Jump Bifurcation and Hysteresis in an Infinite-Dimensional Dynamical System of Coupled Spins

1990 ◽  
Vol 50 (1) ◽  
pp. 108-124 ◽  
Author(s):  
Renato E. Mirollo ◽  
Steven H. Strogatz
Keyword(s):  
Dynamical System ◽  
Infinite Dimensional ◽  
Dimensional Dynamical System ◽  
Infinite Dimensional Dynamical System

On the equation ut = ∆uα + uβ

1993 ◽  
Vol 123 (2) ◽  
pp. 373-390 ◽  
Author(s):  
Yuan-Wei Qi
Keyword(s):  
Dynamical System ◽  
Blow Up ◽  
Uniqueness Result ◽  
Fast Diffusion ◽  
Negative Solution ◽  
Fast Diffusion Equation ◽  
Infinite Dimensional ◽  
The Cauchy Problem ◽  
Dimensional Dynamical System ◽  
Infinite Dimensional Dynamical System

SynopsisThe Cauchy problem of ut, = ∆uα + uβ, where 0 < α < l and α>1, is studied. It is proved that if 1< β<α + 2/n then every nontrivial non-negative solution is not global in time. But if β>α+ 2/n there exist both blow-up solutions and global positive solutions which decay to zero as t–1/(β–1) when t →∞. Thus the famous Fujita result on ut = ∆u + up is generalised to the present fast diffusion equation. Furthermore, regarding the equation as an infinite dimensional dynamical system on Sobolev space W1,s (W2.s) with S > 1, a non-uniqueness result is established which shows that there exists a positive solution u(x, t) with u(., t) → 0 in W1.s (W2.s) as t → 0.

scholarly journals Forced Oscillations of a Damped Korteweg-de Vries Equation in a Quarter Plane

2003 ◽  
Vol 05 (03) ◽  
pp. 369-400 ◽  
Author(s):  
Jerry L. Bona ◽  
S. M. Sun ◽  
Bing-Yu Zhang
Keyword(s):  
Dynamical System ◽  
Limit Cycle ◽  
Small Amplitude ◽  
Vries Equation ◽  
Initial Condition ◽  
Infinite Dimensional ◽  
De Vries ◽  
Exponentially Stable ◽  
Dimensional Dynamical System ◽  
Infinite Dimensional Dynamical System

Laboratory experiments have shown that when nonlinear, dispersive waves are forced periodically from one end of an undisturbed stretch of the medium of propagation, the signal eventually becomes temporally periodic at each spatial point. It is our purpose here to establish this as a fact at least in the context of a damped Korteweg-de Vries equation. Thus, consideration is given to the initial-boundary-value problem [Formula: see text] For this problem, it is shown that if the small amplitude, boundary forcing h is periodic of period T, say, then the solution u of (*) is eventually periodic of period T. More precisely, we show for each x > 0, that u(x, t + T) - u(x, t) converges to zero exponentially as t → ∞. Viewing (*) (without the initial condition) as an infinite dimensional dynamical system in the Hilbert space Hs(R+) for suitable values of s, we also demonstrate that for a given, small amplitude time-periodic boundary forcing, the system (*) admits a unique limit cycle, or forced oscillation (a solution of (*) without the initial condition that is exactly periodic of period T). Furthermore, we show that this limit cycle is globally exponentially stable. In other words, it comprises an exponentially stable attractor for the infinite-dimensional dynamical system described by (*).

scholarly journals The Cosmological Semiclassical Einstein Equation as an Infinite-Dimensional Dynamical System

2021 ◽  
Author(s):  
Hanno Gottschalk ◽  
Daniel Siemssen
Keyword(s):  
Dynamical System ◽  
Einstein Equation ◽  
Global Solutions ◽  
Standard Theory ◽  
Infinite Dimensional ◽  
Infinite Dimensional Dynamical Systems ◽  
Point Splitting ◽  
Dimensional Dynamical System ◽  
Infinite Dimensional Dynamical System ◽  
Comprehensive Framework

AbstractWe develop a comprehensive framework in which the existence of solutions to the semiclassical Einstein equation (SCE) in cosmological spacetimes is shown. Different from previous work on this subject, we do not restrict to the conformally coupled scalar field and we admit the full renormalization freedom. Based on a regularization procedure, which utilizes homogeneous distributions and is equivalent to Hadamard point splitting, we obtain a reformulation of the evolution of the quantum state as an infinite-dimensional dynamical system with mathematical features that are distinct from the standard theory of infinite-dimensional dynamical systems (e.g., unbounded evolution operators). Nevertheless, applying methods closely related to Ovsyannikov’s method, we show existence of maximal/global solutions to the SCE for vacuum-like states and of local solutions for thermal-like states. Our equations do not show the instability of the Minkowski solution described by other authors.

Analog Techniques for Modeling and Controlling the Mackey-Glass System

1997 ◽  
Vol 07 (04) ◽  
pp. 957-962 ◽  
Author(s):  
A. Namajūnas ◽  
K. Pyragas ◽  
A. Tamaševičius
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Mathematical Model ◽  
Delay Differential Equation ◽  
Analog Circuit ◽  
Glass System ◽  
Infinite Dimensional ◽  
Dimensional Dynamical System ◽  
Infinite Dimensional Dynamical System ◽  
Delay Differential

An analog circuit which models an infinite-dimensional dynamical system is described. It provides an easy and rapid way to simulate periodic and chaotic behaviors in a delay differential equation, the Mackey-Glass mathematical model. Several methods of controlling unstable steady states are considered. Analog techniques for stabilizing non-zero stationary point in the Mackey-Glass system are developed.

scholarly journals Homeostatic Plasticity for Single Node Delay-Coupled Reservoir Computing

2015 ◽  
Vol 27 (6) ◽  
pp. 1159-1185 ◽  
Author(s):  
Hazem Toutounji ◽  
Johannes Schumacher ◽  
Gordon Pipa
Keyword(s):  
Homeostatic Plasticity ◽  
Reservoir Computing ◽  
Computational Power ◽  
Biologically Inspired ◽  
Single Node ◽  
Plasticity Mechanism ◽  
Learning Mechanism ◽  
Infinite Dimensional ◽  
Dimensional Dynamical System ◽  
Infinite Dimensional Dynamical System

Supplementing a differential equation with delays results in an infinite-dimensional dynamical system. This property provides the basis for a reservoir computing architecture, where the recurrent neural network is replaced by a single nonlinear node, delay-coupled to itself. Instead of the spatial topology of a network, subunits in the delay-coupled reservoir are multiplexed in time along one delay span of the system. The computational power of the reservoir is contingent on this temporal multiplexing. Here, we learn optimal temporal multiplexing by means of a biologically inspired homeostatic plasticity mechanism. Plasticity acts locally and changes the distances between the subunits along the delay, depending on how responsive these subunits are to the input. After analytically deriving the learning mechanism, we illustrate its role in improving the reservoir’s computational power. To this end, we investigate, first, the increase of the reservoir’s memory capacity. Second, we predict a NARMA-10 time series, showing that plasticity reduces the normalized root-mean-square error by more than 20%. Third, we discuss plasticity’s influence on the reservoir’s input-information capacity, the coupling strength between subunits, and the distribution of the readout coefficients.

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