PERIOD-DOUBLING SCENARIO WITHOUT FLIP BIFURCATIONS IN A ONE-DIMENSIONAL MAP

2005 ◽  
Vol 15 (04) ◽  
pp. 1267-1284 ◽  
Author(s):  
V. AVRUTIN ◽  
M. SCHANZ
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Symmetry Breaking ◽  
Period Doubling ◽  
Pitchfork Bifurcation ◽  
Period Doubling Bifurcation ◽  
Coexisting Attractors ◽  
One Dimensional ◽  
Poincaré Return Map ◽  
Smooth Dynamical System

In this work a one-dimensional piecewise-smooth dynamical system, representing a Poincaré return map for dynamical systems of the Lorenz type, is investigated. The system shows a bifurcation scenario similar to the classical period-doubling one, but which is influenced by so-called border collision phenomena and denoted as border collision period-doubling bifurcation scenario. This scenario is formed by a sequence of pairs of bifurcations, whereby each pair consists of a border collision bifurcation and a pitchfork bifurcation. The mechanism leading to this scenario and its characteristic properties, like symmetry-breaking and symmetry-recovering as well as emergence of coexisting attractors, are investigated.

scholarly journals Non-existence of wandering intervals and structure of topological attractors of one dimensional dynamical systems 2. The smooth case

1989 ◽  
Vol 9 (4) ◽  
pp. 751-758 ◽  
Author(s):  
A. M. Blokh ◽  
M. Yu. Lyubich
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Critical Points ◽  
One Dimensional ◽  
Smooth Case ◽  
Smooth Dynamical System

AbstractWe prove that an arbitrary one dimensional smooth dynamical system with non-degenerate critical points has no wandering intervals.

Chaos from a hysteresis and switched circuit

1995 ◽  
Vol 353 (1701) ◽  
pp. 47-57 ◽  
Keyword(s):  
Symmetry Breaking ◽  
Piecewise Linear ◽  
Period Doubling ◽  
Analytic Formula ◽  
Period Doubling Bifurcation ◽  
Sufficient Condition ◽  
Laboratory Measurements ◽  
One Dimensional ◽  
Bifurcation Phenomena ◽  
Theoretical Results

This paper considers a simple piecewise linear hysteresis circuit. We define one-dimensional return map and derive its analytic formula. It enables us to give a sufficient condition for chaos generation and to analyse bifurcation phenomena rigorously. Especially, we have discovered period-doubling bifurcation with symmetry breaking. Some of theoretical results are verified by laboratory measurements.

THE ROLE OF CODIMENSION IN DYNAMICAL SYSTEMS

1992 ◽  
Vol 03 (06) ◽  
pp. 1295-1321 ◽  
Author(s):  
JASON A.C. GALLAS
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Period Doubling ◽  
Main Body ◽  
Hénon Map ◽  
Henon Map ◽  
Dynamical Parameters

Isoperiodic diagrams are used to investigate the topology of the codimension space of a representative dynamical system: the Hénon map. The codimension space is reported to be organized in a simple and regular way: instead of “structures-within-structures” it consists of a “structures-parallel-to-structures” sequence of shrimp-shaped isoperiodic islands immersed on a via caotica. The isoperiodic islands consist of a main body of principal periodicity k=1, 2, 3, 4, …, which bifurcates according to a period-doubling route. The Pk=k×2n, n=0, 1, 2, … shrimps are very densely concentrated along a main α-direction, a neighborhood parallel to the line b=−0.583a+1.025, where a and b are the dynamical parameters in Eq. (1). Isoperiodic diagrams allow to interpret and unify some apparently uncorrelated phenomena, such as ‘period-bubbling’, classes of reverse bifurcations and antimonotonicity and to recognize that they are in fact signatures of the complicated way in which period-doubling occurs in higher codimensional systems.

Period-1 to Period-8 Motions in a Nonlinear Jeffcott Rotor System

2020 ◽  
Author(s):  
Yeyin Xu ◽  
Albert C.J. Luo
Keyword(s):  
Period Doubling ◽  
Pitchfork Bifurcation ◽  
Mapping Method ◽  
Rotor System ◽  
Period Doubling Bifurcation ◽  
Periodic Motions ◽  
Bifurcation Tree ◽  
Rotor Systems ◽  
Period 2 ◽  
Jeffcott Rotor

