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 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.

"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.

scholarly journals Methodological and applied aspects of classification and typology of international geopolitical conflicts

2020 ◽  
pp. 38-53
Author(s):  
Aleksandr Aleksandrovich Tikhov
Keyword(s):  
Integral Approach ◽  
Modern World ◽  
Key Factors ◽  
Diagnostic Features ◽  
Geographical Environment ◽  
The Matrix ◽  
Conflict Interaction ◽  
Qualitative Changes

This article is dedicated to the development of methods and applied aspects of the typology of international geopolitical conflicts in modern world. Relevance of this topic is substantiated by intensification of intergovernmental conflicts on the regional and transregional levels, as well as escalation of sociopolitical tension. The author offers a method of classification of geopolitical conflicts that considers spatial and time specificity of such forms of cooperation and is based on the comprehensive analysis of all components of the process. As one of the key factors, the author highlights the causes and prerequisites for formation of a conflict. An integral approach towards typology of the forms of conflict interaction between the countries is being developed on the basis of three-way classification matrix. The matrix is based on the three diagnostic features that characterize the key factors of formation and development of a conflict, and substantiate a subsequent model of intensification of a conflict. This allows achieving the necessary level of objectivity for conducting further complex diagnostics of geopolitical conflict and precision of the result of typology using the instruments and methodology of different scientific approaches towards studying such type of processes. The proposed method may be used in studying certain types of geopolitical conflicts of the past and present, forecasting the development and qualitative changes of a certain conflict in future, as well as comprehensive assessment of conflict potential of a particular territory of region as a whole. In the course of this work, the author established and confirmed interpretation and interdependence between the political decisions of one or another country and the corresponding geographical environment. Determination of causal lings allow viewing the method as a foundation for future examination of conflicts and forecast their development.

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.

Lateral resolution of the electron microbeam analysis

1978 ◽  
Vol 36 (1) ◽  
pp. 542-543
Author(s):  
H.J. Dudek
Keyword(s):  
Quantitative Analysis ◽  
Depth Resolution ◽  
Lateral Resolution ◽  
Cylindrical Particle ◽  
Correction Procedure ◽  
X Ray ◽  
Element Concentrations ◽  
Microbeam Analysis ◽  
The Matrix

The chemical inhomogenities in modern materials such as fibers, phases and inclusions, often have diameters in the region of one micrometer. Using electron microbeam analysis for the determination of the element concentrations one has to know the smallest possible diameter of such regions for a given accuracy of the quantitative analysis.In th is paper the correction procedure for the quantitative electron microbeam analysis is extended to a spacial problem to determine the smallest possible measurements of a cylindrical particle P of high D (depth resolution) and diameter L (lateral resolution) embeded in a matrix M and which has to be analysed quantitative with the accuracy q. The mathematical accounts lead to the following form of the characteristic x-ray intens ity of the element i of a particle P embeded in the matrix M in relation to the intensity of a standard S

Determination of the phase structure of polymerblends by electron microscopy

1978 ◽  
Vol 36 (1) ◽  
pp. 492-493
Author(s):  
Dr. G. Kaemof
Keyword(s):  
Screw Extruder ◽  
Electron Microscopic ◽  
Thin Sections ◽  
Single Screw Extruder ◽  
Transmission Electron ◽  
The Matrix ◽  
Twin Screw ◽  
Special Case ◽  
Microscopic Investigations

A mixture of polycarbonate (PC) and styrene-acrylonitrile-copolymer (SAN) represents a very good example for the efficiency of electron microscopic investigations concerning the determination of optimum production procedures for high grade product properties.The following parameters have been varied:components of charge (PC : SAN 50 : 50, 60 : 40, 70 : 30), kind of compounding machine (single screw extruder, twin screw extruder, discontinuous kneader), mass-temperature (lowest and highest possible temperature).The transmission electron microscopic investigations (TEM) were carried out on ultra thin sections, the PC-phase of which was selectively etched by triethylamine.The phase transition (matrix to disperse phase) does not occur - as might be expected - at a PC to SAN ratio of 50 : 50, but at a ratio of 65 : 35. Our results show that the matrix is preferably formed by the components with the lower melting viscosity (in this special case SAN), even at concentrations of less than 50 %.

Polishing process effect on the image of dispersed precipitates

1984 ◽  
Vol 42 ◽  
pp. 534-535
Author(s):  
C.T. Hu ◽  
C.W. Allen
Keyword(s):  
Matrix Material ◽  
Specimen Preparation ◽  
Second Phase ◽  
Precipitate Particle ◽  
Polishing Process ◽  
Second Phase Particles ◽  
The Matrix ◽  
Surface Profiles ◽  
Thinning Process

One important problem in determination of precipitate particle size is the effect of preferential thinning during TEM specimen preparation. Figure 1a schematically represents the original polydispersed Ni3Al precipitates in the Ni rich matrix. The three possible type surface profiles of TEM specimens, which result after electrolytic thinning process are illustrated in Figure 1b. c. & d. These various surface profiles could be produced by using different polishing electrolytes and conditions (i.e. temperature and electric current). The matrix-preferential-etching process causes the matrix material to be attacked much more rapidly than the second phase particles. Figure 1b indicated the result. The nonpreferential and precipitate-preferential-etching results are shown in Figures 1c and 1d respectively.

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