Critical Bifurcation Surfaces of 3 D Discrete Dynamics

This paper deals with the analytical representation of bifurcations of each 3D discrete dynamics depending on the set of bifurcation parameters. The procedure of bifurcation analysis proposed in this paper represents the 3D elaboration and specification of the general algorithm ofthe n-dimensional linear bifurcation analysis proposed by the author earlier. It is proven that 3D domain of asymptotic stability (attraction) of the fixed point for a given 3D discrete dynamics is bounded by three critical bifurcation surfaces: the divergence, flip and flutter surfaces. The analytical construction of these surfaces is achieved with the help of classical Routh-Hurvitz conditions of asymptotic stability. As an application the adjustment process proposed by T. Puu for the Cournot oligopoly model is considered in detail.


INTRODUCTION
The purpose of this paper is to construct the 3D analytical representation of the general procedure of linear bifurcation analysis developed by Sonis (1993Sonis ( , 1997)).The bifurcation analysis describes the changes in the qualitative properties of the orbits on non-linear discrete dynamics under the changes of the (external) parameters of these dynamics.
The bifurcation phenomena are defined by the position of the boundaries of attraction of the fixed point.It will be proven further that the domain of attraction of the fixed point of 3D discrete dynamics is bounded by three critical bifurcation surfaces: the divergence surface corresponding to the case in which one of the eigenvalues of the E-mail: sonism@mail.biu.ac.il.333 Jacobi matrix of the linear approximation of the dynamics equals to 1; theflip surface corresponding to the existence of the eigenvalue 1, and theflutter surface corresponding to the pair of complex con- jugated eigenvalues with absolute values equal to 1.The crossing of these surfaces by the movement of the fixed point will generate the plethora of all possible bifurcation phenomena.
2. 3D LINEAR BIFURCATION ANALYSIS 2.1.Fixed Points of 3D Discrete Non-Linear Dynamics Let us consider 3D discrete dynamics represented by the following system of autonomous difference M. SONIS equations: Xt+ F(xt, y,,z,; A) (= F,) Y,+l G(xt,Yt, Z,; A) (= zt+ H(xt, y,,zt; A) (= where A {a, a2, a3,..., ak} is a set of external bifurcation parameters and Ft, Gt, H are the differ- entiable functions of x,, Yt, zt.For the fixed set A all fixed points (x, y, z) of the system (2.1) are given by the solutions of the system of algebraic equations x F(x,y,z; A); y G(x,y,z; A); z=H(x,y,z;A) (2.2) Let us now present in detail the 3D procedure of linear bifurcation analysis.First step of the analysis includes the construction of the Jacobi matrix of a linear approximation of the non-linear dynamics (2.1): (2.4) where functions F, G, H are results of substitution (x,, yt, z,) --, (x, y, z) in F,, G, Ht: The eigenvalues of the Jacobi matrix J are the solutions of the characteristic polynomial equation:  Eq. (2.6) then as well known: (2.10) By the yon Neiman theorem the fixed point (x,y,z) is asymptotically stable iff for all eigen- values of the Jacobi matrix J the condition holds: I#l < (2.11) Condition (2.11) defines the space of parameters Cl, C2, C3 (space of eigenvalues) the geometrical domain of asymptotic stability (domain of attrac- tion).The analytical description of this domain can be given with the help of classical Routh-Hurwitz algorithm in the form of non-linear inequalities (see, Samuelson, 1983, pp. 435-437;Sonis, 1997).For the derivation of these inequalities, for 3D discrete dynamics we construct first the new parameters bo, b, b2, b3 such that: b0= 1+c1+c2+c3; be 3 c c2 + 3c3; bl 3 + cl c2 3c3 b3= 1-c+c.-c3 (2.12) b3 0 b2 0 bl b3 b3/N2 (2.14) b2--0 lies on the triangle BCD; and the plane b3 -0 lies on the triangle ABC (see Fig. 1).
In addition, the inequality /2 bl b2 bob3 > 0 (2.19) The classical condition of asymptotic stability are b 0 > 0; /k > 0; /k 2 > 0; A3 > 0 (2.15) which means that: defines the part of the pyramid which includes the point (0, O, O) and lies under the saddle 'X2 bib2 bob3 c2 q-c13 q-c 0 (2.20) bo > 0; b > 0; b2 > 0; b3 > 0; A2 bb2 bob3 > 0 (2.16) 2.4.Geometrical Structure of the Domain of Attraction: Critical Bifurcation Surfaces Thus, the domain of attraction of the fixed point (x, y, z) is defined by three inequalities: bo + c + 6'2 @ 6'3 > 0; b3 c + c2 6'3 > 0 A2 c 2 qt_ c1 (23 c > 0 (2.21)It is possible to see that the inequalities b0 > 0; bl > 0; b2 > 0; b3 > 0 (2.17) define in the space of eigenvalues C1,2, the interior of the pyramid with the vertices A(-1, 1, 1), B(1, 1, 1), C(3, 3, 1), D(-3, 3, 1) (2.18) The boundaries of the domain of attraction are two planes, bo (2.22) and a saddle A2 c2 q-ClC3 c 0 (2.23) such that the plane bo-0 lies on the triangle ADB; the plane b 0 lies on the triangle A CD; the plane The plane bo 0 intersects the saddle by the sides AD and BD; and the plane b3=0 intersects the saddle by the sides A C and BC; these two planes and the saddle will be called critical bifurcation surfaces.It is important to note that the domain of attraction is divided into two parts by a saddle-like surface in such a manner that one part includes three real eigenvalues and the other one includes two complex conjugated eigenvalues and one real eigenvalue.
For the construction of such a surface let us use the classical Thchirnhausen transformation: This transformation converts the characteristic polynomial Eq. ( 2.6) into /3 @ p/ _[_ q 0 (2.25) where p 2 ?1C2 2 --; q 3 3 27 C13 (2.26)The well-known classical condition for Eq.(2.25) (and same as for Eq.(2.6)) to have only real roots is the negativeness of the discriminant: Thus, the surface A_(p q 2 is dividing the domain of attraction into the above mentioned two parts.
In the space of eigenvalues this surface has an equation:

