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

In other area of parameters 1*c* and 2*c* the Kuramoto-Tsuzuki (Ginzburg-Landau) equation (53) has a stable automodel solution *Wr Fr i ar* ( , ) ( )exp( ( ( ))) . If *a r kr* ( ) then 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 areas of parameters - spatio-temporal or diffusion chaos.

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. But further investigations of solutions of the Kuramoto-Tsuzuki (Ginzburg-Landau) equation (53) directly in its phase space showed that in reality subharmonic cascade of bifurcations of stable two-dimensional tori with arbitrary period in accordance with the Sharkovskii order in every frequency and in two frequencies simultaneously takes place in this equation.

Universality of Transition to Chaos in All Kinds of Nonlinear Differential Equations 167

Feigenbaum toroidal singular attractor (b), period three torus (c) and more complex toroidal singular

Note that in monograph (Magnitskii & Sidorov, 2006) one can find full bifurcation diagram of existence of various subharmonic cascades of bifurcations of two-dimensional invariant tori in the second boundary value problem for the Kuramoto-Tsuzuki (Ginzburg-Landau)

For the analysis of running waves and spatio-temporal chaos in autooscillating active mediums we apply the method used in the Section 4.1.2 for the analysis of mechanisms of formation of running waves, impulses and diffusion chaos in nonlinear excitable mediums. Let's show, that in case of autooscillating active mediums role of cascades of bifurcations of limit cycles converging to a separatrix loop of singular point is plaid by cascades of bifurcations of two-dimensional tori of four-dimensional system of ordinary differential equations converging to singular two-dimensional homoclinic structure, being the Cartesian product of a singular limit cycle on a separatrix loop of singular point. Thus the four-dimensional system has infinite number of subharmonic and homoclinic toroidal singular attractors, generating spatio-temporal chaos in original autooscillating system of partial differential equations. The solutions of four-dimensional system specifying movement on the singular homoclinic structure, tend to the periodic singular solution at

 . Thus, formation of running waves and spatio-temporal chaos in autooscillating active mediums also is described by the universal bifurcation Feigenbaum-Sharkovskii-

Rewrite the Cauchy problem on a straight line for the Kuramoto-Tsuzuki (Ginzburg-Landau) equation with complex-valued function *W x t u x t iv x t* ( ,) ( ,) ( ,) as system of two

> 1 2 1 2 0 0

We shall search a solution of system of Eqs. (54) as a running wave

down the system of (54) as the system of two ordinary differential equations of the second

concerning the second derivatives *u* and *v* and passing to phase variables *uu zvv r* , , ,

*t xx xx <sup>t</sup> xx xx u uu с v u cv u v v v с u v с u vu v x ux u x vx v x t* 

, ( ,0) ( ), ( ,0) ( ), 0 .

2 2 2 2 1 2 1 2 *сuuu с v u cv u v* ( )( ), *сv v с u v* ( )( ), *с u vu v* (55)

2 2 2 2

. Resolving the system of Eqs. (55)

*x ct* and write

( )( ), ( )( ),

(54)

attractor (d).

order

Magnitskii theory.

equation (53) in the space of parameters 1 2 (,) *c c* .

*4.2.2. Running waves and chaos in autooscillating mediums* 

parabolic equations with real variables *uxt* ( ,) and *vxt* ( ,)

where the derivative undertakes on a variable

*u x t u x ct v x t v x ct* ( , ) ( ), ( , ) ( ) . Let's enter an automodel variable

we shall receive four-dimensional system of ordinary differential equations

It was considered the second boundary value problem on a segment [0, ]*l* for equation (53) and it was constructed four-dimensional subspace ( (0), (0), ( / 2), ( / 2)) *u v ul vl* of infinitelydimensional phase space of the problem. Then for different values of bifurcation parameters 1*c* and 2*c* the section of four-dimensional subspace has been carried out by the plane *u l*( / 2) 0 and there were considered projections of this section on the plane ( (0), ( / 2)) *u vl* . Such method of the analysis of phase space of solutions of Kuramoto-Tsuzuki (Ginzburg-Landau) equation (53) appeared extremely fruitful and has enabled to find in the equation all cascades of bifurcations of two-dimensional tori in accordance with the theory FSM (see Figs. 25-26).

**Figure 25.** Bifurcation cascade on internal frequency in the equation (53). Projections of section *u l*( / 2) 0 on the plane ( (0), ( / 2)) *u vl* of two-dimensional invariant tori: period four torus (a), period eight torus (b), Feigenbaum toroidal singular attractor (c) and more complex toroidal singular attractor (d).

