# 7.5: Dampening of Chaos

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ Figure $$\PageIndex{1}$$. Bifurcations with changing parameters. In all parts s = −r−1. (A) r = 2.84, period 3, (B) r = 2.575, near period 4, (C) r = 2.48, period 4, (D) r = 2.26, period 2. Figure $$\PageIndex{2}$$. Changing nature of equilibrium, period 1. In all parts s = −r−1. (A) r = 2.00, (B) r = 1.86, (C) r = 1.24, (D) r = 0.60.

If the growth rate r diminishes, the amount that the population can overshoot its carrying capacity also diminishes, meaning that the size and severity of the ﬂuctuations should diminish as well. Figures $$\PageIndex{1}$$ and $$\PageIndex{2}$$ show this happening.

When r diminishes from 3 to 2.84, for example, as in Part A of Figure $$\PageIndex{1}$$, chaos vanishes and the oscillations become regular, jumping from a specific low value to a specific medium value to a specific high value, then dropping back to repeat the cycle. This is called “period three.” Sensitive dependence on initial conditions has also vanished; slight changes in the starting population will not produce different patterns, as in Figure 7.3.1, but will end up approaching exactly the same three levels as before. The pattern is stable. Moreover, changing parameter r slightly will not change the period-three pattern to something else. The exact values of the three levels will shift slightly, but the period-three pattern will remain.

But when r is changed more than slightly—to 2.575, for example, as in Part B— the period-three pattern vanishes and, in this case, a chaos-like pattern appears. The population fluctuates among four distinct bands, with a complex distribution within each, as shown on the right in Part B. With r somewhat lower—at 2.48, for example, as in Part C— the bands coalesce into a period-four pattern, which is stable like the period-three pattern in Part A. With further reductions in r, the period-four pattern is cut in half to a period-two pattern, as in Part D, and finally to a period-one, an equilibrium pattern.

Figure $$\PageIndex{2}$$ shows the progression from r = 3 downward, as it changes from an oscillation toward the equilibrium, as in Parts A and B, and to a smooth approach, as in Parts C and D. This smooth approach begins when the growth rate is small enough that the population does not overshoot its carrying capacity.

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