From intermittency to transitivity in neuropsychobiological flows

1983 ◽  
Vol 245 (4) ◽  
pp. R484-R494 ◽  
Author(s):  
A. J. Mandell
Keyword(s):  
Nonlinear Dynamics ◽  
Brain Function ◽  
Chaotic Behavior

Nonlinear dynamics offers a language for describing many aspects of brain function. Intermittency, alternating periods of periodic and chaotic behavior, and transitivity, the indecomposability of a flow, are discussed here in detail. Applications are suggested to neuropsychobiological phenomena, such as the effects of drugs and other agents.

DEVELOPMENT OF THERAPEUTIC BRAIN STIMULATION TECHNIQUES WITH METHODS FROM NONLINEAR DYNAMICS AND STATISTICAL PHYSICS

2006 ◽  
Vol 16 (07) ◽  
pp. 1889-1911 ◽  
Author(s):  
PETER A. TASS ◽  
CHRISTIAN HAUPTMANN ◽  
OLEKSANDR V. POPOVYCH
Keyword(s):  
Nonlinear Dynamics ◽  
Parkinson’S Disease ◽  
Parkinson's Disease ◽  
Essential Tremor ◽  
Brain Stimulation ◽  
Brain Function ◽  
Statistical Physics ◽  
Delayed Feedback ◽  
Phase Resetting ◽  
Feedback Mechanisms

Synchronization processes may severely impair brain function, for instance, in Parkinson's disease, essential tremor or epilepsies. We present three different effectively desynchronizing stimulation techniques which have been developed with methods from nonlinear dynamics and statistical physics. These techniques exploit either stochastic phase resetting principles or complex delayed feedback mechanisms. We explain how these methods work and how they can be applied to therapeutic brain stimulation.

On the Motion of the Pendulum in an Alternating, Sawtooth Force Field

2020 ◽  
Vol 30 (09) ◽  
pp. 2050135
Author(s):  
Alexander A. Burov ◽  
Vasily I. Nikonov
Keyword(s):  
Nonlinear Dynamics ◽  
Iterative Method ◽  
Force Field ◽  
Periodic Solutions ◽  
Chaotic Behavior ◽  
Invariant Tori ◽  
Melnikov Method ◽  
Stability Conditions ◽  
Poincaré Maps ◽  
Separatrix Splitting

The motion of the pendulum in a variable sawtooth force field is considered. For the “lower” equilibrium, the necessary stability conditions are investigated numerically, the results are presented in the form of an Ince–Strutt diagram. Using the Poincaré–Melnikov method separatrix splitting is studied analytically. Numerically, for some values of parameters, the nonlinear dynamics is studied using Poincaré maps, the regions of regular and chaotic behavior are revealed. The iterative method earlier proposed is used for the localization of periodic solutions, located inside the numerically identified “invariant tori”.

scholarly journals Reflection of Solar Activity Dynamics in Radionuclide Data

Radiocarbon ◽  
1992 ◽  
Vol 34 (2) ◽  
pp. 207-212 ◽  
Author(s):  
A. V. Blinov ◽  
M. N. Kremliovskij
Keyword(s):  
Nonlinear Dynamics ◽  
Solar Activity ◽  
Numerical Methods ◽  
Chaotic Behavior ◽  
Magnetic Activity ◽  
Data Sets ◽  
Cosmogenic Nuclide ◽  
Proxy Records ◽  
Solar Magnetic Activity ◽  
Activity Dynamics

Variability of solar magnetic activity manifested within sunspot cycles demonstrates features of chaotic behavior. We have analyzed cosmogenic nuclide proxy records for the presence of the solar activity signals. We have applied numerical methods of nonlinear dynamics to the data showing the contribution of the chaotic component. We have also formulated what kind of cosmogenic nuclide data sets are needed for investigations on solar activity.

NONLINEAR DYNAMICS OF DIGITAL PHASE-LOCKED LOOPS WITH DELAY

1994 ◽  
Vol 04 (03) ◽  
pp. 715-726 ◽  
Author(s):  
MARIA DE SOUSA VIEIRA ◽  
ALLAN J. LICHTENBERG ◽  
MICHAEL A. LIEBERMAN
Keyword(s):  
Nonlinear Dynamics ◽  
Chaotic Behavior ◽  
Nonlinear Oscillators ◽  
Coupled Systems ◽  
Phase Locked Loops ◽  
Bifurcation Diagrams ◽  
Digital Phase ◽  
Coupled Nonlinear Oscillators ◽  
Digital Phase Locked Loops ◽  
The Stability

We investigate numerically and analytically the nonlinear dynamics of a system consisting of two self-synchronizing pulse-coupled nonlinear oscillators with delay. The particular system considered consists of connected digital phase-locked loops. We find mapping equations that govern the system and determine the synchronization properties. We study the bifurcation diagrams, which show regions of periodic, quasiperiodic and chaotic behavior, with unusual bifurcation diagrams, depending on the delay. We show that depending on the parameter that is varied, the delay will have a synchronizing or desynchronizing effect on the locked state. The stability of the system is studied by determining the Liapunov exponents, indicating marked differences compared to coupled systems without delay.

