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Chapter 9: Problem 2

(Pecora and Carroll's approach) In the pioneering work of Pecora and Carroll (1990), one of the receiver variables is simply set equal to the corresponding transmitter variable. For instance, if \(x(t)\) is used as the transmitter drive signal, then the receiver equations are $$ \begin{aligned} &x_{r}(t)=x(t) \\ &\dot{y}_{f}=r x(t)-y_{r}-x(t) z_{r} \\ &\dot{z}_{r}=x(t) y_{r}-b z_{r} \end{aligned} $$ where the first equation is not a differential equation. Their numerical simulations and a heuristic argument suggested that \(y_{r}(t) \rightarrow y(t)\) and \(z_{r}(t) \rightarrow z(t)\) as \(t \rightarrow \infty\), even if there were differences in the initial conditions. Here is a simple proof of that result, due to He and Vaidya (1992). a) Show that the error dynamics are $$ \begin{aligned} &e_{1} \equiv 0 \\ &\dot{e}_{2}=-e_{2}-x(t) e_{3} \\ &\dot{e}_{3}=x(t) e_{2}-b e_{3} \end{aligned} $$ where \(e_{1}=x-x_{+}, e_{2}=y-y\), and \(e_{3}=z-z_{r}\). b) Show that \(V=e_{2}^{2}+e_{3}^{2}\) is a Liapunov function. c) What do you conclude?

Short Answer

Expert verified
We derived the error equations given by \(\dot{e}_{2}=-e_{2}-x(t) e_{3}\) and \(\dot{e}_{3}=x(t) e_{2}-b e_{3}\), and verified the function \(V=e_{2}^{2}+e_{3}^{2}\) is a Lyapunov function for the system, showing that \(\dot{V} \le 0\). This confirms that the transmitter and receiver synchronize, as \(y_r(t) \rightarrow y(t)\) and \(z_r(t) \rightarrow z(t)\) for \(t \rightarrow \infty\).

Step by step solution

01

Obtain error equations

Recall the given receiver equations: $$ \begin{aligned} &x_{r}(t)=x(t)\\ &\dot{y}_{r}=r x(t)-y_{r}-x(t) z_{r}\\ &\dot{z}_{r}=x(t) y_{r}-b z_{r} \end{aligned} $$ Now we need to find the equations for the errors \(e_1 = x - x_r\), \(e_2 = y - y_r\), and \(e_3 = z - z_r\). Since \(x_r = x\), \(e_1 = 0\). To obtain other error equations, note that: - \(\dot{e}_2 = \dot{y} - \dot{y}_r\) - \(\dot{e}_3 = \dot{z} - \dot{z}_r\) Now substitute \(\dot{y}\), \(\dot{y}_r\), \(\dot{z}\), and \(\dot{z}_r\) from the given system, and simplify: $$ \begin{aligned} \dot{e}_2 &= \bigl( r x - y - xz \bigr) - \bigl( r x - y_r - x z_r \bigr) \\ &= -e_2 - x(t)e_3, \\ \dot{e}_3 &= \bigl( x y - b z \bigr) - \bigl( x y_r - b z_r \bigr) \\ &= x(t) e_2 - b e_3. \end{aligned} $$ As expected, we got the error equations: $$ \begin{aligned} &e_{1} \equiv 0 \\ &\dot{e}_{2}=-e_{2}-x(t) e_{3} \\ &\dot{e}_{3}=x(t) e_{2}-b e_{3} \end{aligned} $$
02

Verify that V is a Lyapunov function

The given function is: $$V = e_{2}^{2} + e_{3}^{2}$$ To show that this is a Lyapunov function, we need to check the derivative of V with respect to time: $$ \begin{aligned} \dot{V} &= \frac{dV}{dt} \\ &= \frac{d}{dt} \bigl( e_{2}^{2} + e_{3}^{2} \bigr) \\ &= 2e_2 \dot{e}_2 + 2e_3 \dot{e}_3. \end{aligned} $$ Now substitute \(\dot{e}_2\) and \(\dot{e}_3\) from the error equations and simplify: $$ \begin{aligned} \dot{V} &= 2e_2 \left(-e_2 - x(t)e_3\right) + 2e_3 \left( x(t)e_2 - be_3 \right) \\ &= -2e_{2}^{2} - 2 e_3^2 b \\ &\le 0. \end{aligned} $$ Since \(\dot{V} \le 0\), V is a Lyapunov function.
03

Conclusion

Since we have found a Lyapunov function V for the given error dynamics, we can infer that the errors \(e_2\) and \(e_3\) go to zero as \(t \rightarrow \infty\). This means that \(y_r(t) \rightarrow y(t)\) and \(z_r(t) \rightarrow z(t)\) as \(t \rightarrow \infty\), confirming the synchronization of the transmitter and receiver as conjectured by Pecora and Carroll.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Error Dynamics
Understanding the concept of error dynamics is pivotal in the field of synchronization in nonlinear dynamics. It involves analyzing how the discrepancy between the states of two systems evolves over time. In the case of synchronized systems, such as those examined by Pecora and Carroll, these errors, denoted by the variables e1, e2, and e3, represent the difference in variables of the sender and the receiver systems. The goal is to show that these errors diminish as time progresses, which would imply synchronization.

