Linear Bifurcation Analysis with Applications to Relative Socio-Spatial Dynamics

for the description the qualitative properties of orbits of the discrete autonomous iteration processes on the basis of linear approximation of the processes. The basic element of this analysis is the geometrical and numerical modification and application of the classical Routhian formalism, which is giving the description of the behavior of the iteration processes near the boundaries of the stability domains of equilibria. The use of the Routhian formalism is leading to the mapping of the domain of stability of equilibria from the space of control bifurcation parameters into the space of orbits of iteration processes. The study of the behavior of the iteration processes near the boundaries of stability domains can be achieved by the converting of coordinates of equilibria into control bifurcation parameters and by the movement of equilibria in the space of orbits. The crossing the boundaries of the stability domain reveals the plethora of the possible ways from stability, periodicity, the Arnold mode-locking tongues and quasi-periodicity to chaos. The numerical procedure of the description of such phenomena includes the spatial bifurcation diagrams in which the bifurcation parameter is the equilibrium itself. In this way the central problem of control of bifurcation can be solved: for each autonomous iteration process with big enough number of external parameters construct the realization of this iteration process with a preset combination of qualitative properties of equilibria. In this study the two-dimensional geometrical and numerical realizations of linear bifurcation analysis is presented in such a form which can be easily extended to multi-dimensional case. Further, a newly developed class of the discrete relative m-population]n-location Socio-Spatial dynamics is described. The proposed algorithm of linear bifurcation analyses is used for the detail analysis of the log-log-linear model of the one population]three location discrete relative dynamics.


INTRODUCTION
In recent decades a new paradigm of bifurcations in behavior of non-linear systems appeared as a scientific approach and as a method to deal with manifestations of chaos and turbulence in different sciences.At present the essence of scien- tific efforts is shifted to further elaboration of conceptual framework of bifurcation analysis, to standardization of numerical methods and to the detailed description of the new important do- mains of applications.The central problem of the linear bifurcation analysis is the problem of con- trol of bifurcations: to construct for each itera- tion process with a big enough number of the control bifurcation parameters the realization of this iteration process with a preset combination of qualitative properties of orbits.In the solution of this problem three main aspects are inter- wined: analytical and numerical aspects and the aspect of geometrical visualization.
The main objective of this research is two folded: to present the linear bifurcation analysis of the behavior of autonomous finite-dimensional discrete iteration processes and to apply the cor- responding algorithm of analysis to the study of a new branch of non-linear dynamic systems stu- dies: the Discrete Relative m-population/n-location Socio-Spatial Dynamics.
All possible equilibria x*-(X*l,X,...,xn) of the iteration process (1) are given by the system of equations x/*-Fi(A;x*), i-1,2,...,n. (2) In this paper we are presenting the analytical and numerial procedure of the bifurcation analysis in the following way: the essence of this pro- cedure is the exchange of a part A of control bifurcation parameters from the set A by compo- nents of the equilibrium x* (X'l, x2,... Xn) with the help of Eqs.(2).In such a way the compo- nents of equilibria became control bifurcation parameters themselves.
As it will be shown further, the remaining part of parameters A2 A\A give the description of the boundaries of the domain of stability of equi- libria within the space of orbits.This means that it is possible to move the equilibrium without of movement of domain of its stability.The move- ment of equilibrium points can be placed on the segments of straight lines.This allows the com- plete computerized description of the appearance of different bifurcation phenomena in the space of orbits.
Thus, the geometrical content of the proposed bifurcation analysis includes the travels of equilibria in the space of orbits which reveal the quali- tative features of the behavior of the trajectories of the iteration process near the boundaries of domain of stability of equilibria.
As is well known the construction of the ana- lytical forms of the coefficients of the charac- teristic polynomial .P(#) can be done with the help of the principal minors of the Jacobi ma- trix J*.Thus, the following analytical objects should be computed: 3. Principal minors of the Jacobi matrix J*.
