The complexity of the Logistic model was mainly investigated in PSpice but Figure 6.5 was plotted in Matlab. This is a bifurcation plot of the Logistic map signal which splits up into smaller self-similar fractals for a range R values.
FIGURE6.5: Logistic map bifurcation diagram plotted in Matlab.
A Logistic map bifurcation diagram plots the chaos variable with the horizontal time parameter replaced by the swept voltage, V(R), representing the growth factor and is discussed in Chapter 7 Page 7.3). To generate random binary sequences the
Logistic map must operate in the chaos region R ∈ [3.57−4]. Bifurcation shows
how a chaos system behaves when a system parameter changes the stability of an FP causing the trajectory to move from one quasi-stable state to another. The Logistic
map eigenvalues determine the stability and are calculated setting f (yn) = yn. The
one-dimensional Logistic system Jacobian matrix is the differential of the function at the FP.
The population dies for R between 0 and 1, but approaches a value of (1- 1/R) for R between 1 and 2. The population decays in an oscillatory manner for R between 2 and 3 to a fixed value, (1-1/R). A growth factor greater than three causes the population to oscillate between two values referred to as period-doubling. The population bifurcates to a high value one year and a low value the next year (it takes two iterations for the population to return to its original value). Similarly, the period doubles again to four when R is 3.45, with a four-year cycle. At R= 3.57. . . , a point called the accumulation point or Feigenbaum’s number) is reached where chaos starts and the population never settles to a fixed population value.
6.3.1
Stability Loss Delay
The stability of continuous chaos system is analysed in the s-plane, but discrete chaos maps apply the unit circle z-plane [Tobin, 2007c]. Bifurcation in chaos maps occurs whenever an FP loses stability. The map is stable if the pole (eigenvalue) loca- tions are within the unit circle, but bounded-input bounded-output (BIBO) unstable if the poles are outside the unit circle and the map trajectory moves away from the FP at the origin. Poles on the unit circle lead to conditional stability, but if the poles cross the unit circle, then a bifurcation occurs. Classification of the bifurcation type depends on where a pole crosses the unit circle. A period bifurcation (also called a flip or fork bifurcation) occurs for R between 3 and 4. A Hopf bifurcation starts oscillatory behaviour from a steady state value where the poles have zero real and complex conjugate imaginary parts. To plot a bifurcation diagram in PSpice requires changing the time-axis in Probe to a voltage that represents the swept variable. In this analysis, a saw-tooth voltage represents the change in the growth factor with time.
Chapter 6. Chaos Maps
The pitchfork shape at each bifurcation point was not observed after simulation but had “tangential distortion” shown in Figure 6.6 [Blackledge, Bezobrazov, and Tobin, 2015]. This is caused because the saw-tooth was swept too fast in time.
FIGURE6.6: (a) Period bifurcation plots (b) Tangential distortion.
The distortion was caused by the slope of the ramp generator signal which is the rate of change of the parameter under investigation. It was discovered that setting the ramp time to a larger value will reduce this distortion but may produce simu- lation convergence problems and longer simulation times. This phenomenon is not observed in software bifurcation plots because ’FOR’ loops allows the ‘Simulation World’ to pause while the parameter is changed. In the Real World, time does not stop, and the bifurcation shape from sweeping a parameter with time is tangential and not pitchfork.
