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.

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.

DYNAMICAL SYSTEMS OF DIFFERENT CLASSES AS MODELS OF THE KICKED NONLINEAR OSCILLATOR

2001 ◽  
Vol 11 (04) ◽  
pp. 1065-1077 ◽  
Author(s):  
A. P. KUZNETSOV ◽  
L. V. TURUKINA ◽  
E. MOSEKILDE
Keyword(s):  
Dynamical Systems ◽  
Parameter Space ◽  
Nonlinear Oscillator ◽  
Period Doubling ◽  
Bifurcation Diagrams ◽  
Scaling Properties ◽  
Critical Curves

Using the nonlinear dissipative kicked oscillator as an example, the correspondence between the descriptions provided by model dynamical systems of different classes is discussed. A detailed study of the approximate 1D map is undertaken: the period doubling is examined and the possibility of non-Feigenbaum period doubling is shown. Illustrations in the form of bifurcation diagrams and sets of iteration diagrams are given, the scaling properties are demonstrated, and the tricritical points (the terminal points of the Feigenbaum critical curves) in parameter space are found. The congruity with the properties of the corresponding 2D map, the Ikeda map, is studied. A description in terms of tricritical dynamics is found to be adequate only in particular areas of parameter space.

BIANCHI COSMOLOGIES AS DYNAMICAL SYSTEMS WITHOUT THE CONSTRAINT

2000 ◽  
Vol 15 (17) ◽  
pp. 2771-2791
Author(s):  
MAREK SZYDŁOWSKI ◽  
ADAM KRAWIEC
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Phase Portraits ◽  
Finite Domain ◽  
Hamiltonian Constraint ◽  
Two Dimensional ◽  
Class A ◽  
Dimensional Dynamical System ◽  
Analysis Of Dynamics

The Bianchi class A cosmology is treated as a nonlinear dynamical system. In the new variables in which Hamiltonian constraint is solved algebraically, the Bianchi class A model assumes the form of autonomous dynamical system in ℝ4 with polynomial form of vector field. It is proposed that the dimension of minimum reduced phase spaces of unconstrained autonomous systems be treated as a measure of generality of solution. The behavior of these models is studied in terms of qualitative analysis of differential equations. It is shown that the more general Bianchi IX and Bianchi VIII models (called Mixmaster models) can be presented as four-dimensional. We argue that the reduced Mixmaster dynamical systems are chaotic in the same sense as the original ones. The Bianchi I and Bianchi II world models are described by one-dimensional and two-dimensional systems, respectively. We also study dynamics of Bianchi VI0 and Bianchi VII0 models as a three-dimensional dynamical system. For two-dimensional dynamical system, the phase portraits are constructed with the Poincaré sphere which allows the analysis of dynamics both in finite domain and at infinity. For the last class of models we find an invariant submanifold on which systems are analyzed in details.

NONINVERTIBLE TWO-DIMENSIONAL MAPS ARISING IN RADIOPHYSICS

1994 ◽  
Vol 04 (02) ◽  
pp. 383-400 ◽  
Author(s):  
VLADIMIR MAISTRENKO ◽  
YURI MAISTRENKO ◽  
IRINA SUSHKO
Keyword(s):  
Period Doubling ◽  
Phase Portraits ◽  
Two Dimensional ◽  
Bifurcation Diagrams ◽  
One Dimensional ◽  
Transition To Chaos ◽  
Nonlinear Amplifiers ◽  
Noninvertible Maps ◽  
Triangular Maps ◽  
Two Parameter

We study a two-parameter family of noninvertible maps modeling a generator which consists of two identical nonlinear amplifiers and two delay circuits. The ratio of the delays determines the dimension of the map and our attention is mainly on the two-dimensional case. The mechanism of transition to chaos appears to be one-dimensional and is realized through a period-doubling cascade. To get a more complete description we suggest the use of so-called triangular maps. Phase portraits are constructed for some types of model triangular maps. Also we get one- and two-dimensional bifurcation diagrams for the maps considered and attractor basins in the case of multistability using computer simulation.