By the well-known theorem of von Neumann the equilibrium x* is asymptotically stable iff for all its eigenvalues # the following condition holds: l# l< 1. (5) Consider the space P of all coefficients of the characteristic polynomials of the order n.Condi- tion (5) defines in this space the geometrical do- main of asymptotical stability.The analytical description of this stability domain can be con- structed with the help of the classical Routh-- Hurwitz procedure in the form of the non-linear inequalities.This procedure can be described as follows (see, Samuelson, 1983, pp. 435-437).(6) Further, construct the matrix bl b3 bs bo b2 b4 0 bl b3 0 bo b2 (7) and its principal minors A1, A2, A n.The conditions of asymptotical stability are: bo>0; Ar>0, r= 1,2,...,n (8) and the boundaries of the stability domain in the space P determined with the help of described above Routhian procedure by the non-linear equalities: bo--0; At=0, r= 1,2,...,n.(9) On the boundaries (9) the absolute values of some eigenvalues of the Jacobi matrix are equal and the plethora of different bifurcation phenomena exist.
In two-and three-dimensional cases the do- mains of stability can be visualized in the follow- ing form: for n 2 b0 + a + a2; bl 2 2a2; (10) b2 a q-a2; and the stability domain in the space of para- meters al, a2 is defined by the linear inequalities :tza <a2 < 1. (11) Geometrically, these inequalities represent a tri- angle of stability with the vertices [-2], [], [_Ol]" For n 3: bo + al + a2 -+-a3; b, 3 + al a2 3a3; b2 3 a a2 -1-3a3; b3 al + a2 a3; (12) and the stability domain is defined by the linear and quadratic inequalities: + al if-a2 q-a3 > 0; al + a2 a3 > 0; a2 -ff ala3 a23 > O. (3) In the three-dimensional space of the coeffi- cients al,a2, a3 this domain has three boundary surfaces: two planes and a saddle (parabolic hyperboloid).More precisely, the plane + al-k- a2 q-a3--0 touches the domain of stability of equilibria by the triangle ABC with the vertices A= B---1 C-3 the plane a + a2 a3 0 touches the do- main of stability of equilibria by the triangle ABD with the vertices A-B= -1 D- The straight lines generated by segments A C, BC, AD, BD lie on the saddle 1-a2 q-ala3- a32 -0o Next, because the components of the Jacobi matrix J* are the functions of the coordinates of the equlibrium x*-(x, x2, xn), it is possi- ble to construct the analytical and geometrical images of the boundaries of the domain of stabi- lity in the space of orbits.It is important to underline, that because the parameters from A can be analytically presented with the help of the coordinates of the fixed points, the boundaries of the domain of stability in the space of orbits depend only on the parameters from A2. There- fore, it is possible to move an equilibrium to the preset given point of the boundary with known bifurcation effect.
In conclusion, the mapping of the domain of stability of equilibria from the space P of all co- efficients of the characteristic polynomials eigen- values into the space of orbits together with the immovability of the boundaries of the domain of stability in the space of orbits give the possibility to describe all admissible qualitative features of the behavior of the iteration process near the boundaries of the stability domain.The travels of the equilibrium in the space of orbits on the segments of straight lines and the crossing the boundaries of the stability domain reveal the plethora of the possible ways from stability, peri- odicity, Arnold horns and quasi-periodicity to chaos.It is important to stress that the travels of equilibria also reveal geometrically and numeri- cally the mechanism by which the mode-locking areas of periodic resonances destroy quasi-peri- odic orbits without using the elaborate analytical techniques.The numerical procedure of the de- scription of such phenomena includes the con- struction of spatial bifurcation diagrams in which the bifurcation parameter is the equilibrium it- self.
The organization of the travels of equilibria in the space of orbits on the segments of straight lines can be done in the following way: it is possible to parametrize the segment of the straight line between the equilibria x and y as x(j)-x(1-)+y, j-0,1,...,T, (14)   where j is a bifurcation parameter and T is a number of bifurcation steps.In such a way a pla- nar bifurcation diagram can be constructed.The usual (linear or one-dimensional) bifurcation dia- gram can be obtained from ( 14) by the fixation of some coordinate of the vectors x(j).
Thus, for each iteration process with a big en- ough number of the control bifurcation para- meters it is possible to construct the realization of this iteration process with a preset combina- tion of qualitative properties of orbits (cf.Sonis,  1990; 1993; 1994).

