ON MULTICORNS AND UNICORNS I: ANTIHOLOMORPHIC DYNAMICS, HYPERBOLIC COMPONENTS AND REAL CUBIC POLYNOMIALS

2003 ◽  
Vol 13 (10) ◽  
pp. 2825-2844 ◽  
Author(s):  
SHIZUO NAKANE ◽  
DIERK SCHLEICHER
Keyword(s):  
Parameter Space ◽  
Bifurcation Diagrams ◽  
The Family ◽  
Real Analytic ◽  
Cubic Polynomials

We investigate the dynamics and the bifurcation diagrams of iterated antiholomorphic polynomials: These are complex conjugates of ordinary polynomials. Their second iterates are holomorphic polynomials, but dependence on parameters is only real-analytic. The structure of hyperbolic components of the family of unicritical antiholomorphic polynomials is revealed. In case of degree two, they arise naturally in the parameter space of real cubic (holomorphic) polynomials, which we investigate as well.

scholarly journals The parameter space of cubic laminations with a fixed critical leaf

2016 ◽  
Vol 37 (8) ◽  
pp. 2453-2486
Author(s):  
ALEXANDER BLOKH ◽  
LEX OVERSTEEGEN ◽  
ROSS PTACEK ◽  
VLADLEN TIMORIN
Keyword(s):  
Parameter Space ◽  
Mandelbrot Set ◽  
Quadratic Invariant ◽  
The Family ◽  
Combinatorial Model ◽  
Cubic Polynomials ◽  
Quadratic Case

Thurston parameterized quadratic invariant laminations with a non-invariant lamination, the quotient of which yields a combinatorial model for the Mandelbrot set. As a step toward generalizing this construction to cubic polynomials, we consider slices of the family of cubic invariant laminations defined by a fixed critical leaf with non-periodic endpoints. We parameterize each slice by a lamination just as in the quadratic case, relying on the techniques of smart criticality previously developed by the authors.

scholarly journals Bifurcations of Oscillatory and Rotational Solutions of Double Pendulum with Parametric Vertical Excitation

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Michal Marszal ◽  
Krzysztof Jankowski ◽  
Przemyslaw Perlikowski ◽  
Tomasz Kapitaniak
Keyword(s):  
Periodic Solutions ◽  
Parameter Space ◽  
Double Pendulum ◽  
Bifurcation Diagrams ◽  
Kinematic Excitation ◽  
Vertical Excitation ◽  
Two Parameter ◽  
Domain Of Periodic Solutions

This paper investigates dynamics of double pendulum subjected to vertical parametric kinematic excitation. It includes detailed bifurcation diagrams in two-parameter space (excitation’s frequency and amplitude) for both oscillations and rotations in the domain of periodic solutions.

Bifurcation Diagrams and Quotient Topological Spaces Under the Action of the Affine Group of a Family of Planar Quadratic Vector Fields

2015 ◽  
Vol 25 (11) ◽  
pp. 1550150 ◽  
Author(s):  
Oxana Cerba Diaconescu ◽  
Dana Schlomiuk ◽  
Nicolae Vulpe
Keyword(s):  
Parameter Space ◽  
Group Action ◽  
Sufficient Conditions ◽  
Phase Portraits ◽  
Topological Spaces ◽  
Canonical Forms ◽  
Affine Transformations ◽  
Bifurcation Diagrams ◽  
Dimensional Parameter ◽  
Quotient Spaces

In this article, we consider the class [Formula: see text] of all real quadratic differential systems [Formula: see text], [Formula: see text] with gcd (p, q) = 1, having invariant lines of total multiplicity four and two complex and one real infinite singularities. We first construct compactified canonical forms for the class [Formula: see text] so as to include limit points in the 12-dimensional parameter space of this class. We next construct the bifurcation diagrams for these compactified canonical forms. These diagrams contain many repetitions of phase portraits and we show that these are due to many symmetries under the group action. To retain the essence of the dynamics we finally construct the quotient spaces under the action of the group G = Aff(2, ℝ) × ℝ* of affine transformations and time homotheties and we place the phase portraits in these quotient spaces. The final diagrams retain only the necessary information to capture the dynamics under the motion in the parameter space as well as under this group action. We also present here necessary and sufficient conditions for an affine line to be invariant of multiplicity k for a quadratic system.

