THE ABSENCE OF NONCLOSED POISSON-STABLE SEMITRAJECTORIES AND TRAJECTORIES DOUBLY ASYMPTOTIC TO A DOUBLE LIMIT CYCLE FOR DYNAMICAL SYSTEMS OF THE FIRST DEGREE OF STRUCTURAL INSTABILITY ON ORIENTABLE TWO-DIMENSIONAL MANIFOLDS

1968 ◽  
Vol 5 (2) ◽  
pp. 205-219 ◽  
Author(s):  
S H Aranson
Keyword(s):  
Dynamical Systems ◽  
Limit Cycle ◽  
Structural Instability ◽  
Two Dimensional ◽  
Double Limit

scholarly journals On resonances under quasi-periodic perturbations of systems with a double limit cycle, close to two-dimensional nonlinear Hamiltonian systems

2021 ◽  
Vol 23 (1) ◽  
pp. 11-27
Author(s):  
Olga S. Kostromina
Keyword(s):  
Numerical Calculation ◽  
Limit Cycle ◽  
Autonomous System ◽  
Theoretical Study ◽  
Type Equation ◽  
Phase Portraits ◽  
Two Dimensional ◽  
Resonant Structures ◽  
Periodic Perturbations ◽  
Double Limit

The effect of multi-frequency quasi-periodic perturbations on systems close to twodimensional nonlinear Hamiltonian ones is studied. It is assumed that the corresponding perturbed autonomous system has a double limit cycle. Analysis of the Poincar´e–Pontryagin function constructed for the autonomous system makes it possible to establish the presence of such a cycle. When the condition of commensurability of the natural frequency of the corresponding unperturbed Hamiltonian system with the frequencies of the quasi-periodic perturbation is fulfilled, the unperturbed level becomes resonant. Resonant structures essentially depend on whether the selected resonance levels coincide with the levels that generate limit cycles in the autonomous system. An averaged system is obtained that describes the topology of the neighborhoods of resonance levels. Possible phase portraits of the averaged system are established near the bifurcation case, when the resonance level coincides with the level in whose neighborhood the corresponding autonomous system has a double limit cycle. To illustrate the results obtained, the results of a theoretical study and of a numerical calculation are presented for a specific pendulum-type equation under two-frequency quasi-periodic perturbations.

scholarly journals On two-frequency quasi-periodic perturbations of systems close to two-dimensional Hamiltonian ones with a double limit cycle

2021 ◽  
Vol 31 (1) ◽  
pp. 35-49
Author(s):  
O.S. Kostromina
Keyword(s):  
Limit Cycle ◽  
Nonlinear Oscillations ◽  
Autonomous Systems ◽  
Type Equation ◽  
Unperturbed System ◽  
Two Dimensional ◽  
Periodic Perturbations ◽  
Averaged Systems ◽  
Theoretical Results ◽  
Double Limit

The problem of the effect of two-frequency quasi-periodic perturbations on systems close to arbitrary nonlinear two-dimensional Hamiltonian ones is studied in the case when the corresponding perturbed autonomous systems have a double limit cycle. Its solution is important both for the theory of synchronization of nonlinear oscillations and for the theory of bifurcations of dynamical systems. In the case of commensurability of the natural frequency of the unperturbed system with frequencies of quasi-periodic perturbation, resonance occurs. Averaged systems are derived that make it possible to ascertain the structure of the resonance zone, that is, to describe the behavior of solutions in the neighborhood of individual resonance levels. The study of these systems allows determining possible bifurcations arising when the resonance level deviates from the level of the unperturbed system, which generates a double limit cycle in a perturbed autonomous system. The theoretical results obtained are applied in the study of a two-frequency quasi-periodic perturbed pendulum-type equation and are illustrated by numerical computations.

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.

scholarly journals Conditional Lie-Bäcklund Symmetries and Differential Constraints of Radially Symmetric Nonlinear Convection-Diffusion Equations with Source

Entropy ◽  
2020 ◽  
Vol 22 (8) ◽  
pp. 873
Author(s):  
Lina Ji ◽  
Rui Wang
Keyword(s):  
Dynamical Systems ◽  
Diffusion Equations ◽  
Two Dimensional ◽  
Nonlinear Convection ◽  
Differential Constraint ◽  
Invariant Surface ◽  
Differential Constraints ◽  
Convection Diffusion ◽  
Constraint Method ◽  
Symmetry Method

A conditional Lie-Bäcklund symmetry method and differential constraint method are developed to study the radially symmetric nonlinear convection-diffusion equations with source. The equations and the admitted conditional Lie-Bäcklund symmetries (differential constraints) are identified. As a consequence, symmetry reductions to two-dimensional dynamical systems of the resulting equations are derived due to the compatibility of the original equation and the additional differential constraint corresponding to the invariant surface equation of the admitted conditional Lie-Bäcklund symmetry.

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