(i) Control of the spatiotemporal dynamics in a generic reaction-diffusion system

Already in the first proposal period, we developed a generic activator-inhibitor reaction-diffusion model, which originally was derived for vertical charge transport through semiconductor structures like the HHED and which is typical for a large class of spatially extended systems in physics, chemistry and biology. Global coupling is achieved through the connected electric load circuit. This model exhibits not only front dynamics but complex and chaotic spatiotemporal scenarios as well [7]. During the report period we have obtained surprising results and have made substantial progress managing to control spatiotemporal patterns using time-delayed feedback methods (Pyragas-control [1,40]).

In dimensionsless units, the model equations for the generic model have the form of a reaction-diffusion system of activator-inhibitor type, with global coupling. We have extended the model by adding the control forces $F_a$ and $F_u$ [15]:

$$\partial_t a(\vec{x},t) = \frac{u-a}{(u-a)^2 + 1} - T a + \Delta a - K F_a(\vec{x},t),$$ (1)

$$\partial_t u(t) = \alpha\left[j_0 - (u-\langle a\rangle)\right] -K F_u(t).$$ (2)

where $a$ is the space dependent activator variable (electron density distribution in the layer vertical to the transport direction) and $u$ is the inhibitor variable (voltage across the device). The first equation represents the continuity equation for electrons that flow through the layer and the second equation is the Kirchhoff-equation for the total current $\sim j_0$ , which causes a global coupling through the voltage drop across the load resistance. The mean spatial charge density distribution $\langle a \rangle$ enters into the current and represents the global coupling. $\alpha$ , $j_0$ and $T$ are system parameters, which represent the time scale, the external control parameter or a parameter determining the bistability regime, respectively.

The spatial variable in the layer $\vec{x}$ can be one- or two-dimensional depending on the problem. In the following we consider the one-dimensional case. The control forces $F_a$ and $F_u$ , each of which is multiplied by a control amplitude $K$ , can be chosen arbitrarily.

Figure 1: Control regimes of the time-delayed feedback in the $K-R$ plane for the generic HHED model with (a) diagonal and (b) local control without inhibitor control. Here $\star$ means successful and $\cdot$ non successful control. The solid lines denote the analytical solution for the boundaries of the control regime according to (5). See [15]

First let $K=0$ (no control). Complex and chaotic behaviour is expected in a parameter range for which, at the same time, conditions for both a spatial instability (current filament) and a temporal one (Hopf bifurcation) are satisfied, as we have shown in the one-dimensional case [7]. The question arises under which circumstances similar behaviour for systems of two spatial dimensions is possible. In collaboration with guest scientist W. Just (London) as well as with project B6 we managed to show, using an amplitude expansion of the supercritical codimension-two-bifurcation, that one should not in general expect a coexistence of Turing- and Hopf instabilities in two dimensions in a (through a second diffusive coupling) locally coupled system unless the system size of the two directions is so small that a quasi-onedimensional dynamics arises. [10].

The generic structure of equations (1) and (2) is underlying for other projects as well. A very interesting collaboration arose with project B4, in which an electrochemical model for pattern formation in electrode surfaces was investigated: a detailed comparison to our globally coupled reaction-diffusion model revealed astonishing similarities in the scenarios of complex spatiotemporal dynamics [11]. Extending the globally coupled two-component reaction-diffusion system by a third diffusive component, one can describe apart form stationary, breathing or spiking current filaments, also moving filaments (or domains, respectively) [27].

Now we consider the case $K\neq 0$ in one spatial dimension.The system parameters are chosen such that for $K=0$ chaotic spatiotemporal spiking arises. Our aim is to stabilize an unstable periodic spatiotemporal orbit, which is characterized by the period $\tau$ and the Floquet exponents $\lambda$ . Furthermore the control forces $F_a$ and $F_u$ should vanish for successful control (noninvasive control).

