## A  (c)  Figure 17 Period-halving bifurcation from chaos to cycles of system  with initial value X(0) = (1, 1,1). (a) Chaos when p = 43; (b) phase portrait of 16T period solution for p = 43.7; (c) phase portrait of 8T period solution for p = 43.75; (d) phase portrait of 4T period solution for p = 44; (e) phase portrait of 2T period solution for p = 45; (f) phase portrait of. 0-12V

0-12V

Figure 17 Period-halving bifurcation from chaos to cycles of system  with initial value X(0) = (1, 1,1). (a) Chaos when p = 43; (b) phase portrait of 16T period solution for p = 43.7; (c) phase portrait of 8T period solution for p = 43.75; (d) phase portrait of 4T period solution for p = 44; (e) phase portrait of 2T period solution for p = 45; (f) phase portrait of.

z z y x eventually becomes chaotic. These are two archetypal sequences of bifurcations, or 'routes to chaos'.

### Spatial Turing Bifurcation

Pattern formation in nonlinear complex systems is one of the central problems of the natural, social, and technological sciences. In particular, starting with the pioneering work of Segel and Jackson in 1972, spatial patterns and aggregated population distributions are common in nature and in a variety of spatiotemporal models with local ecological interactions. Promulgated by the theoretical paper of Turing, the field of research on pattern formation modeled by reaction-diffusion systems, providing a general theoretical framework for describing pattern formation in systems from many diverse disciplines, including biology, chemistry, physics, and so on, seems to be a new increasingly interesting area, particularly during the last decade. For more details, see Spatial Models and Geographic Information Systems .

Here, we consider the following spatiotemporal reaction-diffusion model:

— = a - (b + 1)« + u2 v + Du V2 u = f (u, v) + Du V2 u

— = bu - u2 v + Du V2 v = g (u, v) + Dv V2 v where a, b are scaled kinetic parameters. The steady state of the model  is (u*, v*) = (a, b/a).

The characteristic equation of system  is \A - k2D - A/| = 0

where and A is given by

qug qvg

0 Dv

fu fv gu gv

Equation  can be solved, yielding the so-called characteristic polynomial of the original problem (eqn ):

where tr* = fN + gP - k2{Du + Dv) = tr0 - k2 (Du + Dv)

Ak = fugv - fvgu - k2 f„Dv + gvDu) + k^DuDv = A0 - k2 fuDv + gvDu) + k4DuDv

The roots of eqn  yield the dispersion relation Mk)= 1(ttk tr2 - 4Ak j The Hopf bifurcation occurs when

Then we can get the critical value of Hopf bifurcation parameter k bH = 1 + a2 

The Turing bifurcation occurs when

We can obtain the critical value of bifurcation parameter b:

With fixed a = 3, Dv = 10, Figure 18 shows the stability diagram of the spatial uniform steady state of the autonomous system in the (Du, b) plane. The Hopf bifurcation line is horizontal at b= 10.0. It crosses the Turing bifurcation line at Du = 5.19. Both bifurcations are supercritical.

We study the behavior of the system in the oscillatory and oscillatory Turing (AT > 0, Re AH > 0) domains at constant supercriticality, k = 10.2. Figure 19 shows the dispersion curves at the designated points in Figure 18.

 II 6 5 4 3 2 1 Hopf III IV

Figure 18 Bifurcation diagram of spatio-temporal model . Figure 19 Dispersion relations showing unstable Hopf mode and transition of Turing and wave mode from stable to unstable, e.g., as decreases.

The Hopf bifurcation line and the Turing bifurcation line intersect at a co-dimension-2 bifurcation point, the Turing-Hopf bifurcation point (in the numerical case, the Turing-Hopf bifurcation point is (5.19, 10.0)). The bifurcation lines separate the parametric space into four distinct domains. In domain IV, located below the two bifurcation lines, the steady state is the only stable solution of the system. Domain II is a region of pure Hopf instabilities, and domain III is pure Turing instabilities. In domain I, which is located above the two bifurcation lines, both Hopf and Turing instabilities occur.

It is well known that at the Hopf threshold, the temporal symmetry of the system is broken and gives rise to uniform oscillations in space and periodic oscillations in time, while at the Turing threshold, the spatial symmetry of the system is broken and stationary in time and oscillatory in space.

Du 0 