User's Guide - Chapter 5

Chapter 5 Algorithms and Theory

[Editor's Note: This chapter contained formatted equations which are in the process of being converted to in-lined GIF images]

5.1 Technical Definitions

The point is called the image (or consequent or forward iterate) of rank i of p.

The preimages of rank i of p are the points which are mapped into p after i applications of the map T.

The critical curve of rank 1 of T, denoted as LC, is the locus of points having at least two coincident preimages of rank 1.

A subset A of the plane is called trapping if it is mapped into itself by T, T(A)A ; it is called invariant if T(A)=A, and backward invariant if (A)=A

An absorbing area d' is a closed subset of R2 bounded by a finite number of arcs of critical curves, which is trapping, T(d')d', and for which a neighborhood exists, the points of which have an image of finite rank in the interior of d'.

Its basin of attraction D(d') is an open set of points having an image of finite rank in d'.

An annular absorbing area is an absorbing area of annular shape, that is, a simply connected area deprived of the points of a hole in its interior.

A chaotic area d is an invariant area of d', bounded by critical arcs or limit points of critical points, which contains a chaotic invariant set.

A chaotic area d may be modified by a non-classical bifurcation, called a contact bifurcation, characterized by a contact between its boundary and the boundary of its basin of attraction, the frontier F.

A fixed point P of a map T is expanding if a neighborhood U of P such that all eigenvalues of DT(p) are greater than 1 in absolute value pU.

A point q is homoclinic to P if a positive integer j such that and a sequence of preimages of q converges to P. P is a snap-back repellor (SBR) if it is expanding and there exists a homoclinic point q of P.

5.2 Properties

In general, an absorbing area d' can be determined with boundary made up of a finite number of critical arcs belonging to the images of an arc, say of .
continuously differentiable LC is generally the image under T of where is the locus of points for which the Jacobian of T vanishes.
The critical curve LC separates the plane into open regions. The points of each region possess the same number of distinct preimages of rank 1.
Its frontier (or boundary) is backward invariant.
If a fixed point P belonging to d' is expanding but is not an SBR, then an annular absorbing area d'a d' can be obtained, with external and internal boundaries made up of a finite number of critical arcs belonging to the images of the arc of .

5.3 Theory

The critical curves are used to define the domains of the inverses of T, determine the absorbing areas, and characterize the global bifurcations. In order to do so, the properties of critical curves and their iterates are used to develop a theory which characterizes bifurcations in terms of the interaction of thecritical curves with other geometric aspects of the dynamics such as the stable and unstable sets of a saddle, the boundary of a basin of attraction or the boundary of a chaotic area. Contacts between these and other objects are indicative of a wide variety of bifurcations. The procedures and algorithms used by endo are based on this theory and useful in its further development.

The algorithm for determining an absorbing area is based on the following procedure discovered by Mira. Let be the point of intersection of and , its preimage of rank 1 belonging to and the image of rank i of . If a positive integer m such that the critical arc intersects , use procedure 2. Otherwise, use procedure 1. For instance, in the Dorband Double Logistic with , use procedure 1. For , use procedure 2.

Procedure 1: Let be a point of such that and a critical arc intersects at . Then the area with boundary is an absorbing area.

Procedure 2: Let m be the first integer such that the critical arc intersects and let be the intersection point furthest from . Then the area with boundary is an absorbing area.

5.4 An Example

Consider the family of two dimensional maps T: R2R2, (x,y)(x',y') as a function of a real parameter defined by :

This "double logistic" map can be considered as a model for a coupled pair of oscillators. The fixed points of T and their local stability can be calculated by solving the equation then calculating the eigenvalues at each fixed point. Varying the parameter , we divide the dynamical behavior into five "regimes". For a complete treatment of this example, see Gardini, L., R.H. Abraham, R. J. Record, D. Fournier-Prunaret, [1994] "A Double Logistic Map", International Journal of Bifurcation and Chaos, Vol. 4, No. 1, 145-176.

Behavior in the first regime, 0.0 < < 0.2, is characterized by a simply connected basin of attraction. There are fixed points on the diagonal, , at the origin, O = (0,0) and S* = (3/4, 3/4). There are fixed points off the diagonal at =() and =(). The origin is a saddle for (0,0.4) and a repelling node for >0.4 while S* is a saddle for (0,0.66...) and a repelling node for >0.66... . are attracting foci for (0,0.4) and repelling foci for >0.4 . The stable set of the origin, Ws(O), consists of two arcs connecting O and its "alternate" inverse O-1 .

In the second regime, 0.2 < < 0.4, the basin of attraction is no longer simply connected but consists of an infinite number of simply connected components . At =0.2 a global basin bifurcation occurs characterized by the contact of the critical curve with the backward invariant sets Ws(O) and Ws(S*). Z4 penetrates the basin of attraction D. Ws(S*), which determines D(), has a more complex structure, containing infinitely many arcs.

The third regime, 0.4 < < * 0.70209, is initiated at =0.4 with a flip bifurcation of the origin and a Neimark-Hopf bifurcation of the fixed points and . The attracting foci each give rise to an attracting closed invariant curve i , i=1,2. The origin bifurcates from a saddle to a repelling node. A repelling 2-cycle saddle appears on the formerly attracting branch of the saddle at the origin. Construction of the absorbing area is now possible using the procedure developed by Mira and outlined above.

