
# 9: Chaos


• 9.1: Chaos in Discrete-Time Models
Figure 8.10 showed a cascade of period-doubling bifurcations, with the intervals between consecutive bifurcation thresholds getting shorter and shorter geometrically as r increased. This cascade of period doubling eventually leads to the divergence of the period to inﬁnity at r ≈ 1.7 in this case, which indicates the onset of chaos. In this mysterious parameter regime, the system loses any ﬁnite-length periodicity, and its behavior looks essentially random. Figure 9.1 shows an example of such ch
• 9.2: Characteristics of Chaos
It is helpful to realize that there are two dynamical processes always going on in any kind of chaotic systems: stretching and folding. Any chaotic system has a dynamical mechanism to stretch, and then fold, its phase space, like kneading pastry dough (Fig. 9.4). Imagine that you are keeping track of the location of a speciﬁc grain of ﬂour in the dough while a pastry chef kneads the dough for a long period of time. Stretching the dough magniﬁes the tiny differences in positions.
• 9.3: Lyapunov Exponent
The Lyapunov exponent is a useful analytical metric that can help characterize chaos. It measures how quickly an inﬁnitesimally small distance between two initially close states grows over time,
• 9.4: Chaos in Continuous-Time Model
Edward Lorenz, an American mathematician and meteorologist, and one of the founders of chaos theory, accidentally found chaotic behavior in the following model (called the Lorenz equations) that he developed to study the dynamics of atmospheric convection in the early 1960s.

Thumbnail: A sample trajectory through phase space is plotted near a Lorenz attractor with σ = 10, ρ = 28, β = 8/3. The color of the solution fades from black to blue as time progresses, and the black dot shows a particle moving along the solution in time. Initial conditions: x(0) = 0, y(0) = 2, z(0) = 20. 0 < t < 35. The 3-dimensional trajectory {x(t), y(t), z(t)} is shown from different angles to demonstrate its structure. (CC BY-SA 3.0 unported; Dan Quinn).