The author presented a paper at a nonlinear mathematics conference in Dublin where this distortion was discussed [Tobin, Blackledge, and Bezobrasov, 2016]. Professor Vassili Gelfreich (Warwick University) suggested this could be sta- bility loss delay (SLD), an important and interesting phenomenon not completely understood. SLD was first observed by M.A.Shishkova [Shishkova, 1973] under the supervision of Lev Pontryagin [Pontriagin and Rodygin, 1960], [Morris and Moss, 1986]. In a private discussion, Gelfreich referred to a paper by Anatoly Neyshtadt in which SLD is examined [Neishtadt, 2009]. The behaviour of systems at the bifurca- tion region depends on parameter factors close to a critical bifurcation point. When a bifurcation parameter changes slowly over time and passes through a value which produces bifurcation, the system delays the bifurcation, hence the name. At the point where it bifurcates quite abruptly, the trajectory heads off at a tangent rather than the cusp bifurcation customarily observed. Gelfreich and Neyshtadt suggested adding small amounts of Gaussian noise to the model to correct the SLD. This was experimented in PSpice and produce some reduction in SLD. Another experiment
eliminated SLD and used a step generator which in a sense emulated a program- ming ‘FOR’ loop and is explained in Chapter 7 Page 122.
6.3.2
Period Three in Bifurcation Plots
James Yorke and T.Y. Li wrote an article “Period Three Implies Chaos” in a 1975 is- sue of American Mathematical Monthly. They stated if a period 3-cycle exists, then it implies cycles of every other period will exist too, and also chaotic cycles with no period at all [Li and Yorke, 1975]. There are Logistic map values of R which pro- duce ‘Period Three’ window regions called intermittencies, as shown in Figure 6.6 (a). For R= 3.6786, an odd-period cycle appears, which according to the Li-Yorke theo- rem, produces period three cycles in chaotic zones, but only after period-doubling bifurcations have ended [Saha and Strogatz, 1995]. The ‘Period Three’ window is observed by magnifying a portion of the attractor at specific values of the growth factor to show the same figtree pattern occurring again at each of the three lines - a fractal self-similarity phenomenon shown in Figure 6.6 (a) [Li and Yorke, 1975], [Saha and Strogatz, 1995].
Figure 6.6 (b) shows the ‘Period Three’ in greater detail around the region R = 3.77 to 3.84 to reveal the fractal nature of the bifurcation diagram. However, the presence of periodic window regions in a bifurcation diagram would make this re- gion unsuitable for generating OTP because it is no longer deterministic and occurs
for R = 1+√8 ≈ 3.83, where the system oscillates between three values. This in-
termittency window shows a self-similar fractal pattern (a pattern that is scale-free), which can be observed in finer detail at each bifurcation point. The same pattern occurs irrespective of the scale.
6.3.3
Feigenbaum Constants
Mitchell Feigenbaum discovered the period-doubling mechanism for systems to be- came chaotic and showed relationships exist between recurring ratios in period- doubling called the Feigenbaum constants nearly as familiar as π and e. Figure 6.7 shows how these constants are obtained from measurements on the plot. The Logis- tic map period-doubling first occurs for R = 3, the second at 3.455, and the third at
3.533, etc., [Briggs, 1991]3.
Chapter 6. Chaos Maps
The second Feigenbaum constant was determined from a PSpice simulation by mea- suring the width of the bifurcation opening at a datum point of 0.5 V between suc- cessive bifurcation period-doubling shown.
FIGURE6.7: Measuring Feigenbaum constants.
δ = lim y→∞ Rn −Rn−1 Rn+1−Rn = Delta3 Delta4 = 3.455−3 3.553−3.455 =4.64. . . (6.4)
This is close to the Feigenbaum constant, 4.6692. . . [Briggs, 1991] but an exact value cannot be measured as it requires evaluating the limit n to infinity. The second
Feigenbaum constant, α= 2.5029. . ., is determined by adding a 0.5 V reference line
to measure the bifurcation ‘opening’. This constant was derived from the ratio of the openings at each period-doubling at the same location on the bifurcation plot. The value measured for this constant in Figure 6.7 is close to the theoretical value of 2.502907875 . . . . α = lim y→∞ δn δn+1 = Delta1 Delta2 = (854 mV− 438 mV) (521 mV− 359 mV =2.54. . . (6.5)
The Feigenbaum constants apply to any NDS and can be explained using a mathe- matical process called renormalization [Stewart, 1997].