# [S2E2] Chaos Theory

The team heads back to the dumpster outside Paige's dorm. Warrick finds some blood inside the dumpster, while Nick finds fresh paint transfer on the outside. Paint chips on the ground nearby are the same color as the transfer on the dumpster, and the team wonders if they're looking at a hit-and-run. Due to the high ground clearance of the transfer, they figure the vehicle in question may be an SUV. This would also explain Paige's blunt force trauma, as the height of the transfer is equal with Paige's abdomen. The theory is that the driver tossed Paige's body in the dumpster to hide their crime. Sharon Woodbury is a viable suspect, but the black paint chips don't match the Woodbury's silver Volvo. Later on, blood analysis confirms that the blood in the dumpster is Paige's.

## [S2E2] Chaos Theory

Grissom suggests that since the solution isn't "neat, plausible, and wrong," then perhaps it's "messy, unlikely, and right." He expounds upon chaos theory and wonders if the team is looking at a series of random events. Paige was in her dorm room and ended up behind a dumpster; Grissom believes the answer is somewhere between those two points.

Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have completely random states of disorder and irregularities.[1] Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals, and self-organization.[2] The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning that there is sensitive dependence on initial conditions).[3] A metaphor for this behavior is that a butterfly flapping its wings in Brazil can cause a tornado in Texas.[4][5][6]

Small differences in initial conditions, such as those due to errors in measurements or due to rounding errors in numerical computation, can yield widely diverging outcomes for such dynamical systems, rendering long-term prediction of their behavior impossible in general.[7] This can happen even though these systems are deterministic, meaning that their future behavior follows a unique evolution[8] and is fully determined by their initial conditions, with no random elements involved.[9] In other words, the deterministic nature of these systems does not make them predictable.[10][11] This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by Edward Lorenz as:[12]

Chaotic behavior exists in many natural systems, including fluid flow, heartbeat irregularities, weather, and climate.[13][14][8] It also occurs spontaneously in some systems with artificial components, such as the road traffic.[2] This behavior can be studied through the analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps. Chaos theory has applications in a variety of disciplines, including meteorology,[8] anthropology,[15] sociology, environmental science, computer science, engineering, economics, ecology, and pandemic crisis management.[16][17] The theory formed the basis for such fields of study as complex dynamical systems, edge of chaos theory, and self-assembly processes.

Chaos theory concerns deterministic systems whose behavior can, in principle, be predicted. Chaotic systems are predictable for a while and then 'appear' to become random. The amount of time for which the behavior of a chaotic system can be effectively predicted depends on three things: how much uncertainty can be tolerated in the forecast, how accurately its current state can be measured, and a time scale depending on the dynamics of the system, called the Lyapunov time. Some examples of Lyapunov times are: chaotic electrical circuits, about 1 millisecond; weather systems, a few days (unproven); the inner solar system, 4 to 5 million years.[18] In chaotic systems, the uncertainty in a forecast increases exponentially with elapsed time. Hence, mathematically, doubling the forecast time more than squares the proportional uncertainty in the forecast. This means, in practice, a meaningful prediction cannot be made over an interval of more than two or three times the Lyapunov time. When meaningful predictions cannot be made, the system appears random.[19]

Chaos theory is a method of qualitative and quantitative analysis to investigate the behavior of dynamic systems that cannot be explained and predicted by single data relationships, but must be explained and predicted by whole, continuous data relationships.

In common usage, "chaos" means "a state of disorder".[20][21] However, in chaos theory, the term is defined more precisely. Although no universally accepted mathematical definition of chaos exists, a commonly used definition, originally formulated by Robert L. Devaney, says that to classify a dynamical system as chaotic, it must have these properties:[22]

If attention is restricted to intervals, the second property implies the other two.[26] An alternative and a generally weaker definition of chaos uses only the first two properties in the above list.[27]

