FROM THE BOX-WITHIN-A-BOX BIFURCATION STRUCTURE TO THE JULIA SET PART II: BIFURCATION ROUTES TO DIFFERENT JULIA SETS FROM AN INDIRECT EMBEDDING OF A QUADRATIC COMPLEX MAP

Part I of this paper has been devoted to properties of the different Julia set configurations, generated by the complex map TZ: z′ = z2 - c, c being a real parameter, -1/4 < c < 2. These properties were revisited from a detailed knowledge of the fractal organization (called "box-within-a-box"), generated by the map x′ = x2 - c with x a real variable. Here, the second part deals with an embedding of TZ into the two-dimensional noninvertible map [Formula: see text]; y′ = γ y + 4x2y, γ ≥ 0. For [Formula: see text] is semiconjugate to TZ in the invariant half plane (y ≤ 0). With a given value of c, and with γ decreasing, the identification of the global bifurcations sequence when γ → 0, permits to explain a route toward the Julia sets, from a study of the basin boundary of the attractor located on y = 0.

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FROM THE BOX-WITHIN-A-BOX BIFURCATION ORGANIZATION TO THE JULIA SET PART I: REVISITED PROPERTIES OF THE SETS GENERATED BY A QUADRATIC COMPLEX MAP WITH A REAL PARAMETER

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127409022877 ◽

2009 ◽

Vol 19 (01) ◽

pp. 281-327 ◽

Cited By ~ 8

Author(s):

CHRISTIAN MIRA ◽

LAURA GARDINI

Keyword(s):

Julia Set ◽

Julia Sets ◽

Real Parameter ◽

Detailed Knowledge ◽

Two Dimensional ◽

Limit Sets ◽

Fatou Sets ◽

Different Types ◽

C Value ◽

Quadratic Complex

Properties of the different configurations of Julia sets J, generated by the complex map TZ: z′ = z2 - c, are revisited when c is a real parameter, -1/4 < c < 2. This is done from a detailed knowledge of the fractal bifurcation organization "box-within-a-box", related to the real Myrberg's map T: x′ = x2 - λ, first described in 1975. Part I of this paper constitutes a first step, leading to Part II dealing with an embedding of TZ into the two-dimensional noninvertible map [Formula: see text]. For γ = 0, [Formula: see text] is semiconjugate to TZ in the invariant half-plane (y ≤ 0). With a given value of c, and with γ decreasing, the identification of the global bifurcations sequence when γ → 0, permits to explain a route toward the Julia sets. With respect to other papers published on the basic Julia and Fatou sets, Part I consists in the identification of J singularities (the unstable cycles and their limit sets) with their localization on J. This identification is made from the symbolism associated with the "box-within-a-box" organization, symbolism associated with the unstable cycles of J for a given c-value. In this framework, Part I gives the structural properties of the Julia set of TZ, which are useful to understand some bifurcation sequences in the more general case considered in Part II. Different types of Julia sets are identified.

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ON SOME GLOBAL BIFURCATIONS OF THE DOMAINS OF FEASIBLE TRAJECTORIES: AN ANALYSIS OF RECURRENCE EQUATIONS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127405012958 ◽

2005 ◽

Vol 15 (05) ◽

pp. 1625-1639 ◽

Cited By ~ 4

Author(s):

EN-GUO GU ◽

JIONG RUAN

Keyword(s):

Fixed Point ◽

Stable Manifold ◽

The Other ◽

Computer Assisted ◽

Two Dimensional ◽

Global Bifurcations ◽

Basin Boundary ◽

Recurrence Equations ◽

Critical Curves ◽

Global Properties

This paper is an attempt to give new results, by a computer-assisted study, on some global bifurcations that change the structure of the domain of feasible trajectories (bounded discrete trajectories having an ecological sense) which can be obtained by the union of all rank preimages of axes. Three two-dimensional recurrence equations (or maps) are analyzed. The two first maps are degenerated invertible maps (i.e. the inverses of them are well defined except a set of zero lebergue measure) for which the basins of attractor are obtained by the backward iteration of a stable manifold of a saddle fixed point belonging to the basin boundary, and the interior domains of feasible trajectories are given by the intersection between the basin of attractor and the first quadrant. The other is a noninvertible map which is investigated by the use of critical curves, a powerful tool for the analysis of global properties of two-dimensional maps.

