Logistic Map - Behavior Dependent On r

Behavior Dependent On r

By varying the parameter r, the following behavior is observed:

• With r between 0 and 1, the population will eventually die, independent of the initial population.
• With r between 1 and 2, the population will quickly approach the value

, independent of the initial population.

• With r between 2 and 3, the population will also eventually approach the same value

, but first will fluctuate around that value for some time. The rate of convergence is linear, except for r=3, when it is dramatically slow, less than linear.

• With r between 3 and (approximately 3.45), from almost all initial conditions the population will approach permanent oscillations between two values. These two values are dependent on r.
• With r between 3.45 and 3.54 (approximately), from almost all initial conditions the population will approach permanent oscillations among four values.
• With r increasing beyond 3.54, from almost all initial conditions the population will approach oscillations among 8 values, then 16, 32, etc. The lengths of the parameter intervals which yield oscillations of a given length decrease rapidly; the ratio between the lengths of two successive such bifurcation intervals approaches the Feigenbaum constant δ = 4.669. This behavior is an example of a period-doubling cascade.
• At r approximately 3.57 is the onset of chaos, at the end of the period-doubling cascade. From almost all initial conditions we can no longer see any oscillations of finite period. Slight variations in the initial population yield dramatically different results over time, a prime characteristic of chaos.
• Most values beyond 3.57 exhibit chaotic behaviour, but there are still certain isolated ranges of r that show non-chaotic behavior; these are sometimes called islands of stability. For instance, beginning at (approximately 3.83) there is a range of parameters r which show oscillation among three values, and for slightly higher values of r oscillation among 6 values, then 12 etc.
• The development of the chaotic behavior of the logistic sequence as the parameter r varies from approximately 3.5699 to approximately 3.8284 is sometimes called the Pomeau–Manneville scenario, which is characterized by a periodic (laminar) phase interrupted by bursts of aperiodic behavior. Such a scenario has an application in semiconductor devices. There are other ranges which yield oscillation among 5 values etc.; all oscillation periods occur for some values of r. A period-doubling window with parameter c is a range of r-values consisting of a succession of sub-ranges. The kth sub-range contains the values of r for which there is a stable cycle (a cycle which attracts a set of initial points of unit measure) of period This sequence of sub-ranges is called a cascade of harmonics. In a sub-range with a stable cycle of period there are unstable cycles of period for all The r value at the end of the infinite sequence of sub-ranges is called the point of accumulation of the cascade of harmonics. As r rises there is a succession of new windows with different c values. The first one is for c = 1; all subsequent windows involving odd c occur in decreasing order of c starting with arbitrarily large c.
• Beyond r = 4, the values eventually leave the interval and diverge for almost all initial values.

For any value of r there is at most one stable cycle. A stable cycle attracts almost all points. For an r with a stable cycle of some period, there can be infinitely many unstable cycles of various periods.

A bifurcation diagram summarizes this. The horizontal axis shows the values of the parameter r while the vertical axis shows the possible long-term values of x.

The bifurcation diagram is a self-similar: if you zoom in on the above mentioned value r = 3.82 and focus on one arm of the three, the situation nearby looks like a shrunk and slightly distorted version of the whole diagram. The same is true for all other non-chaotic points. This is an example of the deep and ubiquitous connection between chaos and fractals.