**Figure 26.** Bifurcation cascade on external frequency in the equation (53). Projections of section *u l*( / 2) 0 on the plane ( (0), ( / 2)) *u vl* of two-dimensional invariant tori: period two torus (a),

Feigenbaum toroidal singular attractor (b), period three torus (c) and more complex toroidal singular attractor (d).

Note that in monograph (Magnitskii & Sidorov, 2006) one can find full bifurcation diagram of existence of various subharmonic cascades of bifurcations of two-dimensional invariant tori in the second boundary value problem for the Kuramoto-Tsuzuki (Ginzburg-Landau) equation (53) in the space of parameters 1 2 (,) *c c* .

## *4.2.2. Running waves and chaos in autooscillating mediums*

166 Nonlinearity, Bifurcation and Chaos – Theory and Applications

But further investigations of solutions of the Kuramoto-Tsuzuki (Ginzburg-Landau) equation (53) directly in its phase space showed that in reality subharmonic cascade of bifurcations of stable two-dimensional tori with arbitrary period in accordance with the Sharkovskii order in

It was considered the second boundary value problem on a segment [0, ]*l* for equation (53) and it was constructed four-dimensional subspace ( (0), (0), ( / 2), ( / 2)) *u v ul vl* of infinitelydimensional phase space of the problem. Then for different values of bifurcation parameters 1*c* and 2*c* the section of four-dimensional subspace has been carried out by the plane *u l*( / 2) 0 and there were considered projections of this section on the plane ( (0), ( / 2)) *u vl* . Such method of the analysis of phase space of solutions of Kuramoto-Tsuzuki (Ginzburg-Landau) equation (53) appeared extremely fruitful and has enabled to find in the equation all cascades of

every frequency and in two frequencies simultaneously takes place in this equation.

bifurcations of two-dimensional tori in accordance with the theory FSM (see Figs. 25-26).

**Figure 25.** Bifurcation cascade on internal frequency in the equation (53). Projections of section *u l*( / 2) 0 on the plane ( (0), ( / 2)) *u vl* of two-dimensional invariant tori: period four torus (a), period eight torus (b),

Feigenbaum toroidal singular attractor (c) and more complex toroidal singular attractor (d).

**Figure 26.** Bifurcation cascade on external frequency in the equation (53). Projections of section *u l*( / 2) 0 on the plane ( (0), ( / 2)) *u vl* of two-dimensional invariant tori: period two torus (a),

For the analysis of running waves and spatio-temporal chaos in autooscillating active mediums we apply the method used in the Section 4.1.2 for the analysis of mechanisms of formation of running waves, impulses and diffusion chaos in nonlinear excitable mediums. Let's show, that in case of autooscillating active mediums role of cascades of bifurcations of limit cycles converging to a separatrix loop of singular point is plaid by cascades of bifurcations of two-dimensional tori of four-dimensional system of ordinary differential equations converging to singular two-dimensional homoclinic structure, being the Cartesian product of a singular limit cycle on a separatrix loop of singular point. Thus the four-dimensional system has infinite number of subharmonic and homoclinic toroidal singular attractors, generating spatio-temporal chaos in original autooscillating system of partial differential equations. The solutions of four-dimensional system specifying movement on the singular homoclinic structure, tend to the periodic singular solution at . Thus, formation of running waves and spatio-temporal chaos in autooscillating active mediums also is described by the universal bifurcation Feigenbaum-Sharkovskii-Magnitskii theory.

Rewrite the Cauchy problem on a straight line for the Kuramoto-Tsuzuki (Ginzburg-Landau) equation with complex-valued function *W x t u x t iv x t* ( ,) ( ,) ( ,) as system of two parabolic equations with real variables *uxt* ( ,) and *vxt* ( ,)

$$\begin{aligned} \mathbf{u}\_t &= \mathbf{u} + \mathbf{u}\_{xx} - \mathbf{c}\_1 \mathbf{v}\_{xx} - (\mathbf{u} - \mathbf{c}\_2 \mathbf{v})(\mathbf{u}^2 + \mathbf{v}^2), \ \mathbf{v}\_t = \mathbf{v} + \mathbf{c}\_1 \mathbf{u}\_{xx} + \mathbf{v}\_{xx} - (\mathbf{c}\_2 \mathbf{u} + \mathbf{v})(\mathbf{u}^2 + \mathbf{v}^2), \\ &- \infty < \mathbf{x} < \infty, \ \mathbf{u}(\mathbf{x}, 0) = \mathbf{u}\_0(\mathbf{x}), \ \mathbf{v}(\mathbf{x}, 0) = \mathbf{v}\_0(\mathbf{x}), \ 0 \le t < \infty. \end{aligned} \tag{54}$$

