Nonlinear dynamical systems: Lyapunov exponents, bifurcations, strange attractors, deterministic chaos, sensitivity to initial conditions. Source of irreducible unpredictability in otherwise deterministic classical systems — a natural…
nonlinear-dynamics
Lyapunov exponent
λ = lim_{t→∞} (1/t)·ln |δx(t)/δx(0)|. Positive λ ⇔ chaotic dynamics with exponential sensitivity to initial conditions. The inverse…
Bifurcation
Qualitative change in the structure of a dynamical system's phase portrait as a parameter is varied: saddle-node, pitchfork, Hopf,…
Strange attractor
Fractal attracting set in phase space toward which chaotic trajectories converge globally while diverging locally. Lorenz, Rössler, Hénon…
Butterfly effect (sensitive dependence)
In a chaotic system, infinitesimal differences in initial conditions amplify exponentially, making long-term prediction infeasible despite…
Self-organized criticality
Dissipative systems driven far from equilibrium can self-tune to a critical point exhibiting power-law avalanches. Bak-Tang-Wiesenfeld…
Turing pattern
Stationary spatial pattern arising in a two-component reaction–diffusion system when the inhibitor diffuses sufficiently faster than the…
Logistic map
Simplest non-trivial discrete-time nonlinear dynamical system: x_{n+1} = r x_n (1 − x_n), defined on [0, 1] with growth parameter r ∈ [0,…
Feigenbaum constants
Two universal constants governing the period-doubling route to chaos in unimodal maps: δ ≈ 4.6692… (ratio of consecutive bifurcation…
Lorenz attractor
Continuous-time chaotic attractor arising in the three-variable ODE system ẋ = σ(y − x), ẏ = x(ρ − z) − y, ż = xy − βz (σ = 10, ρ = 28, β =…
Hopf bifurcation
Pair of complex eigenvalues crosses imaginary axis; limit cycle is born. Supercritical vs subcritical. Belousov-Zhabotinsky, cardiac…
KAM theorem
Weak perturbation of integrable Hamiltonian preserves most invariant tori — sufficiently irrational. Kolmogorov 1954, Arnold 1963, Moser…
Fractal dimension
Hausdorff/box-counting D non-integer for fractal sets. Koch (log4/log3), Sierpinski (log3/log2), strange attractor (~2.06 for Lorenz).…
Bifurcation theory
Saddle-node, transcritical, pitchfork, Hopf, period-doubling codim-1 bifurcations; center-manifold reduction to normal form.
Feigenbaum period doubling
Universal δ ≈ 4.6692, α ≈ 2.5029 for period-2^n cascade to chaos in 1D unimodal maps; self-similar bifurcation diagram.
Lorenz system & strange attractor
ẋ=σ(y-x), ẏ=ρx-y-xz, ż=xy-βz; chaotic for ρ=28, fractal dim ≈ 2.06; butterfly effect origin.
Hénon map & Smale horseshoe
(x,y)→(1-ax²+y, bx); strange attractor for a=1.4, b=0.3; embeds Smale horseshoe, symbolic dynamics.
Fractal & correlation dimensions
Hausdorff d_H; box-counting d_B; correlation C(r)~r^d₂; Grassberger–Procaccia estimator from time series.
Lyapunov exponents spectrum
λ₁≥...≥λ_n; chaos iff λ₁>0; Kaplan-Yorke dim d_KY = j + Σᵢ₌₁ʲ λᵢ/|λⱼ₊₁|; Pesin ID identity for entropy.
Kuramoto synchronization
θ̇_i = ω_i + (K/N) Σsin(θ_j-θ_i); phase transition to synchrony at K_c = 2/(πg(ω̄)); explains firefly flashing, neural sync.
Chimera states
Coexistence of synchronized & incoherent clusters in coupled identical oscillators; pioneered by Kuramoto-Battogtokh (2002).
Reaction-diffusion pattern formation
Turing instability when D_inhibitor > D_activator; spots, stripes, labyrinths; Belousov-Zhabotinsky waves.
Integrable systems & Lax pairs
∂_tL=[M,L] with Lax pair; eigenvalues of L conserved; inverse scattering transform solves KdV, NLS, sine-Gordon.
Nonlinear Schrödinger equation
i∂_tψ + ½∂_x²ψ + |ψ|²ψ = 0; integrable; bright/dark solitons; models BEC dynamics, optical fibers.
Chaos control (OGY method)
Small parametric perturbations stabilize unstable periodic orbits embedded in strange attractor; applied to lasers, arrhythmia.
Stochastic resonance
Noise enhances weak-signal detection in bistable systems; optimal at σ matching Kramers escape rate; neuroscience, climate.
Poincaré–Bendixson theorem
Non-empty compact ω-limit set of planar flow without fixed points must be a periodic orbit; precludes chaos in 2D autonomous.