“CROSSROAD AREA- DISSYMMETRICAL SPRING AREA — SYMMETRICAL SPRING AREA,” AND “DOUBLE CROSSROAD AREA — DOUBLE SPRING AREA” TRANSITIONS

1993 ◽  
Vol 03 (02) ◽  
pp. 429-435 ◽  
Author(s):  
REZK ALLAM ◽  
CHRISTIAN MIRA
Keyword(s):  
Parameter Plane ◽  
Flip Bifurcation ◽  
Two Dimensional ◽  
Cusp Point ◽  
Symmetrical Configuration ◽  
Spring Area

In a parameter plane, crossroad areas and spring areas are two typical organizations of fold and flip bifurcation curves centred at a fold cusp point. Till now only spring areas in a “symmetrical” configuration have been described. This letter introduces another type of spring area for which such a “symmetry” does not exist. It is called a dissymmetrical spring area. When a third parameter is varied, qualitative modifications of the parameter plane are considered, and an example of a two-dimensional diffeomorphism is given.

ON THE "CROSSROAD AREA–SADDLE AREA" AND "CROSSROAD AREA–SPRING AREA" TRANSITIONS

1991 ◽  
Vol 01 (03) ◽  
pp. 641-655 ◽  
Author(s):  
C. MIRA ◽  
J. P. CARCASSÈS
Keyword(s):  
Three Dimensional ◽  
Parameter Plane ◽  
Flip Bifurcation ◽  
Two Dimensional ◽  
Cusp Point ◽  
Dimensional Representation ◽  
One Dimensional ◽  
Transition Mechanism ◽  
Three Dimensional Representation ◽  
Spring Area

Let T be a one-dimensional or two-dimensional map. The three considered areas are related to three different configurations of fold and flip bifurcation curves, centred at a cusp point of a fold curve in the T parameter plane (b, c). The two transitions studied here occur via a codimension-three bifurcation defined in each case, when varying a third parameter a. The transition "mechanism," from an area type to another one, is given with a three-dimensional representation describing the sheet configuration of the parameter plane.

"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.

"CROSSROAD AREA–SPRING AREA" TRANSITION (I) PARAMETER PLANE REPRESENTATION

1991 ◽  
Vol 01 (01) ◽  
pp. 183-196 ◽  
Author(s):  
J. P. CARCASSES ◽  
C. MIRA ◽  
M. BOSCH ◽  
C. SIMÓ ◽  
J. C. TATJER
Keyword(s):  
Parameter Plane ◽  
Flip Bifurcation ◽  
Two Dimensional ◽  
Bifurcation Structure ◽  
One Dimensional ◽  
Codimension Two ◽  
Transition Mechanism ◽  
Spring Area ◽  
Cusp Points ◽  
Codimension Two Bifurcation

This paper is devoted to the bifurcation structure of a parameter plane related to one- and two-dimensional maps. Crossroad area and spring area correspond to a characteristic organization of fold and flip bifurcation curves of the parameter plane, involving the existence of cusp points (fold codimension-two bifurcation) and flip codimension-two bifurcation points. A transition "mechanism" (among others) from one area type to another one is given from a typical one-dimensional map.

DETERMINATION OF DIFFERENT CONFIGURATIONS OF FOLD AND FLIP BIFURCATION CURVES OF A ONE OR TWO-DIMENSIONAL MAP

1993 ◽  
Vol 03 (04) ◽  
pp. 869-902 ◽  
Author(s):  
JEAN-PIERRE CARCASSES
Keyword(s):  
Flip Bifurcation ◽  
Two Dimensional ◽  
Bifurcation Curve ◽  
Cusp Point ◽  
Contour Lines ◽  
Spring Area ◽  
The Matrix ◽  
Point Of Intersection ◽  
Qualitative Changes

Three different configurations of fold and flip bifurcation curves of maps, centered round a cusp point of a fold curve, are considered. They are called saddle area, spring area and crossroad area. For one and two-dimensional maps, this paper uses the notion of contour lines in a parameter plane. A given contour line is related to a constant value of a "reduced multiplier" constructed from the trace and the Jacobian of the matrix associated with a given periodic point. The singularities of such lines define the configuration type of the areas indicated above. When a third parameter varies, the qualitative changes of such areas are directly identified. These singularities also enable the determination of a point of intersection of two bifurcation curves of the same nature (flip or fold), and, when a third parameter varies, the appearance (or disappearance) of a closed fold or flip bifurcation curve.

SINGULARITIES OF THE PARAMETRIC PLANE OF AN n-DIMENSIONAL MAP.

1995 ◽  
Vol 05 (02) ◽  
pp. 419-447 ◽  
Author(s):  
JEAN-PIERRE CARCASSES
Keyword(s):  
Jacobian Matrix ◽  
Flip Bifurcation ◽  
Bifurcation Curve ◽  
Cusp Point ◽  
Contour Lines ◽  
Fold Bifurcation ◽  
Spring Area ◽  
Parametric Plane ◽  
Point Of Intersection ◽  
Qualitative Changes

This paper uses the notion of “contour lines” in a parameter plane. A given contour line is related to a constant value of a “reduced multiplier” constructed from the elements of the Jacobian matrix associated with a given periodic point. The singularities type of such lines permit to determine a point of intersection of two bifurcation curves of same nature (flip or fold) and a point of tangency between a fold bifurcation curve and a flip bifurcation curve. When a third parameter varies, these singularities permit to determine the appearance (or disappearance) of a closed fold or flip bifurcation curve. Three different configurations of fold and flip bifurcation curves, centred round a cusp point of a fold curve, are considered. They are called saddle area, spring area, and crossroad area. The singularities type of the contour lines define the configuration types of these areas and, when a third parameter varies, the qualitative changes of such areas are directly identified.

