TERBIUM MOLYBDATE: GROUP THEORY AND FLUCTUATIONS

1987 ◽  
Vol 01 (05n06) ◽  
pp. 239-244
Author(s):  
SERGE GALAM
Keyword(s):  
Fixed Point ◽  
Group Theory ◽  
Parameter Space ◽  
Landau Theory ◽  
Two Dimensional ◽  
Stable Fixed Point ◽  
Theory Analysis ◽  
Continuous Transition ◽  
Dimensional Parameter ◽  
First Order

A new mechanism to explain the first order ferroelastic—ferroelectric transition in Terbium Molybdate (TMO) is presented. From group theory analysis it is shown that in the two-dimensional parameter space ordering along either an axis or a diagonal is forbidden. These symmetry-imposed singularities are found to make the unique stable fixed point not accessible for TMO. A continuous transition even if allowed within Landau theory is thus impossible once fluctuations are included. The TMO transition is therefore always first order. This explanation is supported by experimental results.

Quasiperiodicity and Chaos in a Generalized Nosé–Hoover Oscillator

2016 ◽  
Vol 26 (10) ◽  
pp. 1650170 ◽  
Author(s):  
Paulo C. Rech
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Harmonic Oscillator ◽  
Parameter Space ◽  
Nonlinear Ordinary Differential Equations ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
First Order ◽  
Chaotic Region ◽  
Quasiperiodic Structures

This paper reports on an investigation of the two-dimensional parameter-space of a generalized Nosé–Hoover oscillator. It is a mathematical model of a thermostated harmonic oscillator, which consists of a set of three autonomous first-order nonlinear ordinary differential equations. By using Lyapunov exponents to numerically characterize the dynamics of the model at each point of this parameter-space, it is shown that dissipative quasiperiodic structures are present, embedded in a chaotic region. The same parameter-space is also used to confirm the multistability phenomenon in the investigated mathematical model.

COMPLEX BIFURCATION STRUCTURES IN THE HINDMARSH–ROSE NEURON MODEL

2007 ◽  
Vol 17 (09) ◽  
pp. 3071-3083 ◽  
Author(s):  
J. M. GONZÀLEZ-MIRANDA
Keyword(s):  
Phase Diagram ◽  
Bifurcation Diagram ◽  
Parameter Space ◽  
Neuron Model ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Rapid Responses ◽  
Bifurcation Structures

The results of a study of the bifurcation diagram of the Hindmarsh–Rose neuron model in a two-dimensional parameter space are reported. This diagram shows the existence and extent of complex bifurcation structures that might be useful to understand the mechanisms used by the neurons to encode information and give rapid responses to stimulus. Moreover, the information contained in this phase diagram provides a background to develop our understanding of the dynamics of interacting neurons.

scholarly journals Chaotic Itinerancy Observed in Mutually Coupled Gaussian Maps

2018 ◽  
Vol 28 (04) ◽  
pp. 1830011
Author(s):  
Mio Kobayashi ◽  
Tetsuya Yoshinaga
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Chaotic Behavior ◽  
Periodic Points ◽  
Two Dimensional ◽  
Stable Fixed Point ◽  
Bifurcation Structure ◽  
Unstable Fixed Point ◽  
Gaussian Map ◽  
Gaussian Maps

A one-dimensional Gaussian map defined by a Gaussian function describes a discrete-time dynamical system. Chaotic behavior can be observed in both Gaussian and logistic maps. This study analyzes the bifurcation structure corresponding to the fixed and periodic points of a coupled system comprising two Gaussian maps. The bifurcation structure of a mutually coupled Gaussian map is more complex than that of a mutually coupled logistic map. In a coupled Gaussian map, it was confirmed that after a stable fixed point or stable periodic points became unstable through the bifurcation, the points were able to recover their stability while the system parameters were changing. Moreover, we investigated a parameter region in which symmetric and asymmetric stable fixed points coexisted. Asymmetric unstable fixed point was generated by the [Formula: see text]-type branching of a symmetric stable fixed point. The stability of the unstable fixed point could be recovered through period-doubling and tangent bifurcations. Furthermore, a homoclinic structure related to the occurrence of chaotic behavior and invariant closed curves caused by two-periodic points was observed. The mutually coupled Gaussian map was merely a two-dimensional dynamical system; however, chaotic itinerancy, known to be a characteristic property associated with high-dimensional dynamical systems, was observed. The bifurcation structure of the mutually coupled Gaussian map clearly elucidates the mechanism of chaotic itinerancy generation in the two-dimensional coupled map. We discussed this mechanism by comparing the bifurcation structures of the Gaussian and logistic maps.

Effects of DC electric fields on neuronal excitability: A bifurcation analysis

2014 ◽  
Vol 28 (18) ◽  
pp. 1450114 ◽  
Author(s):  
Yanqiu Che ◽  
Huiyan Li ◽  
Chunxiao Han ◽  
Xile Wei ◽  
Bin Deng ◽  
...  
Keyword(s):  
Electric Field ◽  
Electric Fields ◽  
Parameter Space ◽  
Bifurcation Analysis ◽  
Neural Control ◽  
Neuronal Excitability ◽  
Modulation Method ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Dc Electric Fields

In this paper, the effects of external DC electric fields on the neuro-computational properties are investigated in the context of Morris–Lecar (ML) model with bifurcation analysis. We obtain the detailed bifurcation diagram in two-dimensional parameter space of externally applied DC current and trans-membrane potential induced by external DC electric field. The bifurcation sets partition the two-dimensional parameter space in terms of the qualitatively different behaviors of the ML model. Thus the neuron's information encodes the stimulus information, and vice versa, which is significant in neural control. Furthermore, we identify the electric field as a key parameter to control the transitions among four different excitability and spiking properties, which facilitates the design of electric fields based neuronal modulation method.