We shall search a solution of system of Eqs. (54) as a running wave *u x t u x ct v x t v x ct* ( , ) ( ), ( , ) ( ) . Let's enter an automodel variable *x ct* and write down the system of (54) as the system of two ordinary differential equations of the second order

$$-c\dot{\boldsymbol{u}} = \boldsymbol{u} + \ddot{\boldsymbol{u}} - c\_1 \ddot{\boldsymbol{v}} - (\boldsymbol{u} - c\_2 \boldsymbol{v})(\boldsymbol{u}^2 + \boldsymbol{v}^2),\\ -c\dot{\boldsymbol{v}} = \boldsymbol{v} + c\_1 \ddot{\boldsymbol{u}} + \ddot{\boldsymbol{v}} - (c\_2 \boldsymbol{u} + \boldsymbol{v})(\boldsymbol{u}^2 + \boldsymbol{v}^2),\tag{55}$$

where the derivative undertakes on a variable . Resolving the system of Eqs. (55) concerning the second derivatives *u* and *v* and passing to phase variables *uu zvv r* , , , we shall receive four-dimensional system of ordinary differential equations

$$\begin{aligned} \dot{u} &= z, \quad \dot{z} = \left( -\mu - cz - c\_1(\upsilon + cr) + \left( (c\_1 c\_2 + 1)\mu + (c\_1 - c\_2)\upsilon \right) (\mu^2 + \upsilon^2) \right) / \left( 1 + c\_1^2 \right), \\ \dot{\upsilon} &= r, \quad \dot{r} = \left( -\upsilon - cr + c\_1(\mu + cz) + \left( (c\_1 c\_2 + 1)\upsilon + (c\_2 - c\_1)\mu \right) (\mu^2 + \upsilon^2) \right) / \left( 1 + c\_1^2 \right), \end{aligned} \tag{56}$$

The greatest interest, as well as in the case of excitable mediums, represents presence in the system of (56) cascades of bifurcations on parameter *c* , not entering obviously to system of the equations (54) and being the value of velocity of perturbation distribution along a spatial axis *x* . This case means, that the system of the Kuramoto-Tsuzuki (Ginzburg-Landau) equations (54) with the fixed parameters 1*c* and 2*c* 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 chaos.

Let's illustrate the last statement with an example of system of Eqs. (56) with the fixed values of parameters 1*c* 2 and 2*c* 0.1 . At these values of parameters the singular periodic solution

$$\alpha = k \cos(\alpha \underline{x}), \quad \upsilon = k \sin(\alpha \underline{x}), \quad \alpha = \left(c + \sqrt{c^2 - 4c\_2(c\_1 - c\_2)}\right) / \left(2(c\_1 - c\_2)\right), \quad k = \sqrt{1 - \alpha^2}$$

Universality of Transition to Chaos in All Kinds of Nonlinear Differential Equations 169

*4.2.3. Spiral waves and chaos in two-dimensional autooscillating mediums* 

1 2

Detailed numerical analysis of the problem with initial conditions

00 0

4

, 0

*m n*

W 0.1 cos cos [1 / ( 1)]

was carried out in the paper (Karamisheva, 2010) (see also (Magnitskii, 2011)) by the method of Poincare sections of finite-dimensional subspaces of infinitely-dimensional phase space. It was shown that for 1*c l* 0.5, 2 spiral waves in the plane (,) *x y* appear at 2*c* -0.65 (see Fig. 28a for 2*c* = 0.68 ). Then for four pairs of points 1 1 (,) *x y* and 2 2 (,) *x y* , laying near the centers of four spiral waves, projections of sections 1 1 *ux y* (,)0 on the plane of coordinates 11 22 ( ( , ), ( , )) *vx y ux y* were constructed. The projection corresponding to a neighborhood of the center of the bottom spiral wave is represented in Fig. 28b. Thus, the Fig. 28 specifies that stable two-dimensional invariant torus is an image of a simple one-coil spiral wave in phase

*mx ny u iv i m*

*l l*

Landau) equation in spatially two-dimensional area:

0

spiral waves, that are functions of a kind

equation (57), chaotic or turbulent regimes.

