scholarly journals A two-dimensional slice through the parameter space of two-generator Kleinian groups

2018 ◽  
Vol 28 (08) ◽  
pp. 1535-1564
Author(s):  
Elena Klimenko ◽  
Natalia Kopteva
Keyword(s):  
Parameter Space ◽  
Kleinian Groups ◽  
Two Dimensional ◽  
Real Trace ◽  
Discreteness Criteria

We describe all real points of the parameter space of two-generator Kleinian groups with a parabolic generator, that is, we describe a certain two-dimensional slice through this space. In order to do this, we gather together known discreteness criteria for two-generator groups with a parabolic generator and present them in the form of conditions on parameters. We complete the description by giving discreteness criteria for groups generated by a parabolic and a [Formula: see text]-loxodromic elements whose commutator has real trace and present all orbifolds uniformized by such groups.

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.

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.

RESONANT GLUING BIFURCATIONS

2000 ◽  
Vol 10 (09) ◽  
pp. 2141-2160 ◽  
Author(s):  
ROBERT W. GHRIST
Keyword(s):  
Saddle Point ◽  
Parameter Space ◽  
Two Dimensional ◽  
Bifurcation Structure ◽  
Centre Manifold ◽  
Principal Eigenvalues ◽  
Homoclinic Connections ◽  
Homoclinic Bifurcations ◽  
Zero Entropy

We consider the codimension-three phenomenon of homoclinic bifurcations of flows containing a pair of orbits homoclinic to a saddle point whose principal eigenvalues are in resonance. We concentrate upon the simplest possible configuration, the so-called "figure-of-eight," and reduce the dynamics near the homoclinic connections to those on a two-dimensional locally invariant centre manifold. The ensuing resonant gluing bifurcations exhibit features of both gluing bifurcations and resonant homoclinic bifurcations. Under certain twist conditions, the bifurcation structure is extremely rich, although describing zero-entropy flows. The analysis carefully exploits the topology of the orbits, the centre manifold and the parameter space.

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.

COMMON DYNAMICAL FEATURES OF PERIODICALLY DRIVEN STRICTLY DISSIPATIVE OSCILLATORS

1993 ◽  
Vol 03 (03) ◽  
pp. 703-715 ◽  
Author(s):  
ULRICH PARLITZ
Keyword(s):  
Periodic Orbits ◽  
Parameter Space ◽  
Nonlinear Oscillators ◽  
Two Dimensional ◽  
Topological Properties ◽  
Basic Pattern ◽  
Bifurcation Structure ◽  
Periodically Driven ◽  
Dynamical Features ◽  
The Way

Periodically driven strictly dissipative nonlinear oscillators in general possess a recurring bifurcation structure in parameter space. It consists of slightly modified versions of a basic pattern of bifurcation curves that was found to be essentially the same for many different oscillators. The periodic orbits involved in these bifurcation scenarios also possess common topological properties characterized in terms of their torsion numbers and the way they are connected when parameters are varied. In this paper, this typical bifurcation structure of periodically driven strictly dissipative oscillators will be presented and discussed in terms of examples from Duffing’s equation. Furthermore a family of two-dimensional maps is given that models (strictly) dissipative oscillators and shows essential features of the bifurcation pattern found.

scholarly journals Analysis of a Class of Low-Dimensional Models of Mutation and Predation

2016 ◽  
Vol 26 (11) ◽  
pp. 1630029 ◽  
Author(s):  
Gavin M. Abernethy ◽  
Mark McCartney
Keyword(s):  
Dynamic Behavior ◽  
Lyapunov Exponents ◽  
Parameter Space ◽  
Numerical Results ◽  
Discrete Models ◽  
Two Dimensional ◽  
Dimensional Models ◽  
Low Dimensional ◽  
Selection Of

We consider a class of simple two-dimensional discrete models representative of a system incorporating both mutation and predation. A selection of analytic and numerical results are presented, classifying the dynamic behavior of the system by means of Lyapunov exponents over a biologically-reasonable region of parameter space, and illustrating the occurrence of hyperchaos and a Neimark–Sacker bifurcation producing regions of quasiperiodicity.

scholarly journals Two-dimensional Veronese groups with an invariant ball

2017 ◽  
Vol 28 (10) ◽  
Author(s):  
Angel Cano ◽  
Luis Loeza
Keyword(s):  
Surface Group ◽  
Compact Surface ◽  
Kleinian Groups ◽  
Hyperbolic Groups ◽  
Two Dimensional

In this paper, we characterize the complex hyperbolic groups that leave invariant a copy of the Veronese curve in [Formula: see text]. As a corollary we get that every discrete compact surface group in [Formula: see text] admits a deformation in [Formula: see text] with a nonempty region of discontinuity which is not conjugate to a complex hyperbolic subgroup. This provides a way to construct new examples of Kleinian groups acting on [Formula: see text].

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.

scholarly journals A Diagrammatic Representation of Phase Portraits and Bifurcation Diagrams of Two-Dimensional Dynamical Systems

2017 ◽  
Vol 27 (13) ◽  
pp. 1730045
Author(s):  
Javier Roulet ◽  
Gabriel B. Mindlin
Keyword(s):  
Dynamical Systems ◽  
Parameter Space ◽  
Partial Information ◽  
Quantitative Information ◽  
Phase Portraits ◽  
Two Dimensional ◽  
Bifurcation Diagrams ◽  
Diagrammatic Representation ◽  
Topological Features

We treat the problem of characterizing in a systematic way the qualitative features of two-dimensional dynamical systems. To that end, we construct a representation of the topological features of phase portraits by means of diagrams that discard their quantitative information. All codimension 1 bifurcations are naturally embodied in the possible ways of transitioning smoothly between diagrams. We introduce a representation of bifurcation curves in parameter space that guides the proposition of bifurcation diagrams compatible with partial information about the system.

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