THE FLUX CREEP AUTOMATON

2004 ◽  
Vol 14 (03) ◽  
pp. 1155-1175 ◽  
Author(s):  
JOHN F. LINDNER ◽  
SCOTT B. HUGHES ◽  
DAVID J. MILLER ◽  
BRADLEY C. THOMAS ◽  
KURT WIESENFELD
Keyword(s):  
Magnetic Field ◽  
Magnetic Flux ◽  
Differential Equations ◽  
Cellular Automaton ◽  
Ordinary Differential Equations ◽  
Flux Creep ◽  
Square Lattice ◽  
Discrete Dynamics ◽  
Two Dimensional ◽  
Granular Superconductors

We study a cellular automaton derived from the phenomenon of magnetic flux creep in two-dimensional granular superconductors. We model the superconductor as an array of Josephson junctions evolving according to a set of coupled ordinary differential equations. In the limit of slowly increasing magnetic field, we reduce these equations to a simple cellular automaton. The resulting discrete dynamics, a stylized version of the continuous dynamics of the differential equations, is equivalent to the dynamics of a gradient sand pile automaton. We study the dynamics as we vary the symmetry of the underlying lattice and the shape of its boundary. We find that the "simplest" realization of the automaton, on a square lattice with commensurate boundaries, results in especially simple dynamics, while "generic" realizations exhibit more complicated dynamics characterized by statistics with broad distributions, even in the absence of noise or disorder.

scholarly journals Chaos in models of double convection

1992 ◽  
Vol 237 ◽  
pp. 209-229 ◽  
Author(s):  
A. M. Rucklidge
Keyword(s):  
Magnetic Field ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fluid Layer ◽  
Period Doubling ◽  
Thermosolutal Convection ◽  
Two Dimensional ◽  
Lorenz Equations ◽  
Third Order ◽  
Horizontal Magnetic Field

In certain parameter regimes, it is possible to derive third-order sets of ordinary differential equations that are asymptotically exact descriptions of weakly nonlinear double convection and that exhibit chaotic behaviour. This paper presents a unified approach to deriving such models for two-dimensional convection in a horizontal layer of Boussinesq fluid with lateral constraints. Four situations are considered: thermosolutal convection, convection in an imposed vertical or horizontal magnetic field, and convection in a fluid layer rotating uniformly about a vertical axis. Thermosolutal convection and convection in an imposed horizontal magnetic field are shown here to be governed by the same sets of model equations, which exhibit the period-doubling cascades and chaotic solutions that are associated with the Shil'nikov bifurcation (Proctor & Weiss 1990). This establishes, for the first time, the existence of chaotic solutions of the equations governing two-dimensional magneto-convection. Moreover, in the limit of tall thin rolls, convection in an imposed vertical magnetic field and convection in a rotating fluid layer are both modelled by a new third-order set of ordinary differential equations, which is shown here to have chaotic solutions that are created in a homoclinic explosion, in the same manner as the chaotic solutions of the Lorenz equations. Unlike the Lorenz equations, however, this model provides an accurate description of convection in the parameter regime where the chaotic solutions appear.

scholarly journals Cell-to-cell Mathematical modeling of arrhythmia phenomena in excitable media

2019 ◽  
Author(s):  
Gabriel López Garza
Keyword(s):  
Mathematical Modeling ◽  
Differential Equations ◽  
Cellular Automaton ◽  
Ordinary Differential Equations ◽  
Excitable Media ◽  
Similar Distribution ◽  
Two Dimensional ◽  
Self Similar

AbstractIn this document are modeled arrhythmias with cellular automaton and ordinary differential equations systems. With an aperiodic, self-similar distribution of two-dimensional arrangement of cells, it is possible to simulate such phenomena as fibrillation, fluttering and a sequence of fibrillation-fluttering. The topology of the cytoarchitecture of a network of cells may determine the initiation and development of arrhythmias.

THE CRITICAL BEHAVIOR OF THE 2D SPIN-1 ISING MODEL WITH THE BILINEAR AND POSITIVE BIQUADRATIC NEAREST NEIGHBOR INTERACTIONS ON A CELLULAR AUTOMATON

2004 ◽  
Vol 15 (10) ◽  
pp. 1425-1438 ◽  
Author(s):  
A. SOLAK ◽  
B. KUTLU
Keyword(s):  
Cellular Automaton ◽  
Phase Boundary ◽  
Critical Behavior ◽  
Nearest Neighbor ◽  
Finite Size ◽  
Square Lattice ◽  
Two Dimensional ◽  
Finite Size Scaling ◽  
Creutz Cellular Automaton ◽  
Beg Model

The two-dimensional BEG model with nearest neighbor bilinear and positive biquadratic interaction is simulated on a cellular automaton, which is based on the Creutz cellular automaton for square lattice. Phase diagrams characterizing phase transitions of the model are presented for comparison with those obtained from other calculations. We confirm the existence of the tricritical points over the phase boundary for D/K>0. The values of static critical exponents (α, β, γ and ν) are estimated within the framework of the finite size scaling theory along D/K=-1 and 1 lines. The results are compatible with the universal Ising critical behavior except the points over phase boundary.

