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.

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.

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.

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.

A STUDY OF BIFURCATIONS IN A CIRCULAR REAL CELLULAR AUTOMATON

1993 ◽  
Vol 03 (02) ◽  
pp. 293-321 ◽  
Author(s):  
JÜRGEN WEITKÄMPER
Keyword(s):  
Hopf Bifurcation ◽  
Dynamical Systems ◽  
Cellular Automata ◽  
Cellular Automaton ◽  
Parallel Computers ◽  
Discrete Dynamical Systems ◽  
Two Dimensional ◽  
Bifurcation Structure ◽  
Logistic Mapping ◽  
Henon Mapping

Real cellular automata (RCA) are time-discrete dynamical systems on ℝN. Like cellular automata they can be obtained from discretizing partial differential equations. Due to their structure RCA are ideally suited to implementation on parallel computers with a large number of processors. In a way similar to the Hénon mapping, the system we consider here embeds the logistic mapping in a system on ℝN, N>1. But in contrast to the Hénon system an RCA in general is not invertible. We present some results about the bifurcation structure of such systems, mostly restricting ourselves, due to the complexity of the problem, to the two-dimensional case. Among others we observe cascades of cusp bifurcations forming generalized crossroad areas and crossroad areas with the flip curves replaced by Hopf bifurcation curves.

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