Sharkovskii-Magnitskii (FSM) theory.

space of solutions of the problem (57).

*t xx yy*

Let's consider the second boundary value problem for Kuramoto-Tsuzuki (Ginzburg-

with complex-valued function *W t u x y t iv x y t* (x,y, ) ( , , ) ( , , ) . Well-known that solutions of the problem of Eqs. (57) can be plane waves, concentric phase waves (peasmakers) and also

( () ) ( ) , cos , sin . *i t ar m W Rre xr yr*

Solutions with 1 *m* correspond to one-coil spiral waves, with 1 *m* - many-coils spiral waves. Spiral waves can be represented on a plane (,) *x y* by two kinds of areas, in one of which (shaded) *uxyt Wxyt* ( , , ) Re ( , , ) 0 , and in another (not shaded) *uxyt Wxyt* ( , , ) Re ( , , ) 0 . It is known also, that in some areas of change of values of parameters 1 2 (,) *c c* the quantity of spiral waves starts to increase, that results finally in their destruction and to a forming in the active autooscillating medium, described by the

We show, that the mechanism of formation of spiral waves and turbulent regimes (spatiotemporal chaos) in the boundary value problem (57) for two-dimensional Kuramoto-Tsuzuki (Ginzburg-Landau) equation is subharmonic and homoclinic cascades of bifurcations of two-dimensional and many-dimensional tori in infinitely-dimensional phase space of variables ( ( , ), ( , )) *uxy vxy* that also satisfy the universal bifurcation Feigenbaum-

 

*W W ic W W ic W W x l y l Wxy W xy W yt W lyt W x t W xlt* 

(1 )( ) (1 ) , 0 , 0 , ( , ,0) ( , ), (0, , ) ( , , ) ( ,0, ) ( , , ) 0

2

(57)

 *x xy y*

of the system of Eqs. (56) is a stable cycle for 1.306 *c* . At smaller values of parameter *c* a stable two-dimensional torus is born from the singular cycle as a result of Andronov-Hopf bifurcation. At the further reduction of values of parameter *c* in system of Eqs. (56) the Feigenbaum cascade of period doubling bifurcations of stable two-dimensional tori on external frequency is realized. Then in system of Eqs. (56) the full subharmonic cascade of bifurcations of stable two-dimensional tori is realized according to the Sharkovskii order and then Magnitskii homoclinic cascade of bifurcations of stable tori is realized converging to the singular homoclinic structure being the Cartesian product of the original singular limit cycle on the separatrix loop of the singular point. Projections of Poincare section ( 0, 0) *u z* of some basic two-dimensional tori and singular toroidal attractors on the plane (,) *r v* are presented in Fig. 27.

**Figure 27.** Projections of Poincare section ( 0, 0) *u z* : period two torus (a), toroidal singular Feigenbaum attractor (b), period three torus (c), period four torus from homoclinic cascade (d) and more complex singular toroidal attractor (e) in the system of Eqs. (56).

### *4.2.3. Spiral waves and chaos in two-dimensional autooscillating mediums*

168 Nonlinearity, Bifurcation and Chaos – Theory and Applications

also infinite number of various regimes of spatio-temporal chaos.

 

periodic solution

(,) *r v* are presented in Fig. 27.

22 2

22 2

2 2

1 1 2 1 2 1

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

The greatest interest, as well as in the case of excitable mediums, represents presence in the system of (56) cascades of bifurcations on parameter *c* , not entering obviously to system of the equations (54) and being the value of velocity of perturbation distribution along a spatial axis *x* . This case means, that the system of the Kuramoto-Tsuzuki (Ginzburg-Landau) equations (54) with the fixed parameters 1*c* and 2*c* can have infinite number of various autowave solutions of any period running along a spatial axis with various velocities, and

Let's illustrate the last statement with an example of system of Eqs. (56) with the fixed values of parameters 1*c* 2 and 2*c* 0.1 . At these values of parameters the singular

21 2 1 2 *uk vk* cos( ), sin( ), ( 4 ( )) / (2( )), 1

**Figure 27.** Projections of Poincare section ( 0, 0) *u z* : period two torus (a), toroidal singular Feigenbaum attractor (b), period three torus (c), period four torus from homoclinic cascade (d) and

more complex singular toroidal attractor (e) in the system of Eqs. (56).