4
A ---0 (2.29) 27 6 27 108 This surface includes the origin.Moreover the plane b0 0 intersects this surface by the sides AD and BD, and the plane b3 0 intersects this surface by the sides A C and BC.The coordinate plane c 0 intersects this surface by the cubic curve c -c23 0 (2.30) the coordinate plane c2-0 intersects this surface by the parabola: The formulae (2.10) imply that bo (2.33) Thus, on the plane b0=0 at least one of the eigenvalues is equal to 1, i.e., dynamics became divergent this is a divergence plane.On the plane b3 0 at least one of the eigenvalues is equal to 1, i.e., dynamics became oscillatory, this is aflip plane.
Each point on the flip triangle ABC (see Fig. 1) cor- responds to the two-periodic cycle, and the move- ment of the fixed point through it generates the Feigenbaum type periodic doubling sequence in three dimensions, leading to chaos (Feigenbaum, 1978).
It is possible to check that on the saddle (2.23) This equality defines the structure of bifurcations on the saddle in the following way.On the saddle all three absolute values of each eigenvalue are not more than 1, since a saddle is the boundary of the domain of asymptotic stability and, moreover, on the saddle we have one real eigenvalue and two complex conjugate eigenvalues, say, #1,2 R(COS 27rf + sin 27rf); (2.40) If parameter is fixed and r is changing, then Eqs. (2.40) describe a straight line in the space of eigenvalues: c--c-r; C2-+cr; c3--r; -1 _< r _< (2.41) This straight line is the straight line generator of the saddle, it lies on the saddle and intersects the side A C in the point (1 c, c, 1) and the side BD in the point (-1 -c, + c,-1) (see Fig. 2).
If f is irrational, one obtains the quasi-periodic orbits.

