Glossary

dynamical systems
A rule (ODE or map) describing how a state moves through a space over time.

phase portraits 🧱
A plot of trajectories (and vector field) in state‐space showing all possible motions.

Lyapunov exponents (λ)
Rate at which nearby trajectories diverge $$(λ>0 ⇒ chaos)$$

Strange Attractors 🧱
A bounded, nonperiodic attractor with fractal dimension and sensitive dependence.

fractal dimensions (D)
Non-integer exponent $D$ in $(N(ε)\propto ε^{-D})$ measuring “space‐filling” complexity.

vector space 🧱
A set with vector addition and scalar multiplication satisfying 9 axioms (commutativity, etc.).

manifold 🧱
A space locally like ℝⁿ; smooth manifold if overlaps are $(C^∞)$.

Hamiltonian system 🧱
Dynamics on phase space $((q,p))$ given by $$(\dot q_i = ∂H/∂p_i,;\dot p_i=-∂H/∂q_i)$$

Liouville’s theorem
Hamiltonian flow is volume‐preserving in phase space (divergence‐free).

symplectic geometry 🧱
Study of closed, nondegenerate 2-form $ω$ on a $(2n)-manifold$; underlies Hamiltonian mechanics.

Noether’s theorem
Each continuous symmetry of the action ↔ a conserved quantity (energy, momentum, …).

quantum mechanics 🧱
States $∈$ Hilbert space; observables = Hermitian operators; evolve via Schrödinger eq.; measurement ⇾ collapse; superposition ≠ entanglement.

Superposition vs. Entanglement
– Superposition: one system in $A + B$
– Entanglement: two systems in $AA + BB$ with non-factorizable joint state