REALIZATION OF THE LINEAR BIFURCATION ANALYSIS FOR TWO-DIMENSIONAL AUTONOMOUS ITERATION PROCESSES
In this section we present in brief a two-dimen- sional realization of the linear bifurcation analysis.The form of this realization can be extended in the same manner to a multi-dimensional case.
The standard linear stability analysis of the general two-dimensional discrete map ( 15) is based on the consideration of the general Jacobi matrix (see, for example, Hsu, 1977; Thompson and  Stewart, 1986, pp.150-161; Sonis, 1990).
By the well-known yon Neumann theorem, the equilibrium (x*, y*) is asymptotically stable if and only if for all its eigenvalues #l, 2 the following conditions hold: (20) The outcome of the general Routh-Hurwitz stability conditions (11) in the case n 2 for the polynomial 12 q_ al# + a2 is bo-+ a + a2; b -2-2a2; (21) b2 a + a2; and the stability domain in the space P of para- meters al, a2 is defined by the linear inequalities" J(t+l t) I OG/Ox OG/Oy 1 OH/Ox OH/Oy (16) and its value J* on the fixed point x*, y* -lal <a2< 1. ( In the plane of the coefficients al, a2 the do- main of stability defined by the conditions ( 22) is the triangle ABC with the vertices (see Fig.  The eigenvalues of the Jacobi matrix J* are the solutions of the characteristic polynomial /,2 at a# + a2 2 Tr J*# + det J* 0, (18)

M. SONIS
The parabola a2--1/4al 2 divides the triangle into two major domains: the eigenvalues are real outside of the parabola and are complex conjugate inside of this parabola.On the parabola itself the eigenvalues are equal.The sides of the triangle of stability are generated by the following straight lines: (25) with f 1/2, f 1 / 4 .Other rational fractions f p/q represent points of weak resonance.
The same periodic behavior is also observed in a small domain of f near p/q.This domain, the mode-locking domain, is the image of the Arnold tongue from the corresponding domain of change in eigenvalues in the complex plane (Arnold, 1977).For strong resonance, the mode-locking domain starts within the domain of stability (Kogan, 1991).
If f is not rational, the quasi-periodic motion of orbits appears.
Presenting al -Tr J* and a2 detJ* through the coordinates x*, y* of the equilibrium one ob- tains in the space of orbits the domain of stabi- lity of equilibria; boundaries of this domain are the following curves: the divergence boundary with the equation Tr J* A* + 1; (27) the flip boundary with the equation On the divergence boundary at least one of the eigenvalues is equal to 1. Crossing of this bound- ary allows for orbits to be repelled from the equi- librium.Such divergence starts from within the domain of stability; this domain is the diver- gence-locking domain.
On the flip boundary at least one of the eigen- values is equal to -1.Each point on the flip boundary corresponds to a two-periodic cycle, and movement outside the domain of stability generates the Feigenbaum type period doubling sequence, leading to chaos (Feigenbaum, 1978).
On the flutter boundary 1#11-1#2]-1.It is easy to describe the type of bifurcations in all points on the flutter boundary.The condition al 121 means that #1 e i2fl, #2 e-i2fl, 0 _< ft <_ 1, and therefore, al Tr J* 1 -4-#2 2 cos 27rf. (26) If f is a rational fraction" f-p/q, then we have q-periodic (resonance) fixed points; between them there are fixed points of strong resonance (29) (It should be mentioned that in three-dimen- sional case we will have the divergence plane, the flip plane and the flutter saddle.It is important to note that for the higher dimensions the invar- iant tori including periodic and quasi-periodic motion appear.This issue will be considered else- where.) In the next sections the ideas of bifurcation ana- lysis will be applied for the specific cases of a new general model of discrete relative multiple population/multiple location socio-spatial dynamics (see Dendrinos and Sonis, 1990).
We will start from one population (stock)/n locations case.Let the vector x(t)=(xl(t),x2(t),...,xn(t)), t=O, 1,2,... be the relative population size distribution at time between n locations.Such a formulation could be specified for any socio-economic quantity, normalized over a regional or national total.
