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A set of chaotic solutions to the Lorenz equations, which describe the two-dimensional flow of a fluid layer heated from below and cooled from above. Characterized by its butterfly-shaped plot.

Chaos in Hamiltonian systems, which are often conservative dynamical systems. Exhibits chaotic behavior despite the energy conservation, generally found in the realm of celestial mechanics.

Given a complex function like $f(z) = z^2 + c$, a Julia set is the boundary of the set of all points that do not escape to infinity under iterations. They are fractal and can be used to study system stability.

Arises from a simplified model of two-dimensional dynamical systems and is used to describe the chaotic evolution of systems over time. Defined by the iterative process: $x_{n+1} = 1 - ax_n^2 + y_n$ and $y_{n+1} = bx_n$.

A type of chaos characterized by irregular alternation between predictable phase and unpredictable chaotic phase. It is common in systems that are close to a periodic doubling route to chaos.

Two mathematical constants that appear in the period-doubling bifurcations of maps and differential equations. They are associated with the transition to chaos.

A route to chaos where a system alternates irregularly between laminar and turbulent flow. This erratic switching is typical in fluid dynamics and cardiac rhythms.

A technique used to reduce the number of dimensions in a dynamical system by taking a cross-section of the phase space that the system intersects transversally, enabling easier visualization and analysis of the system's global behavior.

Sensitive dependence on initial conditions where small changes can have large consequences, as popularly exemplified by the idea that a butterfly flapping its wings in Brazil can cause a tornado in Texas.

Kolmogorov-Arnold-Moser theorem explains the persistence of quasi-periodic orbits in Hamiltonian systems despite small perturbations, implying that not all orbits in perturbed systems are chaotic.

A visual summary of successive period doubling as a system parameter is varied, showing how a system transitions from ordered to chaotic behavior.

A ratio providing a statistical index of complexity comparing how detail in a pattern changes with the scale at which it is measured. Fractals have non-integer dimensions.

A measure that describes the rate of separation of infinitesimally close trajectories. Positive Lyapunov exponents are indicative of chaos.

The set of complex numbers $c$ for which the function $f_c(z) = z^2 + c$ does not diverge when iterated from $z = 0$. Represents a complex map of the stability of corresponding Julia sets.