Double points in families of map germs from ℝ2 to ℝ3

2020 ◽  
pp. 1-18
Author(s):  
J. A. Moya-Pérez ◽  
J. J. Nuño-Ballesteros
Keyword(s):  
Double Point ◽  
Parameter Family ◽  
Milnor Number ◽  
Double Points ◽  
The Family ◽  
Number Formula ◽  
Real Analytic

We show that a 1-parameter family of real analytic map germs [Formula: see text] with isolated instability is topologically trivial if it is excellent and the family of double point curves [Formula: see text] in [Formula: see text] is topologically trivial. In particular, we deduce that [Formula: see text] is topologically trivial when the Milnor number [Formula: see text] is constant.

BUILDING ELECTRONIC BURSTERS WITH THE MORRIS–LECAR NEURON MODEL

2006 ◽  
Vol 16 (12) ◽  
pp. 3617-3630 ◽  
Author(s):  
ALEXANDRE WAGEMAKERS ◽  
MIGUEL A. F. SANJUÁN ◽  
JOSÉ M. CASADO ◽  
KAZUYUKI AIHARA
Keyword(s):  
Phase Space ◽  
Parameter Space ◽  
Neuron Model ◽  
Electronic Circuit ◽  
Initial Point ◽  
Bifurcation Diagrams ◽  
Design And Implementation ◽  
Bursting Neurons ◽  
Fast Spiking ◽  
Dynamical Features

We propose a method for the design of electronic bursting neurons, based on a simple conductance neuron model. A burster is a particular class of neuron that displays fast spiking regimes alternating with resting periods. Our method is based on the use of an electronic circuit that implements the well-known Morris–Lecar neuron model. We use this circuit as a tool of analysis to explore some regions of the parameter space and to contruct several bifurcation diagrams displaying the basic dynamical features of that system. These bifurcation diagrams provide the initial point for the design and implementation of electronic bursting neurons. By extending the phase space with the introduction of a slow driving current, our method allows to exploit the bistabilities which are present in the Morris–Lecar system to the building of different bursting models.

scholarly journals The iteration of cubic polynomials Part I: The global topology of parameter space

1988 ◽  
Vol 160 (0) ◽  
pp. 143-206 ◽  
Author(s):  
Bodil Branner ◽  
John H. Hubbard
Keyword(s):  
Parameter Space ◽  
Global Topology ◽  
Cubic Polynomials

scholarly journals Slices of the parameter space of cubic polynomials

2021 ◽  
Author(s):  
Alexander Blokh ◽  
Lex Oversteegen ◽  
Vladlen Timorin
Keyword(s):  
Parameter Space ◽  
Cubic Polynomials

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.

THE PARAMETER SPACE OF A QUADRATIC FAMILY OF POLYNOMIAL MAPS OF ℂ2

2008 ◽  
Vol 19 (05) ◽  
pp. 541-556
Author(s):  
ALINA ANDREI
Keyword(s):  
Lyapunov Exponents ◽  
Parameter Space ◽  
Mandelbrot Set ◽  
General Context ◽  
One Dimensional ◽  
Polynomial Maps ◽  
Regular Maps ◽  
The Family ◽  
The One ◽  
Polynomial Family

In this paper, we study the parameter space of the quadratic polynomial family fλ,μ(z, w) = (λz + w2, μw + z2), which exhibits interesting dynamics. Two distinct subsets of the parameter space are studied as appropriate analogs of the one-dimensional Mandelbrot set and some of their properties are proved by using Lyapunov exponents. In the more general context of holomorphic families of regular maps, we show that the sum of the Lyapunov exponents is a plurisubharmonic function of the parameter, and pluriharmonic on the set of expanding maps. Moreover, for the family fλ,μ, we prove that the sum of the Lyapunov exponents is continuous.

Sign in / Sign up

Export Citation Format

Share Document