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

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-Nagumo equations can be written down in the following general form

$$u\_t = \mathcal{D}u\_{xx} + f(u, v, \mu), \ v\_t = \mathcal{g}(u, v, \mu). \tag{48}$$

Universality of Transition to Chaos in All Kinds of Nonlinear Differential Equations 163

cycle is born from a zero singular point as a result of Andronov-Hopf bifurcation. The singular point *O* becomes a saddle - focus. At the further reduction of values of parameter *c* the cascade of Feigenbaum double period bifurcations of stable limit cycles takes place in system of Eqs. (49)-(50) up to formation of the first singular attractor - Feigenbaum attractor. At the further reduction of values of parameter *c* in system of Eqs. (49)-(50) the full subharmonic cascade of bifurcations of stable cycles is realized according to the Sharkovskii order and then incomplete homoclinic cascade of bifurcations of stable cycles is realized. Last cycles converge to the homoclinic contour - the separatrix loop of the saddle-focus *O*

**Figure 24.** Projections of period two cycle (a), period four cycle (b), singular attractor (c), period three

Obtained result means, that the system of the FitzHugh-Nagumo equations (48) with fixed values of parameters can have infinite number of various autowave solutions of any period running along a spatial axis with various velocities and also infinite number of various

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

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

cycle (d) and homolcinic period four cycle (e) in the system of Eqs. (49).

regimes of spatio-temporal (diffusion) chaos.

*4.1.3. Cycles and chaos in distributed market economy* 

(Fig. 24).

equation.

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

The analysis of solutions of system of Eqs. (48) on a straight line can be carried out by replacement *x ct* and transition to three-dimensional system of ordinary differential equations

$$\dot{\mathbf{u}} = \mathbf{y}, \ \dot{\mathbf{y}} = -(\mathbf{c}\,\mathbf{y} + f(\mathbf{u}, \mathbf{v}, \mu)) / D, \ \dot{\mathbf{v}} = -\mathbf{g}(\mathbf{u}, \mathbf{v}, \mu) / \mathbf{c},\tag{49}$$

where the derivative undertakes on a variable . Thus the switching wave in system of Eqs. (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 described by limit cycle and separatrix loop of a singular point of the system (49).

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

$$f(\mu, \upsilon, \mu) = -(\mu - 1)(\mu - \delta \upsilon) / \,\varepsilon, \quad g(\mu, \upsilon, \mu) = \text{arctg}\,(a \,\upsilon\text{)} - \upsilon,\tag{50}$$

where parameter is a small parameter. Note, that system of Eqs. (48) with polynomial function *f*(,, ) *u v* and function *g*(,, ) *u v* having at everyone *v* final limiting values at *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 values of parameters. Besides that, for 1 / system of Eqs. (49)-(50) has two more singular points \* \* *Ou u* ( ,0, / ) , where value \* *u* is a positive solution of the equation \* \* *arctg u u* ( ) .

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 perturbations distribution along an axis *x* . For *c* 1 ( 1) / (1 ) the limit stable cycle is born from a zero singular point as a result of Andronov-Hopf bifurcation. The singular point *O* becomes a saddle - focus. At the further reduction of values of parameter *c* the cascade of Feigenbaum double period bifurcations of stable limit cycles takes place in system of Eqs. (49)-(50) up to formation of the first singular attractor - Feigenbaum attractor. At the further reduction of values of parameter *c* in system of Eqs. (49)-(50) the full subharmonic cascade of bifurcations of stable cycles is realized according to the Sharkovskii order and then incomplete homoclinic cascade of bifurcations of stable cycles is realized. Last cycles converge to the homoclinic contour - the separatrix loop of the saddle-focus *O* (Fig. 24).

**Figure 24.** Projections of period two cycle (a), period four cycle (b), singular attractor (c), period three cycle (d) and homolcinic period four cycle (e) in the system of Eqs. (49).

Obtained result means, that the system of the FitzHugh-Nagumo equations (48) with fixed values of parameters can have infinite number of various autowave solutions of any period running along a spatial axis with various velocities and also infinite number of various regimes of spatio-temporal (diffusion) chaos.
