Lyapunov methods to show the trajectories that do not converge to the origin

Question

Consider the map $f(z) = z^2$, where $z$ represents complex numbers. what is the function that $f$ corresponds to the map $p(r, \theta) = (r^2, \theta)$ in polar coordinates. And is it true that all initial points whose trajectories are bounded and do not converge to the origin? Could we prove it or show it by Lyapunov Function? And if we could, can we say that $f$ have chaotic orbits?

Answer 1

Using Euler's formula, we can write a complex number $ z $ in polar form as $ z = re^{i\theta} $, where $ r $ is the magnitude and $ \theta $ is the argument of $ z $. Euler's formula states that $ e^{i\theta} = \cos(\theta) + i\sin(\theta) $.

Then we need to compute $ f(z) = z^2 $:

$ f(z) = (re^{i\theta})^2 = r^2e^{i2\theta} $

In polar coordinates, $ p(r, \theta) = (r^2, 2\theta) $. We can see that $ f(z) $ corresponds to $ p(r, \theta) $ because $ r^2 $ is the magnitude squared and $ 2\theta $ is twice the argument of $ z $. To find the initial points whose trajectories are bounded and do not converge to the origin, let's consider the behavior of $ f(z) = z^2 $ in polar coordinates.

The magnitude of $ f(z) $ is $ r^2 $, which means that the trajectories are bounded if $ r $ is bounded. The argument of $ f(z) $ is $ 2\theta $, so $ \theta $ determines the angle in polar coordinates.

If $ \theta $ is a rational multiple of $ \pi $, then $ f(z) $ will generate a periodic trajectory that does not converge to the origin. This is because $ e^{i2\theta} $ will repeat its values after a certain number of iterations.

On the other hand, if $ \theta $ is an irrational multiple of $ \pi $, then $ f(z) $ will generate trajectories that densely fill the unit circle but do not converge to the origin.

Thus we yield the conclusion that the trajectories of $ f(z) = z^2 $ are bounded and do not converge to the origin if $ \theta $ is a rational multiple of $ \pi $. The Lyapunov exponents provide information about the rate of exponential separation of initially nearby trajectories in a dynamical system. For the map $ f(z) = z^2 $ in polar coordinates $ p(r, \theta) = (r^2, 2\theta) $, let's analyze the Lyapunov exponents.

The iteration of the map in polar coordinates is $ p_n = (r_n^2, 2\theta_n) $, where $ p_n $ represents the $ n $-th iterate. The Lyapunov exponents $ \lambda_i $ can be computed as follows:

$ \lambda_i = \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} \ln \left|\frac{d}{dx} f_i(p_k)\right| $

Here, $ f_i(p_k) $ represents the $ i $-th component of the map at the $ k $-th iterate.

For the map $ p(r, \theta) = (r^2, 2\theta) $, the components are $ f_1(r, \theta) = r^2 $ and $ f_2(r, \theta) = 2\theta $. Let's compute the partial derivatives:

$ \frac{\partial f_1}{\partial r} = 2r \quad \text{and} \quad \frac{\partial f_1}{\partial \theta} = 0 $

$ \frac{\partial f_2}{\partial r} = 0 \quad \text{and} \quad \frac{\partial f_2}{\partial \theta} = 2 $

Now, compute the Jacobian matrix $ J $ using these partial derivatives:

$ J = \begin{bmatrix} 2r & 0 \\ 0 & 2 \end{bmatrix} $

The Lyapunov exponents are the eigenvalues of $ J $. Since $ J $ is diagonal, the eigenvalues are simply the diagonal elements:

$ \lambda_1 = 2r, \quad \lambda_2 = 2 $

For bounded orbits, $ r $ is bounded, so $ \lambda_1 $ is also bounded. The Lyapunov exponents are positive, indicating an expansion of nearby trajectories. However, they are constant and not sensitive to initial conditions, so the orbits are not chaotic. To show that $ f(z) = z^2 $ has chaotic orbits, we need to demonstrate sensitivity to initial conditions, a dense set of periodic orbits, and topological transitivity.

For this map in polar coordinates $ p(r, \theta) = (r^2, 2\theta) $, you would need to show that small changes in initial conditions lead to significantly different trajectories over time (sensitivity to initial conditions). Additionally, you need to find a dense set of periodic orbits and demonstrate topological transitivity, which means that you can reach any point in the phase space from any other point.

The map $ f(z) = z^2 $ itself may not always exhibit chaotic behavior. Chaotic behavior often arises in certain parameter regimes or under certain conditions. Analyzing chaos typically involves numerical simulations and bifurcation diagrams to explore the behavior of the system under different initial conditions and parameter values.