**2. Case study 1: Evolutionary identification of bifurcations**

This part introduces an overview of possible use of SA on bifurcation, i.e. catastrophic events, detection (for a more detailed description, see [43] ). Catastrophic events here means Thom's catastrophes that can be used under certain conditions to model chaotic dynamics and bifurcations, that appear in the nonlinear behavior of various dynamical systems. The main aim of this work is to show that SA are capable of the bifurcations of chaotic system identification without any partial knowledge of internal structure, i.e. based only on measured data. In this part we will discuss mainly SA results. The system selected for numerical experiments here is the well-known system logistic equation derived from predator - prey system. For each algorithm and its version, simulations have been repeated 50 times. Our world, mostly consisting of nonlinear systems, is full of our i.e. human, technology that is less or more reliable. Technological systems are mostly, like their natural counterparts, nonlinear and complex and very often show chaotic as well as catastrophic behavior according to Thom's catastrophe theory, [1], [3] and [4], that describes sudden changes in the dynamical system (well developed and described for so called gradient systems) behavior under slightly changing (usually) external conditions. These changes, depending on one or more parameters, can be modeled like the special N dimensional surfaces in the so-called parameter space, see for example [1]. As an example of such systems (and catastrophic events), we can mention systems such as electrical networks (blackout,...), economic systems (black Friday, NY stock market 1929,...), weather systems (Lorenz model of weather born via series of bifurcations modeled by Thom's catastrophes, see [1]), civil construction failure (bridge collapse, etc), complex systems (self-criticality and spontaneous system reconstruction leading to the better energetic stability) and more. Different mathematical models, of which one possibility is the aforementioned Thom's catastrophe theory, model such events. Our aim in this paper is to show that it is possible to use SA to identify such events on mathematical models of such systems and/or it is possible to use SA to design technological systems in such a way that the possibility to reach regimes exhibiting sudden changes in their behavior (i.e. catastrophe events) is minimized. This study is an extension of our previous research in [9] and [10].

point in the parameter space represents one of the possible system configurations. Those points which are part of the so-called catastrophic fold (surface, plane, ...) are related to system parameter configuration when a system is changing its behavior (moment when bifurcation occurs). When a system control parameter is changed then the point in the parameter space moves and the moment when it crosses through the catastrophic fold changes, then behavior of the system is changed. This change can mean in reality changes in periodicity as well as switching to chaotic dynamic and/or also more drastic changes in the systemâ's physical structure and behavior. Such changes then can lead, in reality, to real catastrophes like aircrafts crashing, dam failure, collapse of a building or power network, etc. As demonstrated in [1] on the Lorenz system (weather model), born of chaos can be understood as a way through the series of bifurcations Thom's catastrophes. Mutual relation between Thom's catastrophes and

Simulated Annealing in Research and Applications 73

Experiments have been designed as described in the Table 1. Each experiment was repeated 50 times and the results are reported in the Results section. All simulations have been done on a special grid computer. This grid computer consists of 16 XServers, each 2x2 GHz Intel

For the experiments described here (SA) [16] had been used. The control parameter settings

bifurcations is thus clear.

**Figure 1.** Catastrophe surface - Fold.

Xeon, 1 GB RAM, 80 GB HD i.e. 64 CPUs.

have been found empirically and are given in Table 1 (SA).

**2.2. Experiment setup**

*2.2.1. Used algorithms*

Identification of bifurcations has been done in this research so that Lyapunov exponent is calculated in its absolute value, see Figure 6. Zero values on this cost function landscape (multiple global extremes) then indicate for what parameter A bifurcations occur. In order to locate all these zero values SA is used. They advantage is to overcome multiple local extremes that are present in used cost function. They cannot be wiped out, because of chaotic nature of studied problem.