The one population/multiple location relative discrete socio-spatial dynamics then is given by: xi(t / 1) Fi(x(t))/Fj(x(t)), j=l 1, 2, n; t--0, 1, 2, ...; (31) J The expression Fi(x(t)) is the locational com- parative advantages enjoyed by the population at (i, t).Functions Fi depend on the relative distri- bution of the population in all locations, and on other environmental parameters.
A specific log-linear formulation for the func- tions Fi with the universality properties may be represented by the following: Fi(x(t)) Ai H xj(t)aij; J -oc<aij+oc; Ai>O, i= 1,2,...,n; (32) where A1, A2,..., An are the composite loca- tional advantages of the locations 1, 2, n, and the matrix [[ai..[[ is the matrix of the compo- site elasticities of relative population growth.This iteration process can reproduce each preset dynamic behavior including stability, periodic motion, quasi-periodicity and various forms of chaotic movement.
A specific log-log-linear formulation for the functions Fi may be represented by the following functions Fi(x(t)) of the exponential form: Fi(x(t)) exp Wi H xj(t)aij J (33) -oe<aij<+oc; i= 1, 2, n; where the matrix Ilaijll is the matrix of the spa- tio-temporal composite elasticities.
It is important to stress that the relative dy- namics (30) can be generated by the following extreme principle (cf.Gontar, 1981; Sonis and  Gontar, 1992): the relative Socio-Spatial dy- namics proceed in such a way that in the transfer from time to time / the information functional i(t,t + 1) xi(t + 1) i=1 x [lnxi(t + 1)-In f.(x(t))-1] (34) reaches its minimum in the space of vectors x(t + 1) subject to the conservation condition: xi(t / 1) 1. i=1 This extreme principle defines a new law of collective non-local population redistribution behavior which is a meso-level counterpart of the utility optimization individual behavior.Moreover, it is possible to formulate a more general extreme principle which will generate the multinomial relative socio-spatial dynamics as well as an arbitrary iteration process with the help of informational functionals of the universal analytical form.Such a principle represents the collective local and non-local synergetic interac- tions between the constituencies of an arbitrary autonomous iteration process (Sonis and Gontar,  1992).It should be mentioned that the informa- tion minimization principle is the discrete analo- gue of the problem stated and solved by Vito Volterra in 1939; to construct the Hamilton var- iational principle for the logistic type system of differential equations describing the "struggle for existence".The analytical form of the informa- tion minimization principle is similar to the gen- eralization of the Volterra principle in modern Innovation Diffusion theory (Sonis, 1992).
First of all let us describe the space of orbits of the dynamics (37).For this purpose the bary- centric coordinates within the Moebius triangle will be used.
1.A Moebius plane as a space of orbits Moebius plane is the two-dimensional space (plane) de- fined by three barycentric coordinates Xl, x2, x3, Xl/Xz/x3=l, of each point within it.The scale element of this plane is the Moebius equi- lateral triangle with the unit scale on its sides.This triangle is generated by three coordinate axes (Fig. 2).It is possible to measure the bary- centric coordinates of each point in Moebius plane by projecting it (parallel to the sides) onto the sides of the Moebius triangle.If the point P lies within the Moebius triangle, then its barycentric coordinates xl, X2, X3 must be between 0 and 1: APPLICATION OF THE CONTROL OF BIFURCATIONS ALGORITHM TO THE STUDY OF THE ONE POPULATION]THREE LOCATION RELATIVE DYNAMICS Consider the following one population/three loca- tion log-log-linear model: If the point Q lies outside the Moebius tri- angle, then one of the barycentric coordinates must be negative and other to be greater than 1,  For the dynamics (37) the Moebius triangle gives the natural way to present the orbits of dynamics and their fixed points.Moreover, be- cause of conditions A2, A3 > 0 the orbits of the relative dynamics occur within the Moebius tri- angle itself.
2. Fixed points Now we will concentrate our- selves on the graphical representation of the be- havior of the non-periodic fixed point Xl, x2, x of the dynamics (37) within the Moebius triangle under arbitrary changes in the parameters A2, A3 > 0 and -oc < 23, 31 < q-CX3.