of the system of Eqs. (56) is a stable cycle for 1.306 *c* . At smaller values of parameter *c* a stable two-dimensional torus is born from the singular cycle as a result of Andronov-Hopf bifurcation. At the further reduction of values of parameter *c* in system of Eqs. (56) the Feigenbaum cascade of period doubling bifurcations of stable two-dimensional tori on external frequency is realized. Then in system of Eqs. (56) the full subharmonic cascade of bifurcations of stable two-dimensional tori is realized according to the Sharkovskii order and then Magnitskii homoclinic cascade of bifurcations of stable tori is realized converging to the singular homoclinic structure being the Cartesian product of the original singular limit cycle on the separatrix loop of the singular point. Projections of Poincare section ( 0, 0) *u z* of some basic two-dimensional tori and singular toroidal attractors on the plane

*uz z u сz с v cr с с u с c vu v c vr r v сr с u cz с с v с cuu v c* 

1 1 2 2 1 1

*с с сс с с с k*

(56)

Let's consider the second boundary value problem for Kuramoto-Tsuzuki (Ginzburg-Landau) equation in spatially two-dimensional area:

$$\begin{aligned} \mathcal{W}\_t &= \mathcal{W} + (1 + ic\_1)(\mathcal{W}\_{xx} + \mathcal{W}\_{yy}) - (1 + ic\_2) \left| \mathcal{W} \right|^2 \mathcal{W}\_\prime \ 0 \le \mathbf{x} \le l\_\prime \ 0 \le y \le l\_\prime \\ \mathcal{W}(\mathbf{x}, y, 0) &= \mathcal{W}\_0(\mathbf{x}, y), \mathcal{W}\_\mathbf{x}(0, y, t) = \mathcal{W}\_\mathbf{x}(l\_\prime y, t) = \mathcal{W}\_y(\mathbf{x}, 0, t) = \mathcal{W}\_y(\mathbf{x}, l\_\prime t) = 0 \end{aligned} \tag{57}$$

with complex-valued function *W t u x y t iv x y t* (x,y, ) ( , , ) ( , , ) . Well-known that solutions of the problem of Eqs. (57) can be plane waves, concentric phase waves (peasmakers) and also spiral waves, that are functions of a kind

$$\mathcal{W} = R(r)e^{i(\alpha\vartheta + a(r) + m\phi)}, \ \mathbf{x} = r\cos\phi, \ y = r\sin\phi.$$

Solutions with 1 *m* correspond to one-coil spiral waves, with 1 *m* - many-coils spiral waves. Spiral waves can be represented on a plane (,) *x y* by two kinds of areas, in one of which (shaded) *uxyt Wxyt* ( , , ) Re ( , , ) 0 , and in another (not shaded) *uxyt Wxyt* ( , , ) Re ( , , ) 0 . It is known also, that in some areas of change of values of parameters 1 2 (,) *c c* the quantity of spiral waves starts to increase, that results finally in their destruction and to a forming in the active autooscillating medium, described by the equation (57), chaotic or turbulent regimes.

We show, that the mechanism of formation of spiral waves and turbulent regimes (spatiotemporal chaos) in the boundary value problem (57) for two-dimensional Kuramoto-Tsuzuki (Ginzburg-Landau) equation is subharmonic and homoclinic cascades of bifurcations of two-dimensional and many-dimensional tori in infinitely-dimensional phase space of variables ( ( , ), ( , )) *uxy vxy* that also satisfy the universal bifurcation Feigenbaum-Sharkovskii-Magnitskii (FSM) theory.

Detailed numerical analysis of the problem with initial conditions

$$\mathbf{W}\_0 = \boldsymbol{\mu}\_0 + i\boldsymbol{v}\_0 = 0.1 \sum\_{m,n=0}^4 \cos\frac{\pi m \chi}{l} \cos\frac{\pi n \chi}{l} [1 + i/(m+1)]^2$$

was carried out in the paper (Karamisheva, 2010) (see also (Magnitskii, 2011)) by the method of Poincare sections of finite-dimensional subspaces of infinitely-dimensional phase space. It was shown that for 1*c l* 0.5, 2 spiral waves in the plane (,) *x y* appear at 2*c* -0.65 (see Fig. 28a for 2*c* = 0.68 ). Then for four pairs of points 1 1 (,) *x y* and 2 2 (,) *x y* , laying near the centers of four spiral waves, projections of sections 1 1 *ux y* (,)0 on the plane of coordinates 11 22 ( ( , ), ( , )) *vx y ux y* were constructed. The projection corresponding to a neighborhood of the center of the bottom spiral wave is represented in Fig. 28b. Thus, the Fig. 28 specifies that stable two-dimensional invariant torus is an image of a simple one-coil spiral wave in phase space of solutions of the problem (57).