Abstract In this paper, a bifurcation tree of period-1 to period-8 motions in a nonlinear Jeffcott rotor system is obtained through the discrete mapping method. The bifurcations and stability of periodic motions on the bifurcation tree are discussed. The quasi-periodic motions on the bifurcation tree are caused by two (2) Neimark bifurcations of period-1 motions, one (1) Neimark bifurcation of period-2 motions and four (4) Neimark bifurcations of period-4 motions. The specific quasi-periodic motions are mainly based on the skeleton of the corresponding periodic motions. One stable and one unstable period-doubling bifurcations exist for the period-1, period-2 and period-4 motions. The unstable period-doubling bifurcation is from an unstable period-m motion to an unstable period-2m motion, and the unstable period-m motion becomes stable. Such an unstable period-doubling bifurcation is the 3rd source pitchfork bifurcation. Periodic motions on the bifurcation tree are simulated numerically, and the corresponding harmonic amplitudes and phases are presented for harmonic effects on periodic motions in the nonlinear Jeffcott rotor system. Such a study gives a complete picture of periodic and quasi-periodic motions in the nonlinear Jeffcott rotor system in the specific parameter range. One can follow the similar procedure to work out the other bifurcation trees in the nonlinear Jeffcott rotor systems.

scholarly journals Dweiltime Analysis of Symmetry-Breaking Dynamical Systems

1995 ◽  
Vol 50 (12) ◽  
pp. 1117-1122 ◽  
Author(s):  
J. Vollmer ◽  
J. Peinke ◽  
A. Okniński
Keyword(s):  
Dynamical Systems ◽  
Symmetry Breaking ◽  
Dwell Time ◽  
Initial Conditions ◽  
Dissipative Systems ◽  
Time Analysis ◽  
Chaotic Scattering ◽  
One Dimensional ◽  
Basin Boundary ◽  
Sensitive Dependence

Abstract Dweiltime analysis is known to characterize saddles giving rise to chaotic scattering. In the present paper it is used to characterize the dependence on initial conditions of the attractor approached by a trajectory in dissipative systems described by one-dimensional, noninvertible mappings which show symmetry breaking. There may be symmetry-related attractors in these systems, and which attractor is approached may depend sensitively on the initial conditions. Dwell-time analysis is useful in this context because it allows to visualize in another way the repellers on the basin boundary which cause this sensitive dependence.

Antimonotonicity, Bifurcation and Multistability in the Vallis Model for El Niño

2019 ◽  
Vol 29 (03) ◽  
pp. 1950032 ◽  
Author(s):  
Karthikeyan Rajagopal ◽  
Sajad Jafari ◽  
Viet-Thanh Pham ◽  
Zhouchao Wei ◽  
Durairaj Premraj ◽  
...  
Keyword(s):  
Hopf Bifurcation ◽  
El Niño ◽  
El Nino ◽  
Equilibrium Points ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
Coexisting Attractors ◽  
The Stability ◽  
First Time ◽  
Parameter Condition

In this paper, the well-known Vallis model for El Niño is analyzed for the parameter condition [Formula: see text]. The conditions for the stability of the equilibrium points are derived. The condition for Hopf bifurcation occurring in the system for [Formula: see text] and [Formula: see text] are investigated. The multistability feature of the Vallis model when [Formula: see text] is explained with forward and backward continuation bifurcation plots and with the coexisting attractors. The creation of period doubling followed by their annihilation via inverse period-doubling bifurcation known as antimonotonicity occurrence in the Vallis model for [Formula: see text] is presented for the first time in the literature.

SYMMETRY AND SYNCHRONY IN COUPLED CELL NETWORKS 1: FIXED-POINT SPACES

2006 ◽  
Vol 16 (03) ◽  
pp. 559-577 ◽  
Author(s):  
FERNANDO ANTONELI ◽  
IAN STEWART
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Dynamical Systems ◽  
Symmetry Breaking ◽  
Nearest Neighbor ◽  
Invariant Subspaces ◽  
Folk Theorem ◽  
Coupled Cell Networks ◽  
Equivariant Dynamical Systems ◽  
Cell Networks

Equivariant dynamical systems possess canonical flow-invariant subspaces, the fixed-point spaces of subgroups of the symmetry group. These subspaces classify possible types of symmetry-breaking. Coupled cell networks, determined by a symmetry groupoid, also possess canonical flow-invariant subspaces, the balanced polydiagonals. These subspaces classify possible types of synchrony-breaking, and correspond to balanced colorings of the cells. A class of dynamical systems that is common to both theories comprises networks that are symmetric under the action of a group Γ of permutations of the nodes ("cells"). We investigate connections between balanced polydiagonals and fixed-point spaces for such networks, showing that in general they can be different. In particular, we consider rings of ten and twelve cells with both nearest and next-nearest neighbor coupling, showing that exotic balanced polydiagonals — ones that are not fixed-point spaces — can occur for such networks. We also prove the "folk theorem" that in any Γ-equivariant dynamical system on Rk the only flow-invariant subspaces are the fixed-point spaces of subgroups of Γ.

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