Center Bifurcation in the Lozi Map

2021 ◽  
Vol 31 (16) ◽  
Author(s):  
Iryna Sushko ◽  
Viktor Avrutin ◽  
Laura Gardini
Keyword(s):  
Fixed Point ◽  
Piecewise Linear ◽  
Parameter Plane ◽  
Bifurcation Curve ◽  
Bifurcation Structure ◽  
Piecewise Linear Map ◽  
Rigorous Description ◽  
Number Formula ◽  
Two Parameters ◽  
Parameter Values

We consider the well-known Lozi map, which is a 2D piecewise linear map depending on two parameters. This map can be considered as a piecewise linear analog of the Hénon map, allowing to simplify the rigorous proof of the existence of a chaotic attractor. The related parameter values belong to a part of the parameter plane where the map has two saddle fixed points. In the present paper, we investigate a different part of the parameter plane, namely, the vicinity of the curve related to a center bifurcation of the fixed point. A distinguishing property of the Lozi map is that it is conservative at the parameter value corresponding to this bifurcation. As a result, the bifurcation structure close to the center bifurcation curve is quite complicated. In particular, an attracting fixed point (focus) can coexist with various attracting cycles, as well as with chaotic attractors, and the number of coexisting attractors increases as the parameter point approaches the center bifurcation curve. The main result of the present paper is related to the rigorous description of this bifurcation structure. Specifically, we obtain, in explicit form, the boundaries of the main periodicity regions associated with the pairs of complementary cycles with rotation number [Formula: see text]. Similar approach can be applied to other periodicity regions. Our study contributes also to the border collision bifurcation theory since the Lozi map is a particular case of the 2D border collision normal form.

THE DOVETAIL BIFURCATION STRUCTURE AND ITS QUALITATIVE CHANGES

1993 ◽  
Vol 03 (04) ◽  
pp. 903-919 ◽  
Author(s):  
C. MIRA ◽  
H. KAWAKAMI ◽  
R. ALLAM
Keyword(s):  
Three Dimensional ◽  
Parameter Plane ◽  
Bifurcation Curve ◽  
Bifurcation Structure ◽  
Fold Bifurcation ◽  
Spring Area ◽  
Two Parameters ◽  
Qualitative Changes

In a parameter plane defined by two parameters of a map, the dovetail bifurcation structure is a bifurcation curve organization, for which two fold cusps have a fold bifurcation segment in common. This gives rise to the association of either a crossroad area with a saddle area or a spring area with a saddle area. This paper describes this structure in the parameter plane for different configurations, and in the corresponding three-dimensional representations of the plane sheets. Then it identifies six different qualitative changes of this structure when a third parameter varies.

Bifurcation Structures in a Bimodal Piecewise Linear Map: Chaotic Dynamics

2015 ◽  
Vol 25 (03) ◽  
pp. 1530006 ◽  
Author(s):  
Anastasiia Panchuk ◽  
Iryna Sushko ◽  
Viktor Avrutin
Keyword(s):  
Parameter Space ◽  
Chaotic Dynamics ◽  
Piecewise Linear ◽  
Chaotic Attractors ◽  
Bifurcation Structure ◽  
Tent Map ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
Replacement Technique ◽  
Bifurcation Structures

In this work, we investigate the bifurcation structure of the parameter space of a generic 1D continuous piecewise linear bimodal map focusing on the regions associated with chaotic attractors (cyclic chaotic intervals). The boundaries of these regions corresponding to chaotic attractors with different number of intervals are identified. The results are obtained analytically using the skew tent map and the map replacement technique.

CENTER BIFURCATION FOR TWO-DIMENSIONAL BORDER-COLLISION NORMAL FORM

2008 ◽  
Vol 18 (04) ◽  
pp. 1029-1050 ◽  
Author(s):  
IRYNA SUSHKO ◽  
LAURA GARDINI
Keyword(s):  
Piecewise Linear ◽  
Two Dimensional ◽  
Invariant Region ◽  
Piecewise Linear Map ◽  
Border Collision Bifurcations ◽  
Linear Maps ◽  
Complex Eigenvalues ◽  
Arnold Tongues ◽  
Pass Through ◽  
Parameter Values