Inside the third regime at 0.487At 0.487 a critical arc becomes tangent to and an annular absorbing area (AAA) appears. While is not a snap-back repellor, a simply connected absorbing area d' and a AAA da' with = and = where : {

At = 1 0.48735 the invariant attracting curve becomes tangent to . is deformed from a smooth oval by the appearance of oscillations as a result of the "folding" of the portion of above . At = 2 0.505 the invariant attracting curve becomes tangent to LC (crossing at the two points and ). After the crossing will be tangent to LC at two points separated by . Contact between the chaotic attractor and its basin boundary at 0.64218 produces a "contact bifurcation".

S*=(3/4, 3/4) bifurcates from a saddle to a repelling node at = 0.66... . Along the diagonal , S* bifurcates from attracting to repelling while spawning a 2-cycle, Q1-Q2 , attracting on but repelling in the orthogonal direction (i.e. Q1-Q2 is a 2-cycle saddle for T)

The fourth regime is the parameter interval * 0.70209 0.1 . At =*an arc of the critical curve on the boundary of the AAA becomes tangent to at the periodic point Q1, causing the reunion of the two disjoint annular chaotic areas into a single connected chaotic area

The hole H(S*) disappears at the SBR bifurcation of S* occurring at = s 0.714 .The two symmetric rank-1 preimages of S* outside fall on the boundary of d' and thus on an arc of a critical curve. All the critical curves LCi, i3, pass through S*. Infinitely many homoclinic orbits of S* exist and persist after the SBR bifurcation for all values of > s .

The SBR bifurcation of the fixed points and occurs at = p 0.737 . After this bifurcation , the holes H() and H () disappear and the chaotic area d coincides with d' and is simply connected. This bifurcation is characterized by the critical curves , passing through the fixed points and .

At 0.7596 there is a pitchfork bifurcation of the 2-cycle saddle R1-R2 which becomes a repelling node and spawns the appearance of a couple of 2-cycle saddles on the Frontier. There is also a pitchfork bifurcation of the 2-cycle saddle Q1-Q2 on which becomes an attracting node.

At 0.84 an 8-cycle undergoes a Neimark-Hopf bifurcation giving rise to 8 closed invariant curves. At 0.845 contact bifurcations produce a 4-cyclic annular chaotic area. At 0.852 an SBR bifurcation gives rise to a 4-cyclic chaotic area, not annular. At 0.854 a contact bifurcation between the 4 basins of attraction and the 4-cyclic chaotic areas produces a single chaotic attractor in the absorbing area d'. This sequence is repeated for the attracting 4-cycle which occurs at =0.88 .

The SBR bifurcation of S* for occurs when the two critical curves and intersect at the fixed point S* at 0.8928 . Prior to this, all the points homoclinic to S* in d ' were outside . After, there are points homoclinic to S* in .

The last contact bifurcation of d' occurs at =1.0, a homoclinic bifurcation in which the origin becomes an SBR. In this, the fifth regime, for >1.0, the point at infinity is the unique attractor. In the unit square there survives a Cantor set , invariant under T, with repelling cycles. That is, is a "strange repellor".

5.5 An Application

In 1981 Graciela Chichilnisky introduced the North-South model of macroeconomic interaction between two regions. For a complete treatment of this application, see Abraham, R. H., G. Chichilnisky, R. J. Record, [1994] "Dynamics of North-South Trade and the Environment" in Environmental Economics, Graciela Chichilnisky, ed., Mattei Foundation, Milan (in press). In that paper, a dynamical model is developed, based on the idea of slowly changing the capital stock variable in the static model and assuming a rapid approach to equilibrium. Capital accumulation through time in each region is described by the equations :

That is, capital stock at time t+1 is the sum of capital stock at time t plus savings minus depreciation. The North-South trade variables are as follows :

B - basic goods

I - industrial goods

K - capital

L - labor (later, labor is replaced by the environment, E)

- price of basic goods

- price of industrial goods

p - price of basic goods as set by the international terms of trade after normalizing, =1

w - price of labor (wages)

r - rental of capital

- property rights of a common resource property

subscripts denote region (N or S)

subscript T denotes total in both regions (e.g. )

superscripts denote supply (S) and demand (D) (e.g. denotes demand for basic goods in the South)

In the dynamical model, the common resource property rights in the South, , is varied. Additional parameters in the North-South model are held fixed at experimentally determined values which are chosen such that the variables p, r, w, L, K, B, I > 0 in each region. The fixed parameter values used are shown in the following table.

parameter               North               South             

                        2                   4.5               

                        0.15                0.02              

                        1.8                 0.01              

                        1.7                 3                 

                        6                   75                

                        -9.7                -0.025            

                        12                  2.7               

                        0.5                 -2

In the North-South dynamical model the equilibrium level of GNP in each region is :

The planar endomorphism used is

with :

Once p is determined, the GNP is obtained from . To determine p, solve the quadratic

where :

and
.

In simulations s=0.12, =0.1 and =0.6 .

Back to Dr. Record's Resume or proceed to Appendix A of Dr. Record's Ph.D. Thesis