*4.1.3. Cycles and chaos in distributed market economy*

162 Nonlinearity, Bifurcation and Chaos – Theory and Applications

replacement

where parameter

function *f*(,, ) *u v*

 *arctg u u* ( ) .

equations

where the derivative undertakes on a variable

consider the system of Eqs. (48)- (49) with nonlinearities

and function *g*(,, ) *u v*

perturbations distribution along an axis *x* . For *c* 1 ( 1) / (1 )

values of parameters. Besides that, for 1 /

singular points \* \* *Ou u* ( ,0, / )

\* \*

*4.1.2. Running waves, impulses and diffusion chaos in excitable mediums* 

Nagumo equations can be written down in the following general form

*u y y cy f u v D v g u v c* , ( ( , , )) / , ( , , ) / ,

Special case of reaction-diffusion systems is the case of systems of the FitzHugh-Nagumo equations describing nonlinear processes occurring in so-called excitable mediums. Examples of such processes are distribution of impulse on a nervous fiber and a cardiac muscle and also various kinds of autocatalytic chemical reactions. The basic property describing a class of excitable mediums is slow diffusion of one variable in comparison with other variable in system of reaction-diffusion (46). Therefore the system of the FitzHugh-

> ( , , ), ( , , ). *t xx <sup>t</sup> u Du f u v v g u v*

It is well-known, that in system of Eqs. (48) in one-dimensional spatial case there can be switching waves, running waves and running impulses, dissipative spatially nongomogeneous stationary structures, and also diffusion chaos - irregular nonperiodic

The analysis of solutions of system of Eqs. (48) on a straight line can be carried out by

(48) is described by separatrix of the system (49) going from its one singular point into another singular point, running wave and running impulse of system of Eqs. (48) are

Let's show, that diffusion chaos in the system of FitzHugh-Nagumo equations (48) is described by singular attractors of the system of ordinary differential equations (49) in accordance with the Feigenbaum-Sharkovskii-Magnitskii (FSM) theory. For this purpose

*u* , describes some kinds of autocatalytic chemical reactions (Zimmermann et al., 1997). It is easy to see that system of Eqs. (49)-(50) has singular point *O*(0,0,0) for any

A case of greatest interest is, naturally, a case when bifurcation parameter is the parameter *c* , not entering obviously in system of the Eqs. (48) and being the value of velocity of

*x ct* and transition to three-dimensional system of ordinary differential

 

is a small parameter. Note, that system of Eqs. (48) with polynomial

, where value \* *u* is a positive solution of the equation

(50)

nonstationary structures named sometimes as biological (or chemical) turbulence.

described by limit cycle and separatrix loop of a singular point of the system (49).

*f*( , , ) ( 1)( ) / , ( , , ) ( ) , *u v u u v g u v arctg u v*

 

  

having at everyone *v* final limiting values at

system of Eqs. (49)-(50) has two more

  the limit stable

. Thus the switching wave in system of Eqs.

(48)

(49)

Another, essentially different example of formation of spatio-temporal chaos in the nonlinear mediums is the distributed model of a market self-developing economy offered by the author and developed then in (Magnitskii & Sidorov, 2006). The model is a system of three nonlinear differential equations, two of which describe the change and intensity of motion (diffusion) of capital and consumer demand in a technology space under the influence of change of profit rate. The last is described by the third ordinary differential equation.

Self-development of market economy is characterized by spontaneous growth of capital and its movement in the technology space in response to differences in profitability. The model describes formation of social wealth, including production, distribution, exchange, and consumption. A distinctive feature of the model is that distribution of profitability (profit rates) determines the direction and the intensity of motion (diffusion) of capital and its spontaneous growth through generation of added value. Three economic agents having their own interests take part in economic processes that are employers, workers and government. In the model, being based on rigorous rules of Karl Marx's theory of added value, self-development of a market economy involves movement and spontaneous growth of capital of employers, which is the result of creation of added value by workers in the circulation process of capital under government control.

Universality of Transition to Chaos in All Kinds of Nonlinear Differential Equations 165

   

Consider now the second boundary-value problem for the system of Eqs. (51) on an interval

Linearize the considered problem in the neighborhood of the thermodynamic branch, one can obtain that it is stable only when 1 2 *d dd* / (Magnitskii & Sidorov, 2006). Thus, we can draw the second important conclusion: high inertia of the capital, slowing down its response to changes in profit rates and consumer demand, also makes the economic system unstable

It is well-known that any solution of the reaction-diffusion system (46) in a neighborhood

1 2 (1 ) (1 ) , *W W ic W ic W W*

oscillations with frequency 2*c* and this solution is stable in some area of parameters 1*c* and