In this work we study some properties associated with the border-collision bifurcations in a two-dimensional piecewise-linear map in canonical form, related to the case where a fixed point of one of the linear maps has complex eigenvalues and undergoes a center bifurcation when its eigenvalues pass through the unit circle. This problem is faced in several applied piecewise-smooth models, such as switching electrical circuits, impacting mechanical systems, business cycle models in economics, etc. We prove the existence of an invariant region in the phase space for parameter values related to the center bifurcation and explain the origin of a closed invariant attracting curve after the bifurcation. This problem is related also to particular border-collision bifurcations leading to such curves which may coexist with other attractors. We show how periodicity regions in the parameter space differ from Arnold tongues occurring in smooth models in case of the Neimark–Sacker bifurcation, how so-called dangerous border-collision bifurcations may occur, as well as multistability. We give also an example of a subcritical center bifurcation which may be considered as a piecewise-linear analogue of the subcritical Neimark–Sacker bifurcation.

scholarly journals How Many Inflections are There in the Lyapunov Spectrum?

2021 ◽  
Author(s):  
O. Jenkinson ◽  
M. Pollicott ◽  
P. Vytnova
Keyword(s):  
Piecewise Linear ◽  
Lyapunov Spectrum ◽  
Stat Phys ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
Expanding Map ◽  
Lyapunov Spectra ◽  
Linear Maps ◽  
Inflection Points ◽  
Piecewise Linear Maps

AbstractIommi and Kiwi (J Stat Phys 135:535–546, 2009) showed that the Lyapunov spectrum of an expanding map need not be concave, and posed various problems concerning the possible number of inflection points. In this paper we answer a conjecture in Iommi and Kiwi (2009) by proving that the Lyapunov spectrum of a two branch piecewise linear map has at most two points of inflection. We then answer a question in Iommi and Kiwi (2009) by proving that there exist finite branch piecewise linear maps whose Lyapunov spectra have arbitrarily many points of inflection. This approach is used to exhibit a countable branch piecewise linear map whose Lyapunov spectrum has infinitely many points of inflection.

Superconductors with spin–orbit interactions

2016 ◽  
Vol 30 (25) ◽  
pp. 1650183 ◽  
Author(s):  
Yu. N. Ovchinnikov
Keyword(s):  
Magnetic Field ◽  
Superconducting Films ◽  
Critical Magnetic Field ◽  
Spin Orbit ◽  
Zero Temperature ◽  
Interband Pairing ◽  
Two Parameters ◽  
Spin Orbit Interactions ◽  
Parameter Values ◽  
Thin Superconducting Films

The effect of spin-orbit (SO) interaction on the formation of the critical states in thin superconducting films in magnetic field oriented along the film is investigated. Hereby, the case of interband pairing is considered. It was found that eight branches exist in the plane of two parameters [Formula: see text] determined by the value of magnetic field and SO interaction. Six modes leads to inhomogeneous states with different values of the impulse [Formula: see text]. Each state is doubly degenerate over direction of impulse [Formula: see text]. The parameter values at critical point are found for all eight branches in explicit form for zero temperature. The optimal two branches are estimated, corresponding to largest critical magnetic field value for given SO interaction.

"CROSSROAD AREA–SPRING AREA" TRANSITION (II) FOLIATED PARAMETRIC REPRESENTATION

1991 ◽  
Vol 01 (02) ◽  
pp. 339-348 ◽  
Author(s):  
C. MIRA ◽  
J. P. CARCASSÈS ◽  
M. BOSCH ◽  
C. SIMÓ ◽  
J. C. TATJER
Keyword(s):  
Complete Information ◽  
Parametric Representation ◽  
Periodic Points ◽  
Parameter Plane ◽  
Flip Bifurcation ◽  
Cusp Point ◽  
Dimensional Representation ◽  
Bifurcation Structure ◽  
Spring Area

The areas considered are related to two different configurations of fold and flip bifurcation curves of maps, centred at a cusp point of a fold curve. This paper is a continuation of an earlier one devoted to parameter plane representation. Now the transition is studied in a thee-dimensional representation by introducing a norm associated with fixed or periodic points. This gives rise to complete information on the map bifurcation structure.

