Periodic solutions in one-dimensional coupled map lattices

2003 ◽  
Vol 24 (5) ◽  
pp. 521-526
Author(s):  
Zheng Yong-ai ◽  
Liu Zeng-rong
Keyword(s):  
Periodic Solutions ◽  
Coupled Map Lattices ◽  
One Dimensional

scholarly journals Infinitely many periodic solutions for a semilinear wave equation with x-dependent coefficients

2020 ◽  
Vol 26 ◽  
pp. 7
Author(s):  
Hui Wei ◽  
Shuguan Ji
Keyword(s):  
Wave Equation ◽  
Periodic Solutions ◽  
Spectral Properties ◽  
Seismic Waves ◽  
Forced Vibrations ◽  
One Dimensional ◽  
Semilinear Wave Equation ◽  
Spatial Interval ◽  
Rational Multiple ◽  
Infinitely Many Periodic Solutions

This paper is devoted to the study of periodic (in time) solutions to an one-dimensional semilinear wave equation with x-dependent coefficients under various homogeneous boundary conditions. Such a model arises from the forced vibrations of a nonhomogeneous string and propagation of seismic waves in nonisotropic media. By combining variational methods with an approximation argument, we prove that there exist infinitely many periodic solutions whenever the period is a rational multiple of the length of the spatial interval. The proof is essentially based on the spectral properties of the wave operator with x-dependent coefficients.

scholarly journals Periodic solutions for the one-dimensional fractional Laplacian

2019 ◽  
Vol 267 (9) ◽  
pp. 5258-5289 ◽  
Author(s):  
B. Barrios ◽  
J. García-Melián ◽  
A. Quaas
Keyword(s):  
Periodic Solutions ◽  
Fractional Laplacian ◽  
One Dimensional ◽  
The One

scholarly journals Quasi-Periodic Bifurcations of Higher-Dimensional Tori

2016 ◽  
Vol 26 (07) ◽  
pp. 1630016 ◽  
Author(s):  
Motomasa Komuro ◽  
Kyohei Kamiyama ◽  
Tetsuro Endo ◽  
Kazuyuki Aihara
Keyword(s):  
Periodic Solutions ◽  
Phase Locking ◽  
Period Doubling ◽  
Double Covering ◽  
Heteroclinic Cycle ◽  
One Dimensional ◽  
Local Bifurcations ◽  
Saddle Node ◽  
Higher Dimensional ◽  
The Relationship

We classify the local bifurcations of quasi-periodic [Formula: see text]-dimensional tori in maps (abbr. MT[Formula: see text]) and in flows (abbr. FT[Formula: see text]) for [Formula: see text]. It is convenient to classify these bifurcations into normal bifurcations and resonance bifurcations. Normal bifurcations of MT[Formula: see text] can be classified into four classes: namely, saddle-node, period doubling, double covering, and Neimark–Sacker bifurcations. Furthermore, normal bifurcations of FT[Formula: see text] can be classified into three classes: saddle-node, double covering, and Neimark–Sacker bifurcations. These bifurcations are determined by the type of the dominant Lyapunov bundle. Resonance bifurcations are well known as phase locking of quasi-periodic solutions. These bifurcations are classified into two classes for both MT[Formula: see text] and FT[Formula: see text]: namely, saddle-node cycle and heteroclinic cycle bifurcations of the [Formula: see text]-dimensional tori. The former is reversible, while the latter is irreversible. In addition, we propose a method for analyzing higher-dimensional tori, which uses one-dimensional tori in sections (abbr. ST[Formula: see text]) and zero-dimensional tori in sections (abbr. ST[Formula: see text]). The bifurcations of ST[Formula: see text] can be classified into five classes: saddle-node, period doubling, component doubling, double covering, and Neimark–Sacker bifurcations. The bifurcations of ST[Formula: see text] can be classified into four classes: saddle-node, period doubling, component doubling, and Neimark–Sacker bifurcations. Furthermore, we clarify the relationship between the bifurcations of ST[Formula: see text]/ST[Formula: see text] and the bifurcations of MT[Formula: see text]/FT[Formula: see text]. We present examples of all of these bifurcations.