As suggested in Lorenz's book entitled "The Essence of Chaos", published in 1993,[5] "sensitive dependence can serve as an acceptable definition of chaos". In the same book, Lorenz defined the butterfly effect as: "The phenomenon that a small alteration in the state of a dynamical system will cause subsequent states to differ greatly from the states that would have followed without the alteration." The above definition is consistent with the sensitive dependence of solutions on initial conditions (SDIC). An idealized skiing model was developed to illustrate the sensitivity of time-varying paths to initial positions.[5] A predictability horizon can be determined before the onset of SDIC (i.e., prior to significant separations of initial nearby trajectories).[29]

In addition to the above property, other properties related to sensitivity of initial conditions also exist. These include, for example, measure-theoretical mixing (as discussed in ergodic theory) and properties of a K-system.[11]

Topological mixing is often omitted from popular accounts of chaos, which equate chaos with only sensitivity to initial conditions. However, sensitive dependence on initial conditions alone does not give chaos. For example, consider the simple dynamical system produced by repeatedly doubling an initial value. This system has sensitive dependence on initial conditions everywhere, since any pair of nearby points eventually becomes widely separated. However, this example has no topological mixing, and therefore has no chaos. Indeed, it has extremely simple behavior: all points except 0 tend to positive or negative infinity.

where x \displaystyle x , y \displaystyle y , and z \displaystyle z make up the system state, t \displaystyle t is time, and σ \displaystyle \sigma , ρ \displaystyle \rho , β \displaystyle \beta are the system parameters. Five of the terms on the right hand side are linear, while two are quadratic; a total of seven terms. Another well-known chaotic attractor is generated by the Rössler equations, which have only one nonlinear term out of seven. Sprott[46] found a three-dimensional system with just five terms, that had only one nonlinear term, which exhibits chaos for certain parameter values. Zhang and Heidel[47][48] showed that, at least for dissipative and conservative quadratic systems, three-dimensional quadratic systems with only three or four terms on the right-hand side cannot exhibit chaotic behavior. The reason is, simply put, that solutions to such systems are asymptotic to a two-dimensional surface and therefore solutions are well behaved.

Under the right conditions, chaos spontaneously evolves into a lockstep pattern. In the Kuramoto model, four conditions suffice to produce synchronization in a chaotic system.Examples include the coupled oscillation of Christiaan Huygens' pendulums, fireflies, neurons, the London Millennium Bridge resonance, and large arrays of Josephson junctions.[62]

An early proponent of chaos theory was Henri Poincaré. In the 1880s, while studying the three-body problem, he found that there can be orbits that are nonperiodic, and yet not forever increasing nor approaching a fixed point.[63][64][65] In 1898, Jacques Hadamard published an influential study of the chaotic motion of a free particle gliding frictionlessly on a surface of constant negative curvature, called "Hadamard's billiards".[66] Hadamard was able to show that all trajectories are unstable, in that all particle trajectories diverge exponentially from one another, with a positive Lyapunov exponent.

Chaos theory began in the field of ergodic theory. Later studies, also on the topic of nonlinear differential equations, were carried out by George David Birkhoff,[67] Andrey Nikolaevich Kolmogorov,[68][69][70] Mary Lucy Cartwright and John Edensor Littlewood,[71] and Stephen Smale.[72] Except for Smale, these studies were all directly inspired by physics: the three-body problem in the case of Birkhoff, turbulence and astronomical problems in the case of Kolmogorov, and radio engineering in the case of Cartwright and Littlewood.[citation needed] Although chaotic planetary motion had not been observed, experimentalists had encountered turbulence in fluid motion and nonperiodic oscillation in radio circuits without the benefit of a theory to explain what they were seeing.

Despite initial insights in the first half of the twentieth century, chaos theory became formalized as such only after mid-century, when it first became evident to some scientists that linear theory, the prevailing system theory at that time, simply could not explain the observed behavior of certain experiments like that of the logistic map. What had been attributed to measure imprecision and simple "noise" was considered by chaos theorists as a full component of the studied systems. In 1959 Chirikov proposed a criterion for the emergence of classical chaos in Hamiltonian systems (Chirikov criterion). He applied this criterion to explain some experimental results on plasma confinement in open mirror traps.[73][74] This is regarded as the very first physical theory of chaos, which succeeded in explaining a concrete experiment. And Chirikov himself is considered as a pioneer in classical and quantum chaos.[75][76][77] 041b061a72