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.

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.

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.

"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.

EXISTENCE OF A CUSP POINT ON A FOLD BIFURCATION CURVE AND STABILITY OF THE ASSOCIATED FIXED POINT: CASE OF AN n-DIMENSIONAL MAP

1999 ◽  
Vol 09 (05) ◽  
pp. 875-894 ◽  
Author(s):  
J. P. CARCASSÈS ◽  
H. KAWAKAMI
Keyword(s):  
Fixed Point ◽  
Center Manifold ◽  
Sufficient Condition ◽  
Bifurcation Curve ◽  
Cusp Point ◽  
Necessary And Sufficient Condition ◽  
Fold Bifurcation ◽  
Numerical Computing ◽  
The Stability ◽  
Necessary And Sufficient

Considering an n–dimensional map T, a necessary and sufficient condition for the existence of a cusp point on a fold bifurcation curve in a parameter plane of T is proposed. In the case of a nondegenerated cusp point, a necessary and sufficient condition for the stability of the associated nonhyperbolic fixed point is established. These conditions, obtained using the classical methods (center manifold, normal form), are expressed in an explicit algebraic well adapted for numerical computing.

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- 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.

scholarly journals Numerical Exploration of Kaldorian Macrodynamics: Enhanced Stability and Predominance of Period Doubling and Chaos with Flexible Exchange Rates

2008 ◽  
Vol 2008 ◽  
pp. 1-23 ◽  
Author(s):  
Toichiro Asada ◽  
Christos Douskos ◽  
Panagiotis Markellos
Keyword(s):  
Exchange Rates ◽  
Stability Region ◽  
Extreme Values ◽  
Period Doubling ◽  
Model Parameters ◽  
Flip Bifurcation ◽  
Bifurcation Curve ◽  
The Stability ◽  
Flexible Exchange Rates ◽  
Enhanced Stability

We explore a discrete Kaldorian macrodynamic model of an open economy with flexible exchange rates, focusing on the effects of variation of the model parameters, the speed of adjustment of the goods marketα, and the degree of capital mobilityβ. We determine by a numerical grid search method the stability region in parameter space and find that flexible rates cause enhanced stability of equilibrium with respect to variations of the parameters. We identify the Hopf-Neimark bifurcation curve and the flip bifurcation curve, and find that the period doubling cascades which leads to chaos is the dominant behavior of the system outside the stability region, persisting to large values ofβ. Cyclical behavior of noticeable presence is detected for some extreme values of a state parameter. Bifurcation and Lyapunov exponent diagrams are computed illustrating the complex dynamics involved. Examples of attractors and trajectories are presented. The effect of the speed of adaptation of the expected rate is also briefly discussed. Finally, we explore a special model variation incorporating the “wealth effect” which is found to behave similarly to the basic model, contrary to the model of fixed exchange rates in which incorporation of this effect causes an entirely different behavior.

"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.

APPEARANCE AND DISAPPEARANCE OF A DOVETAIL STRUCTURE IN THE PARAMETER PLANE OF AN n-DIMENSIONAL MAP

1999 ◽  
Vol 09 (04) ◽  
pp. 769-783 ◽  
Author(s):  
J. P. CARCASSÈS ◽  
H. KAWAKAMI
Keyword(s):  
Dynamical System ◽  
Singular Point ◽  
Discrete Dynamical System ◽  
Parameter Plane ◽  
Bifurcation Curve ◽  
The Third ◽  
Fold Bifurcation ◽  
Cusp Points ◽  
The Creation ◽  
Definition Of

A dovetail structure is made up of two cusp points located on a same fold bifurcation curve in a parameter plane of a discrete dynamical system defined by a differentiable map. When a third parameter varies, an existing dovetail structure may disappear by the merging of the two cusp points, or a dovetail structure may appear by the creation of two cusp points on a locally smooth fold bifurcation curve. This paper presents a method permitting to determine the value of the third parameter at which a dovetail structure may appear or disappear in n-dimensional systems. The exposed method is based on the definition of a new singular point, called E-point, belonging to a fold bifurcation curve.

scholarly journals Stability of the equilibria in a discrete-time sivs epidemic model with standard incidence

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2393-2408 ◽  
Author(s):  
Mahmood Parsamanesh ◽  
Saeed Mehrshad
Keyword(s):  
Discrete Time ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Total Population ◽  
Reproduction Number ◽  
Flip Bifurcation ◽  
Sis Epidemic Model ◽  
Fold Bifurcation ◽  
The Stability ◽  
Conditions Of Stability

A discrete-time SIS epidemic model with vaccination is presented and studied. The model includes deaths due to disease and the total population size is variable. First, existence and positivity of the solutions are discussed and equilibria of the model and basic reproduction number are obtained. Next, the stability of the equilibria is studied and conditions of stability are obtained in terms of the basic reproduction number R0. Also, occurrence of the fold bifurcation, the flip bifurcation, and the Neimark-Sacker bifurcation is investigated at equilibria. In addition, obtained results are numerically discussed and some diagrams for bifurcations, Lyapunov exponents, and solutions of the model are presented.

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