phase space of a dynamical system can be well understood. In the qualitative study of dynamical systems, the approach is to show that there is a change of coordinates usually unspecified, but computable that makes the dynamical system as simple as possible. A flow in most small patches of the phase space can be made very simple.

This is known as the rectification theorem. The rectification theorem says that away from singular points the dynamics of a point in a small patch is a straight line. The patch can sometimes be enlarged by stitching several patches together, and when this works out in the whole phase space M the dynamical system is integrable. In most cases the patch cannot be extended to the entire phase space. The more subtle reason is a global constraint, where the trajectory starts out in a patch, and after visiting a series of other patches comes back to the original one.

If the next time the orbit loops around phase space in a different way, then it is impossible to rectify the vector field in the whole series of patches. In general, in the neighborhood of a periodic orbit the rectification theorem cannot be used. This is known as the conjugation equation.

## ISBN 13: 9780444002464

Finding conditions for this equation to hold has been one of the major tasks of research in dynamical systems. The results on the existence of a solution to the conjugation equation depend on the eigenvalues of J and the degree of smoothness required from h. As J does not need to have any special symmetries, its eigenvalues will typically be complex numbers. When the eigenvalues of J are not in the unit circle, the dynamics near the fixed point x 0 of F is called hyperbolic and when the eigenvalues are on the unit circle and complex, the dynamics is called elliptic. The hyperbolic case is also structurally stable.

At this point the phase space changes qualitatively and the dynamical system is said to have gone through a bifurcation. At the bifurcation point the structure may change its stability, split into new structures, or merge with other structures. By using Taylor series approximations of the maps and an understanding of the differences that may be eliminated by a change of coordinates, it is possible to catalog the bifurcations of dynamical systems.

For a flow, it will occur when there are eigenvalues on the imaginary axis. For more information, see the main article on Bifurcation theory. Some bifurcations can lead to very complicated structures in phase space. For example, the Ruelle—Takens scenario describes how a periodic orbit bifurcates into a torus and the torus into a strange attractor. In another example, Feigenbaum period-doubling describes how a stable periodic orbit goes through a series of period-doubling bifurcations.

In the Hamiltonian formalism , given a coordinate it is possible to derive the appropriate generalized momentum such that the associated volume is preserved by the flow. The volume is said to be computed by the Liouville measure. In a Hamiltonian system, not all possible configurations of position and momentum can be reached from an initial condition.

Because of energy conservation, only the states with the same energy as the initial condition are accessible. The volume of the energy shell, computed using the Liouville measure, is preserved under evolution. Then almost every point of A returns to A infinitely often. One of the questions raised by Boltzmann's work was the possible equality between time averages and space averages, what he called the ergodic hypothesis. The ergodic hypothesis turned out not to be the essential property needed for the development of statistical mechanics and a series of other ergodic-like properties were introduced to capture the relevant aspects of physical systems.

Koopman approached the study of ergodic systems by the use of functional analysis. An observable a is a function that to each point of the phase space associates a number say instantaneous pressure, or average height. This idea has been generalized by Sinai, Bowen, and Ruelle SRB to a larger class of dynamical systems that includes dissipative systems.

SRB measures replace the Boltzmann factor and they are defined on attractors of chaotic systems. Simple nonlinear dynamical systems and even piecewise linear systems can exhibit a completely unpredictable behavior, which might seem to be random, despite the fact that they are fundamentally deterministic. This seemingly unpredictable behavior has been called chaos.

Hyperbolic systems are precisely defined dynamical systems that exhibit the properties ascribed to chaotic systems. In hyperbolic systems the tangent space perpendicular to a trajectory can be well separated into two parts: one with the points that converge towards the orbit the stable manifold and another of the points that diverge from the orbit the unstable manifold. This branch of mathematics deals with the long-term qualitative behavior of dynamical systems.

Here, the focus is not on finding precise solutions to the equations defining the dynamical system which is often hopeless , but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible attractors? Note that the chaotic behavior of complex systems is not the issue.

Meteorology has been known for years to involve complex—even chaotic—behavior. Chaos theory has been so surprising because chaos can be found within almost trivial systems. The logistic map is only a second-degree polynomial; the horseshoe map is piecewise linear. Dynamical systems are defined over a single independent variable, usually thought of as time.

## Large Scale Dynamic Systems : Stability and Structure by D. D. Siljak | eBay

For other uses, see Dynamical disambiguation. Main article: Dynamical system definition. Main article: Linear dynamical system. Main article: Bifurcation theory. Main article: Ergodic theory. Anderson, P. More is different. Science , — Lopes da Silva, F. Neural mechanisms underlying brain waves: from neural membranes to networks. Omurtag, A. On the simulation of large populations of neurons. Fourcaud, N. Dynamics of the firing probability of noisy integrate-and-fire neurons. Neural Comput. El Boustani, S. A master equation formalism for macroscopic modeling of asynchronous irregular activity states.