Eqs. ( 37) and (38) imply that the coordinates of the non-periodic fixed point x, x, x; satisfy the system of equations: x/x* A2 exp(#23x); x/x* A3 exp(#31x); X q-X 2 q-X This system implies that (40) x x exp[#23A3x' exp(#31x)], x XlA exp(#31xl), (41) and x + xA2 exp[#23A3x exp(#31x)] + X*lA3 exp(#31x)-1. (42) The dynamics (37) have only one non-periodic fixed point.For the proof consider a function It is easy to see that the derivative of this function is positive, and f(x*l) tends to -1 if x tends to 0 + 0, and f(x*) tends to some positive value C if x tends to 0. Thus, the function f(x*l) increases monotonically from -1 to C > 0, and, therefore, there is only one point x between 0 and such thatf(x) 0. This fixed point it is easy to calculate from Eqs. ( 41), ( 42) with the help of the computation of the values of the left part of Eq. ( 42) in two points of the xi-axis.Refinement of the mesh size near suspected fixed point by dividing it in two makes it possible to pin down the location of any fixed point.
3. Changes in the model parameters and linear bifurcation analysis Consider now all models (37) with the fixed positive parameters Az, A3 and changeable parameters #23, #31.It will be shown further, that the position of the domain of stabi- lity and the flip, flutter and divergence bound- aries are prescribed by the values Az, A3 only, while the position of the equilibrium depends on all parameters A2, A3,#23,31.By changing the appropriate parameters 23,#31 one can put the non-periodic equilibrium into an arbitrary place within the domain of stability.Thus, the para- meters #23, #31 are plying a role of external bifur- cation parameters.Eqs.(40) give the following dependence of these external bifurcation para- meters on the coordinates of the fixed point: (43) These relationships allow to convert the fixed point of the dynamics (37) into the internal bifur- cation parameters.The preset choice of the movement of fixed point in the space of orbits (for example, on the straight line between two points of the Moebius triangle) can be converted with the help of the formulas (43) into the change of the external parameters 23,/z31 con- trolling the model bifurcations.(49) J(t + 1, t)-Ilso.(t+ 1, t)ll.
to the flip, flutter and divergence boundaries.The travel of equilibria along the straight line be- tween the points (0.1,0.1,0.8) and (0.1,0.85, 0.05) is chosen with the purpose to show the transfer from stability to flutter and to flip bifurcations (see Fig. 3). Figure 4 presents the usual bifurca- tion diagram for the first coordinates x(t) of the orbits.This diagram shows the following sequence of qualitative phenomena: stable two- periodic cycle, stable attractor, series of resonances /' i : ; ' x ., , segment of movement ,/" ' : : . ,: f",;' of equilibria /" . ." ' . ,.including the Arnold tongue for three-period strong resonance, stable attractor and the mode- locking tongue for two-periodic cycle starting within the domain of stability.The corresponding planar bifurcation diagram is presented in Fig. 5 where the two locuses of invariant curves are clearly visible.The reader can find other examples of the ap- plications of linear bifurcation analysis to the labor-capital core-periphery relative discrete dynamics and to the analysis of new bifurcation phenomena in the classical Henon map in Sonis (1993; 1994; 1996).

CONCLUSIONS
This study presents three-tier vision of the recent developments in the discrete non-linear dynamics: the level of new mathematical models of the dis- crete non-linear dynamics recently developed in different social and natural fields of inquiry; the level of unified conceptual framework of the information minimizing or entropy maximizing principles for discrete non-linear dynamics and the level of linear bifurcation analysis defining the domains of structural stability and bound- aries of structural changes in the qualitative properties of orbits.The development of the spe- cific "calculus of bifurcations" obtains at present the theoretical and practical importance espe- cially in connection with the new emerging inter- est to the analysis of the sustainability properties of economic, social and societal dynamics.

FIGURE 4
FIGURE 4 Bifurcation diagram for the population share Xl.

FIGURE 5
FIGURE 5 Planar bifurcation diagram: one population/ three location relative dynamics.
4. The Jacobi matrix Consider the slope-response