This can be achieved using variants of time-delayed autosynchronization (Pyragas-control) [1,2,3]. Our goal here was to extend this method to spatiotemporal patterns. As starting point we use a delay feedback loop of the form $F_a=F_{\text{loc}}$ , $F_u=F_{\text{vf}}$ (diagonal coupling), with

$$F_{\text{loc}}(x,t) = a(x,t)- a(x,t-\tau) + R F_{\text{loc}}(x,t-\tau),$$ (3)

$$F_{\text{vf}}(t) = u(t)-u(t-\tau) + R F_{\text{vf}}(t-\tau),$$ (4)

where $R$ is a memory parameter. For the diagonal control the Floquet-exponent $\Lambda$ of the controlled orbit satisfies the exact implicit equation [4],

$$\Lambda+K \frac{1-e^{-\Lambda \tau}}{1-Re^{-\Lambda \tau}} = \lambda.$$ (5)

As shown in Fig. 1(a) , the resulting control domain in the K-R plane is numerically reproduced with high accuracy for the generic model [15]. The control regime is bounded by a flip bifurcation for small $K$ values, and a Hopf bifurcation for large $K$ values. Now it is interesting to examine how the control range deforms when applying other control schemes. One example is local control without inhibitor control, which arises for $F_u=0$ , $F_a=F_{\text{loc}}$ . In Fig. 1(b) one can recognize that thereby new control boundaries arise. A systematic comparison of different local and global control schemes has been performed by collaboration with W. Just (London) and J. Socolar (Duke University, USA). There we could show, for instance by calculating the Floquet spectra, that in Fig. 1(b) for large $R$ and $K$ , the control regime is bounded by a subcritical flip bifurcation, and that for global control the control regime gets even bigger if the inhibitor control is omitted.

Figure 2: (a) Floquet-left eigenmodes $\phi _u(t)$ and $\phi _a(x,t)$ for the largest Floquet-exponents of a periodic orbit, as well as the corresponding Floquet-right eigenmode (b) $\psi _u(t)$ and $\psi _a(x,t)$ . See [14]

Figure: Comparison of Floquet-eigenmode control (solid line) and diagonal control (dashed line). The spatiotemporal average $\epsilon = \langle \vert a(x,t) - a(x,t-\tau)\vert +\vert u(t) - u(t-\tau)\vert\rangle_{x,t}$ versus the control amplitude is plotted $K$ . See [14]

For applications, particularly interesting are control schemes which work with the smallest possible amplitude factor $K$ . In this regard, a new control scheme which we developed in collaboration with W. Just [14], has proven surprisingly efficient. For this control scheme, the Floquet-left eigenmode $\phi_{u/a}$ and the, associated with the adjoint problem, Floquet-right eigenmode $\psi_{u/a}$ are calculated for the largest Floquet-exponent of the orbit to be stabilized. A concrete example of such modes is depicted in Fig. 2. The control forces are then constructed as follows:

$$F_u(t) = \psi_u(t) s(t), \quad\quad F_a(x,t) = \psi_a(x,t) s(t),$$ (6)

$$\quad s(t) = \int_0^L \phi_a(x',t)\left[a(x',t)-a(x',t-\tau)\right] x' + \phi_u(t)\left[u(t) -u(t-\tau) \right]$$ (7)

By applying this control force to the generic model, we discovered that control works even for extremely small $K$ -values, i.e. the control threshold decreases by six orders of magnitude, as demonstrated in Fig. 3 in comparison to diagonal control. Investigating this phenomenon in more detail, we determined that a phase shift $\delta$ of the controlled orbit over the phase of the Floquet modes plays an important role which can be treated by perturbation theory [14]. We also applied this new type of Floquet mode control to the Rössler-Model [24]. In this low dimensional system we succeeded in analysing the dependence of the minimal control amplitudes of the phase shift $\delta$ far beyond the perturbation theory Ansatz.

With a suitable expansion of the Floquet mode control in two unstable modes, we succeeded for the first time to place localized spatiotemporal patterns (spikes) aimed at a chosen position of the system [14]. Whereas normally, stable or unstable spatiotemporal spikes in a globally coupled reaction-diffusion system with Neumann boundary conditions are pinned on the boundary of the system, we could stabilize the spikes for vanishing control force, with tended Floquet mode control, in the centre of the system. This involves the control of an unstable orbit on the repellor.