Time-periodic solutions to the one-dimensional wave equation with periodic or anti-periodic boundary conditions

2007 ◽  
Vol 137 (2) ◽  
pp. 349-371 ◽  
Author(s):  
Shuguan Ji ◽  
Yong Li
Keyword(s):  
Wave Equation ◽  
Weak Solution ◽  
Boundary Conditions ◽  
Periodic Solutions ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
One Dimensional ◽  
Time Periodic Solutions ◽  
Time Periodic ◽  
Dimensional Wave

This paper is devoted to the study of time-periodic solutions to the nonlinear one-dimensional wave equation with x-dependent coefficients u(x)ytt – (u(x)yx)x + g(x,t,y) = f(x,t) on (0,π) × ℝ under the periodic boundary conditions y(0,t) = y(π,t), yx(0,t) = yx(π,t) or anti-periodic boundary conditions y(0, t) = –y(π,t), yx[0,t) = – yx(π,t). Such a model arises from the forced vibrations of a non-homogeneous string and the propagation of seismic waves in non-isotropic media. Our main concept is that of the ‘weak solution’. For T, the rational multiple of π, we prove some important properties of the weak solution operator. Based on these properties, the existence and regularity of weak solutions are obtained.

FRACTAL EVOLUTION IN DETERMINISTIC AND RANDOM MODELS

1996 ◽  
Vol 06 (11) ◽  
pp. 1977-1995
Author(s):  
SIEGFRIED FUSSY ◽  
GERHARD GRÖSSING ◽  
HERBERT SCHWABL
Keyword(s):  
Order Parameter ◽  
Specific Model ◽  
Coupled Map Lattices ◽  
Normalization Procedure ◽  
Quantum Cellular Automata ◽  
Time Step ◽  
One Dimensional ◽  
Random Models ◽  
Approximate Estimation ◽  
The Stability

One-dimensional coupled map lattices or quantum cellular automata with any additionally implemented temporal feedback operations (involving some memory of the system’s states) and a normalization procedure after each time step exhibit a universal dynamic property called fractal evolution [Fussy & Grössing, 1994]. It is characterized by a power-law behavior of a system’s order parameter with regard to a resolution-like parameter which controls the deviation from an undisturbed (i.e., feedback-less) system’s evolution and provides a dynamically invariant measure for the emerging spatiotemporal patterns. By comparison with another, simpler model without memory, where the patterns are generated randomly, the underlying principles of fractal evolution are studied. It is shown that our system evolving entirely deterministically, exhibits properties occurring usually only in random models, where the global measures, up to a certain degree, are calculable. Other properties like the fractal evolution exponent remain in general computationally irreducible due to the self-referential feedback dynamics. A specific model with an approximate estimation of the fractal evolution exponent is discussed. The stability of fractal evolution with respect to the dependence of pattern formation on the systems variables is also analyzed.

One-dimensional dynamics for traveling fronts in coupled map lattices

2000 ◽  
Vol 61 (2) ◽  
pp. 1329-1336 ◽  
Author(s):  
R. Carretero-González ◽  
D. K. Arrowsmith ◽  
F. Vivaldi
Keyword(s):  
Coupled Map Lattices ◽  
One Dimensional ◽  
Traveling Fronts

ON THE STRUCTURE OF REAL-VALUED ONE-DIMENSIONAL CELLULAR AUTOMATA

2011 ◽  
Vol 21 (05) ◽  
pp. 1265-1279 ◽  
Author(s):  
XU XU ◽  
STEPHEN P. BANKS ◽  
MAHDI MAHFOUF
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Fixed Points ◽  
Periodic Solutions ◽  
Cellular Systems ◽  
One Dimensional ◽  
Dynamical Behaviors ◽  
New Type ◽  
Complex Patterns

It is well-known that binary-valued cellular automata, which are defined by simple local rules, have the amazing feature of generating very complex patterns and having complicated dynamical behaviors. In this paper, we present a new type of cellular automaton based on real-valued states which produce an even greater amount of interesting structures such as fractal, chaotic and hypercyclic. We also give proofs to real-valued cellular systems which have fixed points and periodic solutions.

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