Corpus evidence — top 10 passages

Most-relevant passages from the entire indexed corpus (67,286 paragraph chunks across YouTube transcripts, PubMed, arXiv, archive.org, Stanford Encyclopedia of Philosophy, OpenAlex, and more) ranked by semantic similarity (bge-small-en-v1.5).

1. 01 · blog0.809

   Defining Chaos: Aperiodicity, Determinism, Nonlinearity, and Sensitive Dependence 1.1 Preliminaries 1.1.1 Dynamical Systems and Determinism 1.1.2 Nonlinear Dynamics 1.1.3 State Space and the Faithful Model Assumption 1.2 A Brief History of Chaos 1.3 Defining Chaos 1.3.1 Qualitative Definitions of Chaos 1.3.2 Quantitative Definitions of Chaos 1.3.3 Lyapunov Exponents and Chaos 1.3.4 Trouble with Definitions 1.4 Taking Stock 2. What is Chaos “Theory”? 3. Nonlinear Models, Faithfulness, and Confirmation 4. Chaos, Determinism, and Quantum Mechanics 5. Questions about Realism and Explanation 5.1 Re…

   blog/plato-stanford-edu/chaos.md

2. 02 · blog0.806

   Different definitions have varying strengths and weaknesses regarding tradeoffs on generality, theorem generation and proving, calculation ease, number of counterexamples, and so forth. The best candidates for necessary conditions for chaos still appear to be (1) something like chaos\(_{d\text{exp}}\) or (2) the presence of stretching and folding mechanisms. Chaos\(_{d\text{exp}}\) may function as a sufficient condition for chaos in many circumstances. These definitions may only hold for our mathematical models, but not be applicable to actual-world systems. Formal definitions seek to fully ch…

   blog/plato-stanford-edu/chaos.md

3. 03 · blog0.795

   Yorke’s (1975) influential “Period Three Implies Chaos” paper that led to the widespread use of the term “chaos” for these mathematical behaviors. 1.3 Defining Chaos To identify systems as chaotic, we need a definition or a list of distinguishing characteristics. 1.3.1 Qualitative Definitions of Chaos The logistic map, \[x _{t+1}= \alpha x_{t}(1-x_{t}), \] where \(\alpha\) is a parameter whose value ranges from one to four and the variable \(x\) ranges from zero to one, is a model chaotic system. A definition of chaos should identify what makes such a dynamical system chaotic, but this turns o…

   blog/plato-stanford-edu/chaos.md

4. 04 · blog0.785

   Do features of our mathematical analyses (e.g., characterizations of instability) turn out to be problematic such that their application to target systems may not be useful (see below)? Furthermore, Kellert’s definition is too broad to pick out only chaotic behaviors. For instance, take the map \(x_{n + 1} = cx_{n}\), a map exhibiting only unstable and aperiodic orbits. For the values \(c = 1.1\) and \(x_{0} = .5\), successive iterations continue to increase and never return near \(x_{0}\). Kellert’s definition would classify this map as chaotic, but the map doesn’t exhibit chaotic behavior. R…

   blog/plato-stanford-edu/chaos.md

5. 05 · blog0.783

   As such, this defining characteristic could be applied to both mathematical models and actual-world systems, though the identification of such mechanisms in target systems may be rather tricky: When dealing with a fluid system, say, we have several nonlinear mechanisms that have been well explored as sources for stretching and folding. In contrast, when we only have time series data (e.g., the hourly price of Chicago Board of Trade hog futures), identifying possible nonlinear mechanisms is difficult. Stretching and folding mechanisms lead to dynamics with attractors, so focusing on such mechan…

   blog/plato-stanford-edu/chaos.md

6. 06 · blog0.782

   First, stretch it in the \(y\) direction by more than a factor of two. Then compress it in the \(x\) direction by more than a factor of two. Now, fold the resulting rectangle and lay it back onto the square so that the construction overlaps and leaves the middle and vertical edges of the initial unit square uncovered. Repeating these stretching and folding operations leads to the Smale attractor. This definition has at least two virtues. First, it can be proven that Chaos\(_{h}\) implies Chaos\(_{d}\). Second, it yields exponential divergence, so we get SD as expected for chaotic systems. Neve…

   blog/plato-stanford-edu/chaos.md

7. 07 · blog0.782

   On the other hand, chaos is usually characterized by a stronger form of sensitive dependence: (SD) \(\exists \lambda\) such that for almost all points \(\bx(0)\), \(\forall \delta \gt 0\) \(\exists t\gt 0\) such that for almost all points \(\by(0)\) in a small neighborhood \((\delta)\) around \(\bx(0)\), \(\abs{\bx(0) - \by(0)}\lt \delta\) and \(\abs{\bx(t) - \by(t)} \approx \abs{\bx(0) - \by(0)}e^{\lambda t}\), where “almost all” is understood as applying for all points in state space except a set of measure zero. Here, \(\lambda\) is often interpreted as the largest global Lyapunov exponent …

   blog/plato-stanford-edu/chaos.md

8. 08 · blog0.779

   If quantum theories were unquestionably indeterministic, and deterministic theories guaranteed repeatability of a strong form, there could conceivably be further experimental input on the question of determinism’s truth or falsity. Unfortunately, the existence of Bohmian quantum theories casts strong doubt on the former point, while chaos theory casts strong doubt on the latter. More will be said about each of these complications below. 3.3 Determinism and Chaos If the world were governed by strictly deterministic laws, might it still look as though indeterminism reigns? This is one of the dif…

   blog/plato-stanford-edu/causal-determinism.md

9. 09 · blog0.774

   A mathematical model is deterministic if it exhibits unique evolution: (Unique Evolution) A given state of a model is always followed by the same history of state transitions. Given a state at a specific time there is only one history of transitions consistent with the relevant laws. Although some popularized discussions of chaos claim it invalidates determinism, chaotic behavior is always deterministic. Much of the confusion over chaos and determinism derives from equating determinism with predictability. While it’s true that apparent randomness can be generated if the state space (see §1.1.3…

   blog/plato-stanford-edu/chaos.md

10. 10 · blog0.773

    Attempts to align definitions such as Chaos\(_{d}\) , SD, Chaos\(_{te}\), or Chaos\(_{\lambda}\) with the ergodic hierarchy yield examples with exponential divergence of trajectories where all past states are perfectly correlated with any future states yielding no randomness associated with chaos. Some cases with positive global Lyapunov exponents have trajectories accelerating off to infinity raising questions about the importance of requiring confinement. Even for bounded systems, the presence of a positive global Lyapunov exponent may not lead to exponential growth in uncertainties. Some sy…

    blog/plato-stanford-edu/chaos.md