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A STUDY OF BIFURCATIONS IN A CIRCULAR REAL CELLULAR AUTOMATON

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127493000234 ◽

1993 ◽

Vol 03 (02) ◽

pp. 293-321 ◽

Cited By ~ 1

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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Chaotic Itinerancy Observed in Mutually Coupled Gaussian Maps

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127418300112 ◽

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.

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Fractal analysis and control of the competition model

International Journal of Biomathematics ◽

10.1142/s1793524516500455 ◽

2016 ◽

Vol 09 (03) ◽

pp. 1650045 ◽

Cited By ~ 6

Author(s):

Mianmian Zhang ◽

Yongping Zhang

Keyword(s):

Feedback Control ◽

Mathematical Models ◽

Fractal Analysis ◽

Fractal Geometry ◽

Julia Set ◽

Julia Sets ◽

Competition Model ◽

Simulation Results ◽

Population Competition ◽

And Control

Lotka–Volterra population competition model plays an important role in mathematical models. In this paper, Julia set of the competition model is introduced by use of the ideas and methods of Julia set in fractal geometry. Then feedback control is taken on the Julia set of the model. And synchronization of two different Julia sets of the model with different parameters is discussed, which makes one Julia set change to be another. The simulation results show the efficacy of these methods.

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BASIN BIFURCATIONS OF TWO-DIMENSIONAL NONINVERTIBLE MAPS: FRACTALIZATION OF BASINS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127494000241 ◽

1994 ◽

Vol 04 (02) ◽

pp. 343-381 ◽

Cited By ~ 67

Author(s):

C. MIRA ◽

D. FOURNIER-PRUNARET ◽

L. GARDINI ◽

H. KAWAKAMI ◽

J.C. CATHALA

Keyword(s):

Phase Plane ◽

The Other ◽

Two Dimensional ◽

Basin Boundary ◽

Critical Curves ◽

Noninvertible Maps

Properties of the basins of noninvertible maps of a plane are studied using the method of critical curves. Different kinds of basin bifurcation, some of them leading to basin boundary fractalization are described. More particularly the paper considers the simplest class of maps that of a phase plane which is made up of two regions, one with two preimages, the other with no preimage.

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Non-smooth homoclinic bifurcation in a conceptual climate model

European Journal of Applied Mathematics ◽

10.1017/s0956792518000153 ◽

2018 ◽

Vol 29 (5) ◽

pp. 891-904 ◽

Cited By ~ 1

Author(s):

JULIE LEIFELD

Keyword(s):

Periodic Orbit ◽

Climate Model ◽

Homoclinic Bifurcation ◽

Global Bifurcations ◽

Bifurcation Structure ◽

Limiting Case

Collision of equilibria with a splitting manifold has been locally studied, but might also be a contributing factor to global bifurcations. In particular, a boundary collision can be coincident with collision of a virtual equilibrium with a periodic orbit, giving an analogue to a homoclinic bifurcation. This type of bifurcation is demonstrated in a non-smooth climate application. Here, we describe the non-smooth bifurcation structure, as well as the smooth bifurcation structure for which the non-smooth homoclinic bifurcation is a limiting case.

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RESONANT GLUING BIFURCATIONS

International Journal of Bifurcation and Chaos ◽

10.1142/s021812740000133x ◽

2000 ◽

Vol 10 (09) ◽

pp. 2141-2160 ◽

Cited By ~ 4

Author(s):

ROBERT W. GHRIST

Keyword(s):

Saddle Point ◽

Parameter Space ◽

Two Dimensional ◽

Bifurcation Structure ◽

Centre Manifold ◽

Principal Eigenvalues ◽

Homoclinic Connections ◽

Homoclinic Bifurcations ◽

Zero Entropy

We consider the codimension-three phenomenon of homoclinic bifurcations of flows containing a pair of orbits homoclinic to a saddle point whose principal eigenvalues are in resonance. We concentrate upon the simplest possible configuration, the so-called "figure-of-eight," and reduce the dynamics near the homoclinic connections to those on a two-dimensional locally invariant centre manifold. The ensuing resonant gluing bifurcations exhibit features of both gluing bifurcations and resonant homoclinic bifurcations. Under certain twist conditions, the bifurcation structure is extremely rich, although describing zero-entropy flows. The analysis carefully exploits the topology of the orbits, the centre manifold and the parameter space.

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