In other area of parameters 1*c* and 2*c* the Kuramoto-Tsuzuki (Ginzburg-Landau) equation

oscillations of the next elements occur with a constant phase lag, that corresponds to movement on space of a phase wave. In a two-dimensional case the equation (53) has also solutions in a kind of leading centers - sequences of running up concentric phase waves, and spiral waves. But equation (53) has also nonperiodic nonhomogeneous solutions in some

From an opinion of most of researchers analysis of such solutions can be successfully fulfilled by using the Galerkin small-mode approximations for reducing the equation (53) to a nonlinear three-dimensional chaotic system of ordinary differential equations. As it was shown in (Magnitskii & Sidorov, 2005b; Magnitskii & Sidorov, 2006), all irregular attractors of reductive three-dimensional system are also singular attractors, and transition to chaos in this system occurs also in accordance with the Feigenbaum-Sharkovskii-Magnitskii (FSM) theory.

*4.2.1. Transition to chaos in Kuramoto-Tsuzuki (Ginzburg-Landau) equation* 

(53) has a stable automodel solution *Wr Fr i ar* ( , ) ( )exp( ( ( )))

of the thermodynamic branch can be approximated by some complex-valued

of the Kuramoto-Tsuzuki (or Time Dependent Ginzburg-

,0 , , 1 2 *с* ,*c* - some real constants. It is evident that for

. Hence, each element of the medium (53) makes harmonious

the equation (53) has a space homogeneous solution

*rr* (53)

. If *a r kr* ( ) then

 \*\*\* 1 11 (,,) , , <sup>1</sup> 1 1 *xyz <sup>d</sup>* 

 

and the thermodynamic branch of this problem

**4.2. Spatio-temporal chaos in autooscillating mediums** 

 

<sup>2</sup>

 

<sup>2</sup>*c* . Such mediums refer to as **autooscillating mediums**.

areas of parameters - spatio-temporal or diffusion chaos.

and lead to its destruction.

solution *W r u r iv r* (,) (,) (,) 

Landau) equation (Kuramoto & Tsuzuki, 1975):

<sup>0</sup> *r x t rR*

 

0

where <sup>2</sup>

arbitrary phase

 , 

<sup>2</sup> *W ic* ( ) exp( ( ))

 

We show that the market economy system can exist in periodic or chaotic regimes only. Periodic regime can have any period in accordance with the theory FSM and any chaotic regime (economic crisis) can be described by some complex cycle or singular attractor.

 The model assumes an unstructured closed economic system that is developing in a finitedimensional Euclidean space *<sup>n</sup> R* , called the technology space. Each point *<sup>n</sup> c R* corresponds to a certain production technology of some commodity and its coordinate , 1,..., *<sup>i</sup> ci n* is the consumption of resource *i* per unit output.

System of market self-developing economy has the form (Magnitskii & Sidorov, 2005; Magnitskii & Sidorov, 2006):

$$\begin{aligned} \frac{\partial \mathbf{x}(t,c)}{\partial t} &= -\text{div}(d\_1(c,\mathbf{x},z)\text{grad}\,z) + b\mathbf{x}((1-\sigma)z-\delta y),\\ \frac{\partial \mathbf{y}(t,c)}{\partial t} &= -\text{div}(d\_2(c,y,z)\text{grad}\,z) + \mathbf{x}(1-(1-\delta)y+\sigma z),\\ \frac{\partial \mathbf{z}(t,c)}{\partial t} &= a(y-d\mathbf{x}).\end{aligned} \tag{51}$$

where *xtc* (,) is a normalized distribution of capital density, *y*(,) *t c* is a normalized distribution of total consumer demand density and *ztc* (,) is a distribution of profit rate at time *t* in the technology space; is government portion of added value (taxis, custom duties, etc.), is employers personal consumption portion of added value and *abd* , , are structural economic parameters. Note that the system of Eqs. (51) is a particular case of systems with multicomponent diffusion, where the activator (the variable providing positive feedback) is the capital and the inhibitor (the variable suppressing capital growth) is the consumer demand.

System of equations describing the variation of macroeconomic variables can be similarly reduced to the form

$$\dot{\mathbf{x}}(t) = b\mathbf{x}((1-\sigma)\mathbf{z} - \delta\mathbf{y}),\\\dot{\mathbf{y}}(t) = \mathbf{x}(1 - (1-\delta)\mathbf{y} + \sigma\mathbf{z}),\\\dot{\mathbf{z}}(t) = \mathbf{a}(\mathbf{y} - d\mathbf{x}).\tag{52}$$

Parameters and are bifurcation parameters in system of Eqs. (52). Increase in values of parameter as well as reduction of values of parameter generate the Feigenbaum cascade of period-doubling bifurcations and then the Sharkovskii subharmonic cascade and chaotic dynamics in system of Eqs. (52) (cycle of period three is presented in Fig.10d). These results gave us possibility to draw the first important conclusion: uncontrolled growth of personal consumption of the employers as well as low government demand for consumer goods (government orders, government support to business, etc.) lead to various crisis phenomena and destroy the economic system.

