**4.1. Diffusion chaos in reaction-diffusion systems**

Wide class of physical, chemical, biological, ecological and economic processes is described by reaction-diffusion systems of partial differential equations

$$u\_t = D\_1 u\_{xx} + f(\mu, \upsilon, \mu), \quad \upsilon\_t = D\_2 \upsilon\_{xx} + g(\mu, \upsilon, \mu), \text{ } 0 \le x \le l,\tag{46}$$

Universality of Transition to Chaos in All Kinds of Nonlinear Differential Equations 161

occurs according to the universal FSM theory both for the

for brusselator system with diffusion

and for coefficients of diffusion

Let us show that the further complication of solutions of brusselator equations (47) at

first and the second boundary value problems. In the beginning we shall consider the first

coefficients 1 2 *D D* 0.15, 0.3 . The Feigenbaum period doubling cascade of bifurcations of stable limit cycles and then the Sharkovskii subharmonic cascade of bifurcations exist in infinitely-dimensional phase space of solutions of the problem. Some main cycles and

**Figure 22.** Singular cycle (a), cycle of the double period (b), the Feigenbaum attractor (c), cycle of period five (d) and one of the singular attractors in the first boundary value problem for the brusselator

In the second boundary value problem singular toroidal attractors were found out in the

1 2 *D D* 0.1, 0.02 . At these fixed values of parameters a two-dimensional stable invariant torus is born from the stable limit cycle in the infinitely-dimensional phase space of the system of Eqs. (47). This torus begins the Feigenbaum period doubling cascade of bifurcations of stable tori on internal frequency generating by the end of cascade the

**Figure 23.** Projections of the section *u l*( / 2) 0 on the plane ( (0), ( / 2)) *u vl* of two-dimensional torus (a), two-dimensional torus of double period on internal frequency (b), two-dimensional torus of period 4 (c) and the Feigenbaum singular toroidal attractor (d) in the second boundary value problems for the

singular attractors of these cascades of bifurcations are presented in Fig. 22.

boundary value problem on the segment [0, ]

brusselator equations for the parameter values *A* 4,*l*

Feigenbaum singular toroidal attractor (see Fig. 23).

growth of values of parameter

equations (47).

brusselator equations (47).

depending on scalar or vector parameter . Such system is very complex system. Behavior of its solutions depends on coefficients of diffusion and their ratio, length of space area and edge conditions. As a rule, there exists a value of the parameter <sup>0</sup> , such that for all 0 reactiondiffusion system has a stable stationary and space homogeneous solution (,) *U V* , denoted as thermodynamic branch. When 0 , then thermodynamic branch loses its stability and after that reaction-diffusion system can have quite different solutions such as periodic oscillations, stationary dissipative structures, spiral waves and nonstationary nonperiodic nonhomogeneous solutions. Last solutions are known as diffusion or spatio-temporal chaos.

## *4.1.1. Diffusion chaos in the brusselator model*

Considered on a segment [0, ]*l* the system of the brusselator equations offered for the first time by the Brussels school of I. Prigoging as a model of some self-catalyzed chemical reaction with diffusion

$$
\mu\_t = D\_1 \mu\_{xx} + A - (\mu + 1)\mu + \mathbf{u}^2 \upsilon\_\prime \quad \upsilon\_t = D\_2 \upsilon\_{xx} + \mu \mu - \mathbf{u}^2 \upsilon\_\prime \tag{47}
$$

It is easily to see, that stationary spatially-homogeneous solution (a thermodynamic branch) of the system of Eqs. (47) is the solution *u Av A* , / . Therefore the first boundary problem for brusselator should satisfy the boundary conditions

$$
\mu(0, t) = \mu(l, t) = A, \ v(0, t) = v(l, t) = \mu \nmid A.
$$

A more detailed analysis shows (Hassard *et al*., 1981; Magnitskii & Sidorov, 2006) that at 0 stable periodic spatially inhomogeneous solutions of the system of Eqs. (47) have the following asymptotic representations for small 1/2 0 ( ) :

$$\ln u(\mathbf{x},t) = A + \varepsilon \cos \alpha t \cdot \sin \frac{\pi \cdot \mathbf{x}}{l} + \mathcal{O}(\varepsilon^2), \\ v(\mathbf{x},t) = \frac{\mu}{A} + \varepsilon \gamma \cos \alpha t \cdot \sin \frac{\pi \cdot \mathbf{x}}{l} + \varepsilon \,\delta \sin \alpha t \cdot \sin \frac{\pi \cdot \mathbf{x}}{l} + \mathcal{O}(\varepsilon^2), \\ \alpha = \frac{\mu}{A} + \mathcal{O}(\varepsilon^2), \quad \varepsilon = \frac{\mu}{A} + \mathcal{O}(\varepsilon^2)$$

where <sup>2</sup> 0 ( ) (1 ( )) *O* , , are some constants, and a kind of spatial harmonics is defined by boundary conditions of a problem. Points of a segment make fluctuations with identical frequency and a constant gradient of a phase. The effect of "wave" running on a segment is created. In the case of the second boundary value problem on a segment with the free ends, the periodic solutions born at 0 will be spatially homogeneous (Magnitskii & Sidorov, 2006).