DIFFERENT BIFURCATION SCENARIOS OF NEURAL FIRING PATTERNS OBSERVED IN THE BIOLOGICAL EXPERIMENT ON IDENTICAL PACEMAKERS

2013 ◽  
Vol 23 (12) ◽  
pp. 1350195 ◽  
Author(s):  
HUAGUANG GU
Keyword(s):  
Parameter Space ◽  
Period Doubling ◽  
Two Dimensional ◽  
Neural Firing ◽  
Biological Experiment ◽  
Dimensional Parameter ◽  
Simple Process ◽  
Firing Patterns ◽  
Complex Process ◽  
Different Levels

Two different bifurcation scenarios of spontaneous neural firing patterns with decreasing extracellular calcium concentrations were observed in the biological experiment on identical pacemakers when potassium concentrations were fixed at two different levels. Six typical experimental scenarios manifesting dynamics closely matching those previously simulated using the Hindmarsh–Rose model and Chay model are provided as representative examples. Bifurcation scenarios from period-1 bursting to period-1 spiking via a complex process and via a simple process, period-doubling bifurcation to chaos, period-adding bifurcation with chaos, and period-adding bifurcation with stochastic burstings were identified. The results not only reveal that an experimental neural pacemaker is capable of generating different bifurcation scenarios but also provide a basic framework for bifurcations in neural firing patterns in a two-dimensional parameter space.

Controlling Hidden Dynamics and Multistability of a Class of Two-Dimensional Maps via Linear Augmentation

2021 ◽  
Vol 31 (03) ◽  
pp. 2150047
Author(s):  
Liping Zhang ◽  
Haibo Jiang ◽  
Yang Liu ◽  
Zhouchao Wei ◽  
Qinsheng Bi
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Numerical Simulations ◽  
Coupling Strength ◽  
Complex Dynamics ◽  
Discrete Dynamical Systems ◽  
Two Dimensional ◽  
Stable Fixed Point ◽  
Hidden Attractor ◽  
Hidden Attractors

This paper reports the complex dynamics of a class of two-dimensional maps containing hidden attractors via linear augmentation. Firstly, the method of linear augmentation for continuous dynamical systems is generalized to discrete dynamical systems. Then three cases of a class of two-dimensional maps that exhibit hidden dynamics, the maps with no fixed point and the maps with one stable fixed point, are studied. Our numerical simulations show the effectiveness of the linear augmentation method. As the coupling strength of the controller increases or decreases, hidden attractor can be annihilated or altered to be self-excited, and multistability of the map can be controlled to being bistable or monostable.

ORIGIN OF CHAOS IN A TWO-DIMENSIONAL MAP MODELING SPIKING-BURSTING NEURAL ACTIVITY

2003 ◽  
Vol 13 (11) ◽  
pp. 3325-3340 ◽  
Author(s):  
ANDREY L. SHILNIKOV ◽  
NIKOLAI F. RULKOV
Keyword(s):  
Fixed Point ◽  
Limit Cycle ◽  
Neural Activity ◽  
Slow Variable ◽  
Two Dimensional ◽  
Stable Fixed Point ◽  
Fast Subsystem ◽  
Slow Subsystem ◽  
Bursting Activity

Origin of chaos in a simple two-dimensional map model replicating the spiking and spiking-bursting activity of real biological neurons is studied. The map contains one fast and one slow variable. Individual dynamics of a fast subsystem of the map is characterized by two types of possible attractors: stable fixed point (replicating silence) and superstable limit cycle (replicating spikes). Coupling this subsystem with the slow subsystem leads to the generation of periodic or chaotic spiking-bursting behavior. We study the bifurcation scenarios which reveal the dynamical mechanisms that lead to chaos at alternating silence and spiking phases.

Exact duality of the dissipative Hofstadter model on a triangular lattice: T-duality and noncommutative algebra

2016 ◽  
Vol 31 (27) ◽  
pp. 1650154
Author(s):  
Taejin Lee
Keyword(s):  
Parameter Space ◽  
Triangular Lattice ◽  
Uniform Magnetic Field ◽  
Quantum Wires ◽  
Temperature Limit ◽  
Two Dimensional ◽  
Zero Temperature ◽  
Dimensional Parameter ◽  
Noncommutative Algebra ◽  
Dual Transformation

We study the dissipative Hofstadter model on a triangular lattice, making use of the [Formula: see text] T-dual transformation of string theory. The [Formula: see text] dual transformation transcribes the model in a commutative basis into the model in a noncommutative basis. In the zero-temperature limit, the model exhibits an exact duality, which identifies equivalent points on the two-dimensional parameter space of the model. The exact duality also defines magic circles on the parameter space, where the model can be mapped onto the boundary sine-Gordon on a triangular lattice. The model describes the junction of three quantum wires in a uniform magnetic field background. An explicit expression of the equivalence relation, which identifies the points on the two-dimensional parameter space of the model by the exact duality, is obtained. It may help us to understand the structure of the phase diagram of the model.

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