Detection of Synergistic Combinatorial Perturbations by a Bifurcation-Based Approach

2021 ◽  
Vol 31 (12) ◽  
pp. 2150175
Author(s):  
Min Luo ◽  
Dasong Huang ◽  
Jianfeng Jiao ◽  
Ruiqi Wang
Keyword(s):  
Gene Expression ◽  
Rational Design ◽  
Epithelial To Mesenchymal Transition ◽  
Regulation Of Gene Expression ◽  
Drug Combinations ◽  
Bifurcation Curve ◽  
Combinatorial Regulation ◽  
Mesenchymal Transition ◽  
Combinatorial Control ◽  
Two Parameters

Drug combination has become an attractive strategy against complex diseases, despite the challenges in handling a large number of possible combinations among candidate drugs. How to detect effective drug combinations and determine the dosage of each drug in the combination is still a challenging task. When regarding a drug as a perturbation, we propose a bifurcation-based approach to detect synergistic combinatorial perturbations. In the approach, parameters of a dynamical system are divided into two groups according to their responses to perturbations. By combining two parameters chosen from two groups, three types of combinations can be obtained. Synergism for different perturbation combinations can be detected by relative positions of the bifurcation curve and the isobole. The bifurcation-based approach can be used not only to detect combinatorial perturbations but also to determine their perturbation quantities. To demonstrate the effectiveness of the approach, we apply it to the epithelial-to-mesenchymal transition (EMT) network. The approach has implications for the rational design of drug combinations and other combinatorial control, e.g. combinatorial regulation of gene expression.

scholarly journals Chaotic Itinerancy Observed in Mutually Coupled Gaussian Maps

2018 ◽  
Vol 28 (04) ◽  
pp. 1830011
Author(s):  
Mio Kobayashi ◽  
Tetsuya Yoshinaga
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Chaotic Behavior ◽  
Periodic Points ◽  
Two Dimensional ◽  
Stable Fixed Point ◽  
Bifurcation Structure ◽  
Unstable Fixed Point ◽  
Gaussian Map ◽  
Gaussian Maps

A one-dimensional Gaussian map defined by a Gaussian function describes a discrete-time dynamical system. Chaotic behavior can be observed in both Gaussian and logistic maps. This study analyzes the bifurcation structure corresponding to the fixed and periodic points of a coupled system comprising two Gaussian maps. The bifurcation structure of a mutually coupled Gaussian map is more complex than that of a mutually coupled logistic map. In a coupled Gaussian map, it was confirmed that after a stable fixed point or stable periodic points became unstable through the bifurcation, the points were able to recover their stability while the system parameters were changing. Moreover, we investigated a parameter region in which symmetric and asymmetric stable fixed points coexisted. Asymmetric unstable fixed point was generated by the [Formula: see text]-type branching of a symmetric stable fixed point. The stability of the unstable fixed point could be recovered through period-doubling and tangent bifurcations. Furthermore, a homoclinic structure related to the occurrence of chaotic behavior and invariant closed curves caused by two-periodic points was observed. The mutually coupled Gaussian map was merely a two-dimensional dynamical system; however, chaotic itinerancy, known to be a characteristic property associated with high-dimensional dynamical systems, was observed. The bifurcation structure of the mutually coupled Gaussian map clearly elucidates the mechanism of chaotic itinerancy generation in the two-dimensional coupled map. We discussed this mechanism by comparing the bifurcation structures of the Gaussian and logistic maps.

scholarly journals The Chaotic Saddle in the Lozi Map, Autonomous and Nonautonomous Versions

2015 ◽  
Vol 25 (13) ◽  
pp. 1550184 ◽  
Author(s):  
Carlos Lopesino ◽  
Francisco Balibrea-Iniesta ◽  
Stephen Wiggins ◽  
Ana M. Mancho
Keyword(s):  
Piecewise Linear ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
The Third ◽  
Autonomous Version ◽  
Area Preserving ◽  
Chaotic Saddle

In this paper, we prove the existence of a chaotic saddle for a piecewise-linear map of the plane, referred to as the Lozi map. We study the Lozi map in its orientation and area preserving version. First, we consider the autonomous version of the Lozi map to which we apply the Conley–Moser conditions to obtain the proof of a chaotic saddle. Then we generalize the Lozi map on a nonautonomous version and we prove that the first and the third Conley–Moser conditions are satisfied, which imply the existence of a chaotic saddle. Finally, we numerically demonstrate how the structure of this nonautonomous chaotic saddle varies as parameters are varied.

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