To obtain the equations for error dynamics, we differentiate the error terms with respect to time, as demonstrated in the textbook exercise. When one system variable is used as the driving signal for another, it results in an interconnected error system governed by differential equations. In our example, the driving signal x(t) remains perfectly synchronized (hence e1 is always zero). The behavior of e2 and e3, however, needs to be examined through their own differential equations. If these errors converge to zero, it indicates that synchronization has been achieved and the receiver is successfully tracking the transmitter.
Liapunov Function
The concept of the Liapunov function is a cornerstone in stability theory, named after the Russian mathematician Aleksandr Lyapunov. In simplistic terms, a Lyapunov function is akin to a mathematical tool that helps us determine the stability of an equilibrium point in a dynamical system. Think of it as a kind of 'energy' that is being measured which, if continually decreasing, implies that the system is stabilizing.

To utilize a Liapunov function, we perform a 'sanity check', verifying if the function is always non-increasing over time. In the discussed exercise, the function V = e22 + e32 serves as our Liapunov function. By calculating the time derivative of V and confirming it is non-positive, we formally show that our system's error dynamics are stable or, in other words, that the errors diminish over time. This provides mathematical evidence that the system will synchronize, as the 'energy' embodied by V decreases.
Pecora and Carroll Method
The Pecora and Carroll method is a foundational approach to achieving synchronization in chaotic systems and was introduced in their seminal 1990 paper. This method involves coupling two previously identical chaotic systems in such a way that one system (the 'transmitter') sends a signal to the other system (the 'receiver'). The coupling is achieved by making one variable of the receiver match the corresponding variable of the transmitter, essentially 'forcing' synchronization.

This method can be visualized as a dance between two partners, where one leads and the other follows their steps perfectly. The synchronization occurs even if there are slight mismatches in their initial conditions, bringing them to step in unison over time. Crucially, the Pecora and Carroll method provided the groundwork for the theoretical understanding of synchronization phenomena and continues to inspire research in the area of chaos theory and its applications.

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Most popular questions from this chapter

(Time horizon) To illustrate the "time horizon" after which prediction becomes impossible, numerically integrate the Lorenz equations for \(r=28, \sigma=10\), \(b=8 / 3\). Start two trajectories from nearby initial conditions, and plot \(x(t)\) for both of them on the same graph.

(Numerical experiments) For each of the values of \(r\) given below, use a computer to explore the dynamics of the Lorenz system, assuming \(\sigma=10\) and \(b=8 / 3\) as usual. In each case, plot \(x(t), y(t)\), and \(x\) vs. \(z\). $$ r=212 \text { (noisy periodicity) } $$

(Tent map, as model of Lorenz map) Consider the map $$ x_{n+1}= \begin{cases}2 x_{n}, & 0 \leq x_{n} \leq \frac{1}{2} \\ 2-2 x_{n}, & \frac{1}{2} \leq x_{n} \leq 1\end{cases} $$ as a simple analytical model of the Lorenz map. a) Why is it called the "tent map"? b) Find all the fixed points, and classify their stability. c) Show that the map has a period-2 orbit. Is it stable or unstable? d) Can you find any period-3 points? How about period-4? If so, are the corresponding periodic orbits stable or unstable?

(Waterwheel's moment of inertia approaches a constant) For the waterwheel of Section \(9.1\), show that \(I(t) \rightarrow\) constant as \(t \rightarrow \infty\), as follows: a) The total moment of inertia is a sum \(I=I_{\text {wheel }}+I_{\text {wasef }}\), where \(I_{\text {wheel }}\) depends only on the apparatus itself, and not on the distribution of water around the rim. Express \(I_{\text {water }}\) in terms of \(M=\int_{0}^{2 \pi} m(\theta, t) d \theta\). b) Show that \(M\) satisfies \(\dot{M}=Q_{\text {total }}-K M\), where \(Q_{\text {notal }}=\int_{0}^{2 A} Q(\theta) d \theta\) c) Show that \(I(t) \rightarrow\) constant as \(t \rightarrow \infty\), and find the value of the constant.

(Transient chaos) Example 9.5.1 shows that the Lorenz system can exhibit transient chaos for \(r=21, \sigma=10, b=\frac{8}{3}\). However, not all trajectories behave this way. Using numerical integration, find three different initial conditions for which there is transient chaos, and three others for which there isn't. Give a rule of thumb which predicts whether an initial condition will lead to transient chaos or not.
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