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.

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.

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.

"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.

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.

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.

BIFURCATION STRUCTURES GENERATED BY THE NONAUTONOMOUS DUFFING EQUATION

1999 ◽  
Vol 09 (07) ◽  
pp. 1363-1379 ◽  
Author(s):  
C. MIRA ◽  
M. TOUZANI-QRIOUET ◽  
H. KAWAKAMI
Keyword(s):  
Parameter Space ◽  
Global Bifurcation ◽  
Duffing Equation ◽  
Damping Term ◽  
Bifurcation Structure ◽  
The Past ◽  
Spring Area ◽  
Bifurcation Structures ◽  
Duffing Model ◽  
Qualitative Changes

This paper deals with some properties of bifurcation structures in the parameter space related to the Duffing equation in the presence of an external periodical excitation B + B0 cos t. So global qualitative modifications of structures in the parameter plane (B, B0) are considered, when a third parameter ε (the damping term of the equation) varies. Complex sets of bifurcation curves are defined. They are based on the global bifurcation structure: crossroad area, saddle area, spring area, lip, quasi-lip, identified in the past with their "foliated representation", and their qualitative changes. For the Duffing model, special associations of the above areas, giving typical patterns called islands, are described with their qualitative modifications.

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.

Applications of Scanning Microscopy in Biomedical Research

1968 ◽  
Vol 26 ◽  
pp. 354-355
Author(s):  
T. L. Hayes
Keyword(s):  
Information Content ◽  
Biomedical Research ◽  
Biomedical Applications ◽  
Three Dimensional ◽  
Dental Enamel ◽  
Resolving Power ◽  
Whole Mount ◽  
Two Parameters ◽  
Content Information ◽  
Sectioned Tissue

Biomedical applications of the scanning electron microscope (SEM) have increased in number quite rapidly over the last several years. Studies have been made of cells, whole mount tissue, sectioned tissue, particles, human chromosomes, microorganisms, dental enamel and skeletal material. Many of the advantages of using this instrument for such investigations come from its ability to produce images that are high in information content. Information about the chemical make-up of the specimen, its electrical properties and its three dimensional architecture all may be represented in such images. Since the biological system is distinctive in its chemistry and often spatially scaled to the resolving power of the SEM, these images are particularly useful in biomedical research.In any form of microscopy there are two parameters that together determine the usefulness of the image. One parameter is the size of the volume being studied or resolving power of the instrument and the other is the amount of information about this volume that is displayed in the image. Both parameters are important in describing the performance of a microscope. The light microscope image, for example, is rich in information content (chemical, spatial, living specimen, etc.) but is very limited in resolving power.

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.

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