Consider now the second boundary-value problem for the system of Eqs. (51) on an interval and the thermodynamic branch of this problem

$$\left( \left( \mathbf{x}^\*, y^\*, z^\* \right) \right) = \left( \frac{1 - \sigma}{d \left( 1 - \delta - \sigma \right)}, \quad \frac{1 - \sigma}{1 - \delta - \sigma}, \quad \frac{1 - \delta}{1 - \delta - \sigma} \right)$$

Linearize the considered problem in the neighborhood of the thermodynamic branch, one can obtain that it is stable only when 1 2 *d dd* / (Magnitskii & Sidorov, 2006). Thus, we can draw the second important conclusion: high inertia of the capital, slowing down its response to changes in profit rates and consumer demand, also makes the economic system unstable and lead to its destruction.

## **4.2. Spatio-temporal chaos in autooscillating mediums**

164 Nonlinearity, Bifurcation and Chaos – Theory and Applications

circulation process of capital under government control.

, 1,..., *<sup>i</sup> ci n* is the consumption of resource *i* per unit output.

(,) ( ).

*ztc a y dx <sup>t</sup>*

phenomena and destroy the economic system.

 

1

2

Magnitskii & Sidorov, 2006):

technology space;

reduced to the form

 and 

Parameters

parameter

their own interests take part in economic processes that are employers, workers and government. In the model, being based on rigorous rules of Karl Marx's theory of added value, self-development of a market economy involves movement and spontaneous growth of capital of employers, which is the result of creation of added value by workers in the

We show that the market economy system can exist in periodic or chaotic regimes only. Periodic regime can have any period in accordance with the theory FSM and any chaotic regime (economic crisis) can be described by some complex cycle or singular attractor.

 The model assumes an unstructured closed economic system that is developing in a finitedimensional Euclidean space *<sup>n</sup> R* , called the technology space. Each point *<sup>n</sup> c R* corresponds to a certain production technology of some commodity and its coordinate

System of market self-developing economy has the form (Magnitskii & Sidorov, 2005;

(,) ( ( , , ) ) ((1 ) ),

*xtc div d c x z grad z bx z y <sup>t</sup>*

(,) ( ( , , ) ) (1 (1 ) ),

where *xtc* (,) is a normalized distribution of capital density, *y*(,) *t c* is a normalized distribution of total consumer demand density and *ztc* (,) is a distribution of profit rate at time *t* in the

employers personal consumption portion of added value and *abd* , , are structural economic parameters. Note that the system of Eqs. (51) is a particular case of systems with multicomponent diffusion, where the activator (the variable providing positive feedback) is the capital and the inhibitor (the variable suppressing capital growth) is the consumer demand.

System of equations describing the variation of macroeconomic variables can be similarly

cascade of period-doubling bifurcations and then the Sharkovskii subharmonic cascade and chaotic dynamics in system of Eqs. (52) (cycle of period three is presented in Fig.10d). These results gave us possibility to draw the first important conclusion: uncontrolled growth of personal consumption of the employers as well as low government demand for consumer goods (government orders, government support to business, etc.) lead to various crisis

*x t bx z y y t x y z z t a y dx* ( ) ((1 ) ), ( ) (1 (1 ) ), ( ) ( ).

as well as reduction of values of parameter

is government portion of added value (taxis, custom duties, etc.),

 

are bifurcation parameters in system of Eqs. (52). Increase in values of

 

> 

(51)

is

(52)

generate the Feigenbaum

*ytc div d c y z grad z x y z <sup>t</sup>*

It is well-known that any solution of the reaction-diffusion system (46) in a neighborhood 0 of the thermodynamic branch can be approximated by some complex-valued solution *W r u r iv r* (,) (,) (,) of the Kuramoto-Tsuzuki (or Time Dependent Ginzburg-Landau) equation (Kuramoto & Tsuzuki, 1975):

$$\mathcal{W}\_{\tau} = \mathcal{W} + (1 + \mathrm{i}\varepsilon\_1)\mathcal{W}\_{rr} - (1 + \mathrm{i}\varepsilon\_2) \left| \mathcal{W} \right|^2 \mathcal{W}\_{\prime} \tag{53}$$

where <sup>2</sup> <sup>0</sup> *r x t rR* , ,0 , , 1 2 *с* ,*c* - some real constants. It is evident that for arbitrary phase the equation (53) has a space homogeneous solution <sup>2</sup> *W ic* ( ) exp( ( )) . Hence, each element of the medium (53) makes harmonious oscillations with frequency 2*c* and this solution is stable in some area of parameters 1*c* and <sup>2</sup>*c* . Such mediums refer to as **autooscillating mediums**.