Let us show that the further complication of solutions of brusselator equations (47) at growth of values of parameter occurs according to the universal FSM theory both for the first and the second boundary value problems. In the beginning we shall consider the first boundary value problem on the segment [0, ] for brusselator system with diffusion coefficients 1 2 *D D* 0.15, 0.3 . The Feigenbaum period doubling cascade of bifurcations of stable limit cycles and then the Sharkovskii subharmonic cascade of bifurcations exist in infinitely-dimensional phase space of solutions of the problem. Some main cycles and singular attractors of these cascades of bifurcations are presented in Fig. 22.

160 Nonlinearity, Bifurcation and Chaos – Theory and Applications

depending on scalar or vector parameter

thermodynamic branch. When 0

reaction with diffusion

0

& Sidorov, 2006).

where <sup>2</sup>

 

0

( ) (1 ( )) *O* ,

free ends, the periodic solutions born at 0

 

 

*4.1.1. Diffusion chaos in the brusselator model* 

**4.1. Diffusion chaos in reaction-diffusion systems** 

by reaction-diffusion systems of partial differential equations

conditions. As a rule, there exists a value of the parameter <sup>0</sup>

 

2 2

of the system of Eqs. (47) is the solution *u Av A* ,

following asymptotic representations for small 1/2

 ,

> 

problem for brusselator should satisfy the boundary conditions

 1 2 ( , , ), ( , , ), 0 , *t xx t xx u Du f uv v Dv guv x l* 

**4. Spatio-temporal chaos in nonlinear partial differential equations** 

Wide class of physical, chemical, biological, ecological and economic processes is described

its solutions depends on coefficients of diffusion and their ratio, length of space area and edge

diffusion system has a stable stationary and space homogeneous solution (,) *U V* , denoted as

that reaction-diffusion system can have quite different solutions such as periodic oscillations, stationary dissipative structures, spiral waves and nonstationary nonperiodic nonhomogeneous solutions. Last solutions are known as diffusion or spatio-temporal chaos.

Considered on a segment [0, ]*l* the system of the brusselator equations offered for the first time by the Brussels school of I. Prigoging as a model of some self-catalyzed chemical

1 2 ( 1) , . *t xx t xx u Du A u uv v Dv u uv*

It is easily to see, that stationary spatially-homogeneous solution (a thermodynamic branch)

*u t ult A v t vlt A* (0, ) ( , ) , (0, ) ( , ) / .

A more detailed analysis shows (Hassard *et al*., 1981; Magnitskii & Sidorov, 2006) that at

<sup>2</sup> <sup>2</sup> ( , ) cos sin ( ), ( , ) cos sin sin sin ( ), *<sup>x</sup> x x uxt A t O vxt <sup>t</sup> t O*

defined by boundary conditions of a problem. Points of a segment make fluctuations with identical frequency and a constant gradient of a phase. The effect of "wave" running on a segment is created. In the case of the second boundary value problem on a segment with the

> 

stable periodic spatially inhomogeneous solutions of the system of Eqs. (47) have the

 ( ) :

 

(46)

, such that for all 0

  reaction-

(47)

 

/ . Therefore the first boundary

. Such system is very complex system. Behavior of

, then thermodynamic branch loses its stability and after

 

 

will be spatially homogeneous (Magnitskii

are some constants, and a kind of spatial harmonics is

0

*l Al l*

 

**Figure 22.** Singular cycle (a), cycle of the double period (b), the Feigenbaum attractor (c), cycle of period five (d) and one of the singular attractors in the first boundary value problem for the brusselator equations (47).

In the second boundary value problem singular toroidal attractors were found out in the brusselator equations for the parameter values *A* 4,*l* and for coefficients of diffusion 1 2 *D D* 0.1, 0.02 . At these fixed values of parameters a two-dimensional stable invariant torus is born from the stable limit cycle in the infinitely-dimensional phase space of the system of Eqs. (47). This torus begins the Feigenbaum period doubling cascade of bifurcations of stable tori on internal frequency generating by the end of cascade the Feigenbaum singular toroidal attractor (see Fig. 23).

**Figure 23.** Projections of the section *u l*( / 2) 0 on the plane ( (0), ( / 2)) *u vl* of two-dimensional torus (a), two-dimensional torus of double period on internal frequency (b), two-dimensional torus of period 4 (c) and the Feigenbaum singular toroidal attractor (d) in the second boundary value problems for the brusselator equations (47).
