codzienność: Chaos theory

Evidence: 100 • authority: 72

Chaos theory is a branch of mathematics and an interdisciplinary area of scientific study. It focuses on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions. These were once thought to have completely random states of disorder and irregularities. The 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. 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…

Evidence: 54 • authority: 72

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its phy…

Evidence: 51 • authority: 92

Hybrid quantum-chaotic key expansion enhances QKD rates using the Lorenz system.. Quantum key distribution (QKD) provides a foundation for information-theoretic security based on quantum mechanics, yet its practical deployment is often constrained by intrinsically low secure key generation rates, particularly in high-bandwidth or low-latency settings. This work introduces a hybrid cryptographic technique that integrates conventional QKD with deterministic chaos, modeled using the Lorenz attractor, to provide a software-based enhancement of the effective key expansion rate. From a short 20-bit QKD seed, the system generates long bitstreams within milliseconds; although these streams exhibit…

Evidence: 42 • authority: 92

Chaotic Dynamics in Financial Markets: Detecting Early-Warning Signals of Systemic Crises (1990–2025, Global Panel). This paper investigates whether financial markets exhibit deterministic chaotic dynamics and whether such dynamics can improve early detection of systemic crises. Using a global dataset covering major equity indices and volatility indicators over the period 1990–2025, we apply a comprehensive nonlinear framework combining maximum Lyapunov exponents, correlation dimension, sample entropy, recurrence quantification analysis (RQA), and nonlinear vector autoregressions. The results provide robust evidence that financial markets are characterized by positive Lyapunov exponents, in…

Evidence: 42 • authority: 72

In chaos theory, the butterfly effect is the sensitive dependence on initial conditions in which a small change in one state of a deterministic nonlinear system can result in large differences in a later state. The term is closely associated with the work of the mathematician and meteorologist Edward Norton Lorenz. He said the butterfly effect is derived from the example of the details of a tornado (the exact time of formation, the exact path taken) being influenced by minor perturbations such as a distant butterfly flapping its wings several weeks earlier. Lorenz originally used a seagull causing a storm but was persuaded to make it more poetic with the use of a butterfly and tornado by 19…

Evidence: 42 • authority: 72

In mathematics, computer science and physics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. A deterministic model will thus always produce the same output from a given starting condition or initial state.

Evidence: 42 • authority: 92

Dynamical system modeling in schizophrenia: a narrative review of computational psychiatry frameworks.. The temporal evolution of schizophrenia symptoms is inherently complex and non-linear. Clinical states fluctuate between relatively stable and highly disorganised phases. These dynamics are only partially captured by conventional diagnostic frameworks. This limitation has motivated the development of models that explicitly account for temporal structure and state transitions. Dynamical systems theory provides a rigorous framework for describing non-linear interactions and evolving symptom trajectories. Recent advances in longitudinal data acquisition and computational analysis have enable…

Evidence: — • authority: 92

Chaos Theory and Child Development: Quantifying Nonlinear Pathways of Growth.. Traditional developmental science has often described child growth as a sequence of stages or linear progressions, yet many phenomena-abrupt spurts and regressions, idiosyncratic pathways, and widening individual differences-resist linear accounts. This article proposes chaos theory as a framework for quantifying developmental trajectories. Chaos theory, which addresses how complex patterns emerge from simple rules in deterministic yet unpredictable ways, aligns with observations of sensitive developmental periods, emergent behaviors, and divergent outcomes. I situate chaos theory alongside dynamic systems theory…

Evidence: — • authority: 92

Phase space volume preserving dynamics for deterministic dynamical systems.. Infinitesimal volumes stretch and contract as they coevolve with classical phase space trajectories according to linearized dynamics. Unless these tangent-space dynamics are modified, chaotic evolution causes the volume spanned by evolving tangent vectors to collapse. However, this collapse is unphysical and due to their exponential alignment along the most expanding direction, independent of the compressibility of the phase-space volume. Here, we propose an alternative linearized dynamics and rectify the generalized Liouville equation to preserve phase space volume, even for non-Hamiltonian systems. Within a class…