Bifurcations

Abstract

This chapter will explore a variety of routes that lead to chaos in dynamical systems, through simulation and FPGA experiments. The goal of this chapter is simply for the reader to understand that a system is chaotic for a certain range of parameters and there are interesting mechanisms that lead to the chaotic behavior.

FPGA realization of the torus breakdown system [6]

Appendices

Problems

4.1

Consider the Langford System, shown in Eqs. (4.26)–(4.28). These equations can be used to describe the motion of turbulent flow in a fluid [9].

$$\begin{aligned} \dot{x}&=xz-\omega y \end{aligned}$$

(4.26)

$$\begin{aligned} \dot{y}&=\omega x + xy \end{aligned}$$

(4.27)

$$\begin{aligned} \dot{z}&=p+z-\frac{1}{3}z^3-(x^2+y^2)(1+qx+\epsilon x) \end{aligned}$$

(4.28)

First, compute the equilibrium points for the system. Now consider the following typical system parameters: $p=1.1, q=0.7$ and $\epsilon = 0.5$. Investigate the route to chaos in this system as a function of parameter $\omega$. In other words, obtain a bifurcation diagram. Implement your design on the FPGA.

4.2

Investigate the route(s) to chaos in the Lorenz system.

4.3

Read [5] and obtain the period-doubling route to chaos in Eqs. (4.8)–(4.10)

4.4

Repeat Problem 4.3 but for the period-adding route.

4.5

Parameterize the pulseFSM in listing E.12 using generics. This will allow us to utilize the pulseFSM for other modules that require different delays.

4.6

We can also examine limit cycles with high periods (such as period-16) on the FPGA as opposed to an analog realization, due to noise immunity on the FPGA. Try to obtain high period limit cycles in any of the system(s) (say R$\ddot{\text {o}}$ssler system) from this chapter.

4.7

Perform an In-system verification using SignalTap, of the period-doubling route to chaos for the R$\ddot{\text {o}}$ssler system.

4.8

Investigate the route to chaos as a function of parameter$\kappa$, in the optically injected laser system in Eqs. (4.29)–(4.31) [10]. Use $\alpha =2.5, \beta =0.015, \varGamma =0.05, \omega =2$.

$$\begin{aligned} \dot{x}_1&=\kappa +\frac{x_1x_3}{2}-\frac{\alpha }{2}x_2x_3 + \omega x_2 \end{aligned}$$

(4.29)

$$\begin{aligned} \dot{x}_2&=-\omega x_1+\frac{\alpha }{2}x_1x_3 + \frac{x_2x_3}{2} \end{aligned}$$

(4.30)

$$\begin{aligned} \dot{x}_3&= -2\varGamma x_3 - (1+2\beta x_3)(x_1^2+x_2^2-1) \end{aligned}$$

(4.31)

4.9

Following Example 4.3, implement the period-adding route to chaos Sect. 4.2.1.

4.10

Following Example 4.3, implement the quasi-periodic route to chaos Sect. 4.3.3.

4.11

Design and implement on the FPGA, the intermittency route to chaos from Sect. 4.2.4,

4.12

Design and implement on the FPGA, chaotic transients from Sect. 4.2.5.

Lab 4: Displaying Bifurcation Parameter(s) on the LCD

Objective: Implement a display module for bifurcation parameter(s).

Lab Exercise:

You should have realized from the design(s) in this chapter that simply pressing the key and mentally keeping track of the increment or decrement of the bifurcation parameter is cumbersome. Hence, utilize the solution from Lab 2 to display the current value of the bifurcation parameter on the LCD display. You can of course display any additional information or even interface to an external monitor using VGA.