MHD STAGNATION POINT FLOW OF HYPERBOLIC TANGENT FLUID WITH VISCOUS DISSIPATION AND CHEMICAL REACTION

2021 ◽  
Vol 21 (2) ◽  
pp. 569-588
Author(s):  
KINZA ARSHAD ◽  
MUHAMMAD ASHRAF
Keyword(s):  
Magnetic Field ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Stagnation Point ◽  
Viscous Dissipation ◽  
Chemical Reaction ◽  
Normal Velocity ◽  
Hyperbolic Tangent ◽  
Linear Ordinary Differential Equations ◽  
Non Linear

In the present work, two dimensional flow of a hyperbolic tangent fluid with chemical reaction and viscous dissipation near a stagnation point is discussed numerically. The analysis is performed in the presence of magnetic field. The governing partial differential equations are converted into non-linear ordinary differential equations by using appropriate transformation. The resulting higher order non-linear ordinary differential equations are discretized by finite difference method and then solved by SOR (Successive over Relaxation parameter) method. The impact of the relevant parameters is scrutinized by plotting graphs and discussed in details. The main conclusion is that the large value of magnetic field parameter and wiessenberg numbers decrease the streamwise and normal velocity while increase the temperature distribution. Also higher value of the Eckert number Ec results in increases in temperature profile.

THE SIMULATION OF 2D SPIN-1 ISING MODEL WITH POSITIVE BIQUADRATIC INTERACTION ON A CELLULAR AUTOMATON

2003 ◽  
Vol 14 (10) ◽  
pp. 1305-1320 ◽  
Author(s):  
BÜLENT KUTLU
Keyword(s):  
Phase Transition ◽  
Ising Model ◽  
Cellular Automaton ◽  
Intermediate Phase ◽  
Finite Size ◽  
Square Lattice ◽  
Two Dimensional ◽  
Finite Size Scaling ◽  
Creutz Cellular Automaton ◽  
Antiferromagnetic Spin

The two-dimensional antiferromagnetic spin-1 Ising model with positive biquadratic interaction is simulated on a cellular automaton which based on the Creutz cellular automaton for square lattice. Phase diagrams characterizing phase transition of the model are presented for a comparison with those obtained from other calculations. We confirm the existence of the intermediate phase observed in previous works for some values of J/K and D/K. The values of the static critical exponents (β, γ and ν) are estimated within the framework of the finite-size scaling theory for D/K<2J/K. Although the results are compatible with the universal Ising critical behavior in the region of D/K<2J/K-4, the model does not exhibit any universal behavior in the interval 2J/K-4<D/K<2J/K.

Some Invariant Solutions of Two-Dimensional Elastodynamics in Linear Homogeneous Isotropic Materials

2013 ◽  
Vol 5 (2) ◽  
pp. 212-221
Author(s):  
Houguo Li ◽  
Kefu Huang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Theoretical Method ◽  
Two Dimensional ◽  
Invariant Solutions ◽  
Partial Differential ◽  
Isotropic Materials ◽  
Group Theoretical Method ◽  
Infinitesimal Operators

AbstractInvariant solutions of two-dimensional elastodynamics in linear homogeneous isotropic materials are considered via the group theoretical method. The second order partial differential equations of elastodynamics are reduced to ordinary differential equations under the infinitesimal operators. Three invariant solutions are constructed. Their graphical figures are presented and physical meanings are elucidated in some cases.

scholarly journals Nonlinear translational symmetric equilibria relevant to the L–H transition

2012 ◽  
Vol 79 (3) ◽  
pp. 257-265 ◽  
Author(s):  
Ap. KUIROUKIDIS ◽  
G. N. THROUMOULOPOULOS
Keyword(s):  
Magnetic Field ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Plasma Flow ◽  
Sufficient Condition ◽  
Sheared Flow ◽  
Equilibrium Configurations ◽  
Shafranov Equation ◽  
The Impact ◽  
The Magnetic Field

AbstractNonlinear z-independent solutions to a generalized Grad–Shafranov equation (GSE) with up to quartic flux terms in the free functions and incompressible plasma flow non-parallel to the magnetic field are constructed quasi-analytically. Through an ansatz, the GSE is transformed to a set of three ordinary differential equations and a constraint for three functions of the coordinate x, in Cartesian coordinates (x,y), which then are solved numerically. Equilibrium configurations for certain values of the integration constants are displayed. Examination of their characteristics in connection with the impact of nonlinearity and sheared flow indicates that these equilibria are consistent with the L–H transition phenomenology. For flows parallel to the magnetic field, one equilibrium corresponding to the H state is potentially stable in the sense that a sufficient condition for linear stability is satisfied in an appreciable part of the plasma while another solution corresponding to the L state does not satisfy the condition. The results indicate that the sheared flow in conjunction with the equilibrium nonlinearity plays a stabilizing role.

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