scholarly journals Maps for analysis of nonlinear dynamics

1998 ◽  
Vol 2 ◽  
pp. 43-58
Author(s):  
Bronislovas Kaulakys
Keyword(s):  
Nonlinear Dynamics ◽  
Quantum Dynamics ◽  
Microwave Field ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Driven Systems ◽  
Multilevel Systems ◽  
Area Preserving Maps ◽  
Area Preserving ◽  
Field Transition

Area preserving maps provide the simplest and most accurate means to visualize and quantify the behavior of nonlinear systems. Convenience of the mapping equations of motion for investigation of transition to chaotic behavior in dynamics of classical atom in microwave field, transition to nonchaotic behavior in randomly driven systems and induced quantum dynamics of simple and multilevel systems is demonstrated.

scholarly journals Study on Nonlinear Dynamics and Chaos Suppression of Active Magnetic Bearing Systems Based on Synchronization

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Shun-Chang Chang
Keyword(s):  
Nonlinear Dynamics ◽  
Chaotic Behavior ◽  
Nonlinear Dynamic Analysis ◽  
Magnetic Bearing ◽  
Active Magnetic Bearing ◽  
Phase Portraits ◽  
Largest Lyapunov Exponent ◽  
Frequency Spectra ◽  
Control Approach ◽  
Chaos Suppression

This study employed a variety of nonlinear dynamic analysis techniques to explore the complex phenomena associated with a nonlinear mathematical model of an active magnetic bearing (AMB) system. The aim was to develop a method with which to assume control over chaotic behavior. The bifurcation diagram comprehensively explicates rich nonlinear dynamics over a range of parameter values. In this study, we examined the complex nonlinear behaviors of AMB systems using phase portraits, Poincaré maps, and frequency spectra. Furthermore, estimates of the largest Lyapunov exponent based on the properties of synchronization confirmed the occurrence of chatter vibration indicative of chaotic motion. Thus, the proposed continuous feedback control approach based on synchronization characteristics eliminates chaotic oscillations. Finally, some simulation results demonstrated the feasibility and efficiency of the proposed control scheme.

Nonlinear dynamics and chaos

2018 ◽  
Author(s):  
Joseph F. Boudreau ◽  
Eric S. Swanson
Keyword(s):  
Nonlinear Dynamics ◽  
Lyapunov Exponent ◽  
Chaotic Behavior ◽  
Phase Portraits ◽  
Nonlinear Ordinary Differential Equations ◽  
Two Dimensional ◽  
Kam Theorem ◽  
Physical Systems ◽  
Nonlinear Dynamics And Chaos ◽  
Iterative Maps

Simple maps and dynamical systems are used to explore chaos in nature. The discussion starts with a review of the properties of nonlinear ordinary differential equations, including the useful concepts of phase portraits, fixed points, and limit cycles. These notions are developed further in an examination of iterative maps that reveal chaotic behavior. Next, the damped driven oscillator is used to illustrate the Lyapunov exponent that can be used to quantify chaos. The famous KAM theorem on the conditions under which chaotic behavior occurs in physical systems is also presented. The principle is illustrated with the Hénon-Heiles model of a star in a galactic environment and billiard models that describe the motion of balls in closed two-dimensional regions.

scholarly journals Synchronization of Chaotic Gyros Based on Robust Nonlinear Dynamic Inversion

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Inseok Yang ◽  
Dongik Lee
Keyword(s):  
Nonlinear Dynamics ◽  
Nonlinear Dynamic ◽  
Sliding Mode ◽  
Chaotic Behavior ◽  
Original System ◽  
Control Technique ◽  
Dynamic Inversion ◽  
Model Uncertainties ◽  
Gain Scheduled ◽  
Nonlinear Dynamic Inversion

A robust nonlinear dynamic inversion (RNDI) technique is proposed in order to synchronize the behavior of chaotic gyros subjected to uncertainties such as model mismatches and disturbances. Gyro is a crucial device that measures and maintains the orientation of a vehicle. By Leipnik and Newton in 1981, chaotic behavior of a gyro under specific conditions was established. Hence, controlling and synchronizing a gyro that shows irregular (chaotic) motion are very important. The proposed synchronization method is based on nonlinear dynamic inversion (NDI) control. NDI is a nonlinear control technique that removes the original system dynamics into the user-defined desired dynamics. Since NDI removes the original dynamics directly, it does not need linearizing and designing gain-scheduled controllers for each equilibrium point. However, achieving perfect cancellation of the original nonlinear dynamics is impossible in real applications due to model uncertainties and disturbances. This paper proposes the robustness assurance method of NDI based on sliding mode control (SMC). Firstly, similarities of the conventional NDI control and SMC are provided. And then the RNDI control technique is proposed. The feasibility and effectiveness of the proposed method are demonstrated by numerical simulations.

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