ON A "CROSSROAD AREA–SPRING AREA" TRANSITION OCCURRING IN A DUFFING–RAYLEIGH EQUATION WITH A PERIODICAL EXCITATION

1993 ◽  
Vol 03 (04) ◽  
pp. 1029-1037 ◽  
Author(s):  
CHRISTIAN MIRA ◽  
MOHAMMED QRIOUET
Keyword(s):  
Excitation Frequency ◽  
Period Doubling ◽  
Qualitative Change ◽  
Parameter Plane ◽  
Rayleigh Equation ◽  
Flip Bifurcation ◽  
External Excitation ◽  
Spring Area ◽  
Bifurcation Structures ◽  
Parameter Coefficient

The bifurcation structures considered in this paper are given by a Duffing–Rayleigh equation in the presence of a periodic external excitation. The first one is related to a cascade of fold lips generated by period doubling at subharmonic oscillations, which is obtained in a parameter plane defined by the excitation frequency and its amplitude. When a third parameter (coefficient of the linear approximation of the damping) varies, a qualitative change of the parameter plane occurs. It is related to a new mechanism of "crossroad area–spring area" transition, the areas corresponding to typical arrangements of fold and flip bifurcation curves around a fold cusp.

On the Parameter Plane of Nonlinear Coupled Oscillators

1993 ◽  
Vol 48 (5-6) ◽  
pp. 655-662
Author(s):  
Wolfgang Metzler ◽  
Achim Brelle ◽  
Klaus-Dieter Schmidt ◽  
Gerrit Danker ◽  
Matthias Köppe ◽  
...  
Keyword(s):  
Coupled Oscillators ◽  
Nonlinear Oscillator ◽  
Mandelbrot Set ◽  
Parameter Plane ◽  
Two Dimensional ◽  
Routes To Chaos ◽  
Nonlinear Coupled Oscillators ◽  
Two Parameter

Abstract Two well-known bifurcation routes to chaos of two-dimensional coupled logistic maps are embedded in a two-parameter plane of a canonical nonlinear oscillator which contains a non-analytic analogon to the Mandelbrot set.

scholarly journals Topological Catastrophe in Massive Current Sheets

1989 ◽  
Vol 104 (2) ◽  
pp. 285-288
Author(s):  
Ph. Peterle ◽  
J. Hoyvaerts
Keyword(s):  
Boundary Conditions ◽  
Mass Flux ◽  
Flux Ratio ◽  
Continuous Variation ◽  
Two Dimensional ◽  
Cusp Point ◽  
Fixed Boundary ◽  
Current Sheets ◽  
Solar Filaments

AbstractA two-dimensional sheet model for solar filaments (Kippenhahn and Schluter configuration) is considered. We investigate, the quasi-static evolution of gravito-magnetohydrostatic equlibria in exploring the response of massive current sheets to a slow continuous variation of the mass/flux ratio with fixed boundary conditions. A catastrophic behavior of the field topology is found to occur in the sequence following the formation of a cusp point (bifurcation).

NONSMOOTH AND SMOOTH BIFURCATIONS IN A DISCONTINUOUS PIECEWISE MAP

2012 ◽  
Vol 22 (05) ◽  
pp. 1250112 ◽  
Author(s):  
ZHIYING QIN ◽  
YUEJING ZHAO ◽  
JICHEN YANG
Keyword(s):  
Fixed Point ◽  
Bifurcation Diagram ◽  
Bifurcation Point ◽  
Peculiar Feature ◽  
Parameter Plane ◽  
Flip Bifurcation ◽  
Border Collision Bifurcation ◽  
Bifurcation Structures ◽  
Piecewise Map ◽  
Two Parameter

In this paper, a piecewise map with singularity of the power (-1/2) is introduced. For this piecewise map, there is an infinite discontinuous gap on the origin. The conditions of nonsmooth border-collision bifurcation and smooth fold or flip bifurcation are analytically derived. For period-1 fixed point, two-parameter-plane can be divided into seven ranges according to different bifurcation structures. For period-n orbits, codimension-2 bifurcation point may lead to different period-increment sequence, and a peculiar feature is found that there are two different period-increment sequences in the same bifurcation diagram.

STUDY OF A TWO-DIMENSIONAL ENDOMORPHISM BY USE OF THE PARAMETRIC SINGULARITIES

2000 ◽  
Vol 10 (12) ◽  
pp. 2853-2862 ◽  
Author(s):  
JEAN-PIERRE CARCASSES ◽  
ABDEL-KADDOUS TAHA
Keyword(s):  
Parameter Plane ◽  
Two Dimensional ◽  
The Third ◽  
Saddle Node ◽  
Qualitative Changes

A two-dimensional cubic endomorphism depending on three parameters is considered. The qualitative changes of the bifurcation curves in a parameter plane are studied when the third parameter varies. More particularly, the crossing through a "saddle-node" singularity of the parameter plane is analytically analyzed.

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