Chaos and nonlinear dynamics

Chaos and nonlinear dynamics study systems whose outputs do not change proportionally with their inputs, including how feedback and nonlinear interactions produce complex behavior.

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Nonlinear system: A system whose response is not proportional to its inputs, often because its variables interact or feed back. Nonlinearity is the defining condition that permits behaviors absent from linear models.
Dynamical system: A mathematical model that describes how a system’s state changes over time. Nonlinear dynamics is a study of a particular class of dynamical systems.
Logistic map: A discrete model of population growth defined by a quadratic recurrence. Its parameter sweep displays period doubling and chaos in a compact example.
Linear system: A system whose governing relations obey superposition and proportional scaling. Linear systems lack many of the interactions that enable nonlinear complexity.
Deterministic chaos: Aperiodic behavior in a deterministic system, typically marked by sensitive dependence on initial conditions. It is the best-known route from simple nonlinear rules to apparently unpredictable motion.
State space: A space whose points represent all possible states of a system. Trajectories and attractors are understood as paths and sets in state space.
Lorenz system: A three-dimensional system of differential equations introduced as a simplified model of atmospheric convection. Its strange attractor became an iconic example of deterministic chaos.
Stochastic process: A process whose evolution includes randomness described by probability distributions. Random irregularity can resemble chaos, but chaos arises from deterministic evolution.
Bifurcation: A qualitative change in a system’s dynamics as a parameter varies. Parameter changes can shift a nonlinear system between fixed points, cycles, and chaotic regimes.
Ordinary differential equation: An equation relating a function of one variable to its derivatives. Many continuous-time nonlinear models specify motion through differential equations.