A method for the complete qualitative analysis of two coupled ordinary differential equations dependent on three parameters

1988 ◽  
Vol 416 (1851) ◽  
pp. 361-389 ◽  
Keyword(s):  
Differential Equations ◽  
Qualitative Analysis ◽  
Ordinary Differential Equations ◽  
Parameter Space ◽  
Bifurcation Theory ◽  
Parameter Plane ◽  
Phase Portraits ◽  
Relevant Parameter ◽  
Bifurcation Diagrams ◽  
Independent Parameters

In this paper, recent advances in bifurcation theory are specialized to systems describable by two coupled ordinary differential equations (ODEs) containing at most three independent parameters. For such systems, by plotting in the relevant parameter plane the locus of successively degenerate singular points, a complete description of all the qualitatively distinct behaviour of the system can be obtained. The description is in terms of phase portraits and bifurcation diagrams. Even though much use is made of existing results obtained via local analyses, the results of this technique cover the entire parameter space. Furthermore, because the information is built up in successive stages the question of whether the parameters universally unfold a given degeneracy does not arise. This can mean a major saving in effort, particularly for degenerate Hopf points. Finally if, as is often the case, the parameters appear in the system in a simple way, the procedure can be applied analytically because the variables (which will appear non-linearly) can be used to parametrize the relevant loci.

Analysis of chemical kinetic systems over the entire parameter space - II. Isothermal oscillators

1988 ◽  
Vol 416 (1851) ◽  
pp. 403-424 ◽  
Keyword(s):  
Parameter Space ◽  
Limit Cycles ◽  
Chemical Kinetic ◽  
Steady States ◽  
Phase Portraits ◽  
Quadratic Model ◽  
Bifurcation Diagrams ◽  
Complex Phase ◽  
Isothermal Kinetic ◽  
Oxidation Of Carbon Monoxide

Two simple but similar isothermal kinetic schemes are studied by using recently developed qualitative techniques with the aim of finding all of the qualitatively distinct behaviour that the system can exhibit. The first scheme, which can be regarded as one of the simplest isothermal models of interest, is based on the ‘quadratic’ autocatalytic step A + B → 2B rate = k q ab , where a and b are the concentrations of A and B respectively, combined with a nonlinear termination step of the form B → C rate = k t b /(1+ rb ), where r is some parameter. The second scheme is a simplified (by treating one parameter as a constant) version of a model that has been proposed for the oxidation of carbon monoxide (CO). This scheme has a termination step of the same form as above, and like the ‘quadratic’ model, the highest nonlinearity apart from this term is quadratic. For the quadratic model we find 10 qualitatively distinct phase portraits and, with our choice of bifurcation parameter, 22 distinct bifurcation diagrams. The most complex phase portraits contain 3 steady states and 2 limit cycles or 2 steady states and 3 limit cycles. For the CO model we find 8 distinct phase portraits but leading to only 5 separate bifurcation diagrams. In both models much of the behaviour occurs over only extremely small regions in parameter space.

scholarly journals Qualitative Theory of Two-Dimensional Polynomial Dynamical Systems

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1884
Author(s):  
Yury Shestopalov ◽  
Azizaga Shakhverdiev
Keyword(s):  
Dynamical Systems ◽  
Feasible Solution ◽  
Qualitative Theory ◽  
Single Parameter ◽  
Quadratic Polynomial ◽  
Phase Portraits ◽  
Two Dimensional ◽  
Polynomial Dynamical Systems ◽  
Specific Solution ◽  
Integrable Dynamical Systems

A qualitative theory of two-dimensional quadratic-polynomial integrable dynamical systems (DSs) is constructed on the basis of a discriminant criterion elaborated in the paper. This criterion enables one to pick up a single parameter that makes it possible to identify all feasible solution classes as well as the DS critical and singular points and solutions. The integrability of the considered DS family is established. Nine specific solution classes are identified. In each class, clear types of symmetry are determined and visualized and it is discussed how transformations between the solution classes create new types of symmetries. Visualization is performed as series of phase portraits revealing all possible catastrophic scenarios that result from the transition between the solution classes.

BIFURCATION ANALYSIS IN A PWM CURRENT-CONTROLLED H-BRIDGE INVERTER

2011 ◽  
Vol 21 (03) ◽  
pp. 985-996 ◽  
Author(s):  
HIROYUKI ASAHARA ◽  
TAKUJI KOUSAKA
Keyword(s):  
Dynamical System ◽  
Parameter Space ◽  
Bifurcation Analysis ◽  
Specific Property ◽  
Two Dimensional ◽  
Bifurcation Diagrams ◽  
Rigorous Analysis ◽  
Bifurcation Phenomena ◽  
The One ◽  
Switched Dynamical System

This paper introduces the complete bifurcation analysis in a PWM current-controlled H-Bridge inverter in a wide parameter space. First, we briefly explain the behavior of the waveform in the circuit in terms of the switched dynamical system. Then, the consecutive waveform during the duration of the clock interval is exactly discretized, and the return map is defined for the rigorous analysis. Using the map, we derive the one- and two-dimensional bifurcation diagrams, and discuss the specific property of each bifurcation phenomena in the circuit.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

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