The dynamic brain: from spiking neurons to neural masses and cortical fields. Harrison, L. Stochastic models of neuronal dynamics. B , — Ma, W. Bayesian inference with probabilistic population codes. Friston, K. The free-energy principle: a unified brain theory? Huys, Q.

## Stability Analysis of Continuous Systems with Structural Perturbations

Computational psychiatry as a bridge from neuroscience to clinical applications. Beggs, J. Neuronal avalanches in neocortical circuits. Friedman, N. Universal critical dynamics in high resolution neuronal avalanche data. The heavy tail of the human brain. Jansen, B. Electroencephalogram and visual evoked potential generation in a mathematical model of coupled cortical columns. Model of brain rhythmic activity. The alpha-rhythm of the thalamus. Kybernetik 15 , 27—37 Marreiros, A. Population dynamics: variance and the sigmoid activation function. Neuroimage 42 , — Larter, R.

A coupled ordinary differential equation lattice model for the simulation of epileptic seizures. Chaos 9 , — Modulation of excitatory synaptic coupling facilitates synchronization and complex dynamics in a nonlinear model of neuronal dynamics. Network 14 , — Stefanescu, R. Reduced representations of heterogeneous mixed neural networks with synaptic coupling.

E 83 , Neural population modes capture biologically realistic large scale network dynamics. Miller, P. A recurrent network model of somatosensory parametric working memory in the prefrontal cortex. Cortex 13 , — Wong, K. A recurrent network mechanism of time integration in perceptual decisions. A novel method for the topographic analysis of neural activity reveals formation and dissolution of 'dynamic cell assemblies'. Dynamics of a neural system with a multiscale architecture.

Stephan, K. Spatial embedding and wiring cost constrain the functional layout of the cortical network of rodents and primates. PLoS Biol. Woolrich, M.

### Bulletin of the American Mathematical Society

Biophysical network models and the human connectome. Neuroimage 80 , — Mejias, J. Feedforward and feedback frequency-dependent interactions in a large-scale laminar network of the primate cortex. Beurle, R. Properties of a mass of cells capable of regenerating pulses. B , 55—94 Amari, S. Dynamics of pattern formation in lateral-inhibition type neural fields. The brain wave equation: a model for the EEG. A derivation of a macroscopic field theory of the brain from the quasi-microscopic neural dynamics.

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Physica D 99 , — Patchy propagators, brain dynamics, and the generation of spatially structured gamma oscillations. E 73 , Waves in spatially-disordered neural fields: a case study in uncertainty quantification. Rennie, C. Unified neurophysical model of EEG spectra and evoked potentials. Eigenmodes of brain activity: neural field theory predictions and comparison with experiment. Neuroimage , 79—98 Muller, L. The stimulus-evoked population response in visual cortex of awake monkey is a propagating wave.

Rubino, D. Propagating waves mediate information transfer in the motor cortex.

Heitmann, S. A dendritic mechanism for decoding traveling waves: principles and applications to motor cortex. Henderson, J. Geometric effects on complex network structure in the cortex. The contribution of geometry to the human connectome. Neuroimage , — Next generation neural mass models. Moran, R. A neural mass model of spectral responses in electrophysiology.

Neuroimage 37 , — Wolf, A. Determining Lyapunov exponents from a time series. Physica D 16 , — Grassberger, P. Characterization of strange attractors. Soong, A. Evidence of chaotic dynamics underlying the human alpha-rhythm electroencephalogram. Pritchard, W. Psychophysiology 29 , — Babloyantz, A. Evidence of chaotic dynamics of brain activity during the sleep cycle. A , — Gregson, R. Comparisons of the nonlinear dynamics of electroencephalograms under various task loading conditions: a preliminary report.

Low-dimensional chaos in an instance of epilepsy. USA 83 , — Theiler, J. Spurious dimension from correlation algorithms applied to limited time-series data. A Gen. Osborne, A. Finite correlation dimension for stochastic systems with power-law spectra. Physica D 35 , — Rapp, P. Filtered noise can mimic low-dimensional chaotic attractors. Topics 47 , — Dimensional analysis of resting human EEG. II: Surrogate-data testing indicates nonlinearity but not low-dimensional chaos. Psychophysiology 32 , — Theory , 1, No. Martynyuk and I. NANU , No. RAN , , No.

Michel, K. Wang, and B. Hu, Qualitative Theory of Dynamical Systems. Benjamin, Inc. Piontkovskii and L. American Control Conf. Peterburgskogo Univ. Martynyuk 1 1. Personalised recommendations. Cite article How to cite?