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# 金融代写|随机控制理论代写STOCHASTIC CONTROL代考|ENEE762 First Self-Synchronization Method

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## 金融代写|随机控制理论代写STOCHASTIC CONTROL代考|First Self-Synchronization Method

Let us now take a single cellular automaton (Ilachinski, 2001; Toffoli \& Margolus, 1987). If $\sigma^{1}(t)$ is the state of the automaton at time $t, \sigma^{1}(t)=\sigma(t)$, and $\sigma^{2}(t)$ is the state obtained from the application of the rule $\Phi$ on that state, $\sigma^{2}(t)=\Phi\left[\sigma^{1}(t)\right]$, then the operator $\Gamma_{p}$ can be applied on the pair $\left(\sigma^{1}(t), \sigma^{2}(t)\right)$, giving rise to the evolution law
$$\sigma(t+1)=\Gamma_{p}\left[\left(\sigma^{1}(t), \sigma^{2}(t)\right)\right]=\Gamma_{p}[(\sigma(t), \Phi[\sigma(t)])] .$$
The application of the $\Gamma_{p}$ operator is as follows. When $\sigma_{i}^{1} \neq \sigma_{i}^{2}$, the sites $i$ of the state $\sigma^{2}(t)$ are updated to the correspondent values taken in $\sigma^{1}(t)$ with a probability $p$. The updated array $\sigma^{2}(t)$ is the new state $\sigma(t+1)$. It is worth to observe that if the system is initialized with a configuration constant in time for the rule $\Phi, \Phi[\sigma]=\sigma$, then this state $\sigma$ is not modified when the dynamic equation (11) is applied. Hence the evolution will produce a pattern constant in time. However, in general, this stability is marginal. A small modification of the initial condition gives rise to patterns variable in time. In fact, as the parameter $p$ increases, a competition among the different marginally stable structures takes place. The dynamics drives the system to stay close to those states, although oscillating continuously and randomly among them. Hence, a complex spatio-temporal behavior is obtained. Some of these patterns can be seen in Fig. 7. However, in rule 18, the pattern becomes stable and, independently of the initial conditions, the system evolves toward this state, which is the null pattern in this case (Sanchez \& Lopez-Ruiz, 2006).

## 金融代写|随机控制理论代写STOCHASTIC CONTROL代考|Second Self-Synchronization Method

Now we introduce a new stochastic element in the application of the operator $\Gamma_{p}$. To differentiate from the previous case we call it $\tilde{\Gamma}{\tilde{p}}$. The action of this operator consists in applying at each time the operator $\Gamma{p}$, with $p$ chosen at random in the interval $(0, \tilde{p})$. The evolution law of the automaton is in this case:
$$\sigma(t+1)=\tilde{\Gamma}{\tilde{p}}\left[\left(\sigma^{1}(t), \sigma^{2}(t)\right)\right]=\tilde{\Gamma}{\bar{p}}[(\sigma(t), \Phi[\sigma(t)])] .$$
The DA density between the present state and the previous one, defined as $\delta(t)=\mid \sigma(t)-$ $\sigma(t-1) \mid$, is plotted as a function of $\tilde{p}$ for different rules $\Phi$ in Fig. 8. Only when the system becomes self-synchronized there will be a fall to zero in the DA density. Let us observe again that the behavior reported in the first self-synchronization method is newly obtained in this case. Rule 18 undergoes a phase transition for a critical value of $\tilde{p}$. For $\tilde{p}$ greater than the critical value, the method is able to find the stable structure of the system (Sanchez \& LopezRuiz, 2006). For the rest of the rules the freezing phase is not found. The dynamics generates patterns where the different marginally stable structures randomly compete. Hence the DA density decays linearly with $\tilde{p}$ (see Fig. 8).

## 金融代写|随机控制理论代写STOCHASTIC CONTROL代考|First Self-Synchronization Method

$t, \sigma^{1}(t)=\sigma(t)$ ，和 $\sigma^{2}(t)$ 是从应用规则获得的状态 $\Phi$ 在那个状态， $\sigma^{2}(t)=\Phi\left[\sigma^{1}(t)\right]$ ，那么运算符 $\Gamma p$ 可以应用于对
$\left(\sigma^{1}(t), \sigma^{2}(t)\right)$, 产生进化规律
$$\sigma(t+1)=\Gamma_{p}\left[\left(\sigma^{1}(t), \sigma^{2}(t)\right)\right]=\Gamma_{p}[(\sigma(t), \Phi[\sigma(t)])] .$$

## 金融代写|随机控制理论代写STOCHASTIC CONTROL代考|Second Self- Synchronization Method

$$\sigma(t+1)=\tilde{\Gamma} \tilde{p}\left[\left(\sigma^{1}(t), \sigma^{2}(t)\right)\right]=\tilde{\Gamma} \bar{p}[(\sigma(t), \Phi[\sigma(t)])] .$$

## MATLAB代写

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