Universality of Transition to Chaos in All Kinds of Nonlinear Differential Equations 171

**Figure 30.** Spatio-temporal chaos at 1*c* 0.5, <sup>2</sup>*c* -0.9 in the plane (,) *x y* (a) and in projection of

At values of parameters 1*c* 0.5, <sup>2</sup>*c* -0.9 already there are no stable spiral waves on a plane (,) *x y* , and in projections of section 1 1 *ux y* (,)0 of anyone four-dimensional subspace of phase space of solutions the continuous spatio-temporal chaotic regime is

In the chapter it is proved and illustrated with numerous analytical and numerical examples that there exists a uniform universal bifurcation mechanism of transition to dynamical chaos in all kinds of nonlinear systems of differential equations including dissipative and conservative, ordinary and partial, autonomous and non-autonomous differential equations and differential equations with delay arguments. This mechanism is working for all nonlinear continuous models describing both natural and social phenomena of a macrocosm surrounding us, including various physical, chemical, biological, medical, economic and sociological processes and laws. And this universal mechanism is described by the Feigenbaum-Sharkovskii-Magnitskii theory - the theory of development of complexity in nonlinear systems through subharmonic and homoclinic cascades of bifurcations of stable limit cycles or stable two-dimensional or many-

Notice, that theory FSM is also applicable for solutions of Navier-Stokes equations, i.e. it solves a problem of turbulence describing various bifurcation scenarios of transition from laminar to turbulent regimes in spatially three-dimensional problem of motion of a viscous incompressible liquid (Evstigneev *et al*., 2009a,b; Evstigneev *et al*., 2010; Evstigneev & Magnitskii, 2010). The solution of this super complex problem is presented in the separate chapter in the present book. Similar scenarios with classical Feigenbaum scenario and

section of four-dimensional subspace of phase space of solutions of the problem (57) (b).

observed (Fig. 30).

**5. Conclusion** 

dimensional invariant tori.

**Figure 28.** Spiral waves in the plane (,) *x y* at 1*c* 0.5, <sup>2</sup>*c* -0.68 (a) and projection of the section 1 1 *ux y* (,)0 of four-dimensional subspace of phase space near center of bottom spiral wave.

At the reduction of negative values of parameter 2*c* there is a complication of structure of spiral waves and solutions corresponding to them in phase space of a boundary value problem (57). In Fig. 29a the picture of spiral waves on a plane (,) *x y* is shown at value 2*c* 0.7 , and in Fig. 29b the projection of one of two parts of section 1 1 *ux y* (,)0 on a plane of coordinates 11 22 ( ( , ), ( , )) *vx y ux y* for two points from a neighborhood of the center of a spiral wave of the greatest radius from Fig. 29a is shown in the increased scale. It is visible, that in phase space of solutions complex two-dimensional torus of the period three from Sharkovskii subharmonic cascade corresponds to a neighborhood of the center of this spiral wave. In Fig. 29c the projection of section 1 1 *ux y* (,)0 in a neighborhood of the other spiral wave located in a right bottom corner in Fig. 29a is presented. The projection represents the shaded ring area. But the second section by the plane 2 2 *ux y* ( , ) 28 of three-dimensional space of points received after carrying out the first section, gives in coordinates 11 22 ( ( , ), ( , )) *vx y vx y* two closed curves. These curves testify the existence of three-dimensional torus in phase subspace of solutions in a neighborhood of the center of the second spiral wave.

**Figure 29.** Spiral waves in the plane (,) *x y* at 1*c* 0.5, <sup>2</sup>*c* -0.7 (a) and projections of parts of sections of four-dimensional subspace of phase space of solutions of the problem (57) in neighborhoods of two spiral waves (b), (c).

**Figure 30.** Spatio-temporal chaos at 1*c* 0.5, <sup>2</sup>*c* -0.9 in the plane (,) *x y* (a) and in projection of section of four-dimensional subspace of phase space of solutions of the problem (57) (b).

At values of parameters 1*c* 0.5, <sup>2</sup>*c* -0.9 already there are no stable spiral waves on a plane (,) *x y* , and in projections of section 1 1 *ux y* (,)0 of anyone four-dimensional subspace of phase space of solutions the continuous spatio-temporal chaotic regime is observed (Fig. 30).