Oy
According to the Routh-Hurwitz procedure, the domain of stability of a fixed point for the 2D dynamics (2.47) is given by the system of inequal- ities (see Hsu, 1977;Sonis, 1993;1996;1997): (2.56) which represents the divergence, flip and flutter inequalities" bo-+ c + c2 > O; b3-1-c + C2 2> O; /2-1-C2 > 0 (2.57) Figure 3 describes the domain of attraction, which is the triangle KLM in the space of eigenvalues {c, c2} with vertices K(-2, 1), L(2, 1), M(0, -1) The sides of the triangle of stability are defined by the following straight lines, the divergence boundary" + cl + C2 0, or Tr j(2)_ det j( 2 This domain (2.56)can be easily obtained from (2.21), putting in (2.21) c_ 0. Geometrically this means that the domain of stability of fixed point for 2D dynamics is the section of 3D domain of stability with the help of the coordinate plane c_=0.The segment KL represents all flutter bifurcations of 2D dynamics.The 3D bifurcation segments (2.40) touch the flutter segment if r 0 and we have all the bifurcations corresponding to a 2 cos 2rrf in the points Cl --O; C2 1; C3 --0; --2 < a <_ 2 (2.62) So, Table I also describes the periodicities on flutter segment KL.
Special Case c l-O" A Dynamics without the Self-Influence The 3D dynamics without self-influence have a form: xt+ F(yt, z,; A) Y,+I G(xt, z,; A) z,+ H(x,, y,; A) (2.63) where x,+ I(Y,+ 1,z,+ 1) does not explicitly depend on x,(y,,z,).In this case Cl =0 and the domain of stability of equilibria is described by divergence, flip and flutter inequalities: bo--+-c2--c3 > 0; b3-+ c2-c3 > 0; A2--1-c2-c 2 >0 (2.64) Figure 4 describes the domain of stability in the space of the parameters c2, c3, whose boundaries are the flutter parabola In his important book "Nonlinear Economic Dynamics" Swedish economist Tonu Puu intro- duced iterative process which leads two oligopolists to their Cournot equilibrium (see Puu, 1997, Chap.5).T. Puu considered the following iteration process: where xt, yt are the supplies at time of two competitors in a duopoly, a, b are their constant marginal costs, A, # are the adjustment speeds.Here it is assumed that a, b > 0; 0 <_ A, # <_ 1.The fixed point (x, y) of this iteration dynamics satisfies the system of algebraic equations" (3.2) and from (2.52) (3.7) The divergence inequality becomes which is always true for positive a, b and A, #, i.e. there are no situations in which iterative dynamics (3.1) diverges.
The flip inequality This inequality is always true, since the left side of (3.11) is positive and the right side is negative.
Thus, the domain of stability of the Cournot equilibrium is defined only by flutter inequality A2 2 > 0 which gives (a-b) Obviously, these roots are reciprocal, since we can use the reciprocal ratio k a/b.
The character of bifurcations in these roots are the same and is defined by the value of c from (2.35) (see also Table I): 3.21) Few examples are of special interest: if the adjustments A,# are unitary, then c=0 and we have (see, Section 2.5) 4-p cycle starting the Feigenbaun double-periodic way to chaos; if A+#= 1, then c= and we have 6-p cycle bifurcations.Moreover, from (3.20) c > 0, thus, Table I indicates that the duopoly adjustment dynamics cannot have the 3-p bifurcation (with c 1), but it can have the 5-p cycle corresponding to c 0.61803, i.e., A + #= 1.38157.

Cournot Equilibrium in Triopoly with Adjustment
The case of triopoly can be considered analogically.

The "Virtual" Duopoly
The "virtual" duopoly is the special case of triopoly with two identical competitors (see Puu, 1997, Chap. 5).Thus, we can obtain from (3.
Domain of Attraction of the Fixed Point and its Routh-Hurwitz Inequalities If #1,#2,#3 are the roots of the characteristic

FIGURE 3
FIGURE 3 Domain of attraction of 2D discrete dynamics.
FIGURE 4 Domain of attraction of 3D non-self-influence discrete dynamics.
points in the 2D eigenvalue space (see Fig.4).The domain of stability of this Cournot equilibrium can be easily calculated from the value of the Jacobi approximation matrix at k approaches the value (3.44) then the attraction of the Cournot equilibrium (3.34) exchanged by the quasi-periodicity defined by the irrational 2.2.The Value of the Jacobi Matrix at aFixed Point

Table I
represents the fixed points generating all