math concept
30 topics use this
Math concept
Dynamical Systems
Core equation
$$\dot{\mathbf{x}} = f(\mathbf{x}), \quad \mathbf{x} \in \mathbb{R}^n$$
Dynamical systems theory studies how systems evolve over time. Fixed points, limit cycles, bifurcations, and chaos are its core phenomena — connecting ecology, economics, neuroscience, climate, and physics.
Linear stability analysis
At an equilibrium $\mathbf{x}^$, linearise: $\dot{\boldsymbol{\xi}} = J\boldsymbol{\xi}$ where $J = Df(\mathbf{x}^)$.
Stability is determined by the eigenvalues of $J$:
- All eigenvalues have negative real part → stable node/focus
- Any eigenvalue with positive real part → unstable
- Purely imaginary eigenvalues → centre (non-linear analysis needed)
Bifurcations
A bifurcation occurs when a small parameter change qualitatively alters the system’s long-run behaviour.
| Type | Description |
|---|---|
| Saddle-node | Two fixed points collide and annihilate |
| Pitchfork | One fixed point splits into three |
| Hopf | Fixed point loses stability, limit cycle born |
| Period-doubling | Route to chaos |
Chaos
A system is chaotic if it is deterministic yet sensitive to initial conditions (positive Lyapunov exponent):
\[\|\delta\mathbf{x}(t)\| \approx e^{\lambda t}\|\delta\mathbf{x}(0)\|, \quad \lambda > 0\]Fields that use this concept
Physical sciences
Astrophysics
Dark Matter
The non-luminous mass component comprising roughly 27% of the universe's energy budget, inferred from gravitational effects.
Friedmann Equations and Cosmology
The dynamical equations governing the expansion history of a homogeneous, isotropic universe derived from general relativity.
Stellar Evolution
The life cycle of stars from gravitational collapse through nuclear burning phases to their final compact remnants.
Earth sciences
Climate modeling
Carbon Cycle Modeling
Box models of carbon exchange between the atmosphere, ocean, and land biosphere, including feedback parameters that modulate the airborne fraction of emissions.
Climate Sensitivity
Quantifying how much global mean temperature responds to a doubling of CO₂, from feedback analysis to paleoclimate and observational constraints.
Energy Balance Models
Zero- and one-dimensional models of Earth's temperature from the balance between absorbed solar radiation and outgoing longwave emission.
Climate Feedbacks
The physical mechanisms by which an initial warming perturbation is amplified or damped, quantified through the feedback framework and linearized forcing-response relationships.
Ocean Circulation
The physical dynamics of large-scale ocean flow, from geostrophic balance and wind-driven gyres to the thermohaline circulation and ENSO variability.
Sea Ice Modeling
Thermodynamic and dynamic modeling of sea ice, including the Stefan freezing condition, viscous-plastic rheology, and Arctic decline under climate change.
Physical sciences
Computational chemistry
Finance & economics
Econometrics
Bargaining
Strategic models of how two parties split a surplus through negotiation.
Evolutionary Game Theory
Population dynamics where strategies spread by fitness rather than rational choice.
Repeated Games
How cooperation can emerge when players interact across multiple rounds.
Earth sciences
Meteorology
Atmospheric Waves
Rossby waves, gravity waves, and their dispersion relations that govern large-scale energy propagation in the atmosphere.
Atmospheric Boundary Layer
The turbulent lowest kilometre of the atmosphere that mediates exchanges of heat, moisture, and momentum between the surface and free troposphere.
Mesoscale Convection
The organisation of thunderstorms into mesoscale convective systems driven by CAPE, wind shear, and cold pool dynamics.
Primitive Equations
The governing equations of atmospheric motion on a rotating sphere, forming the backbone of all global weather and climate models.
Tropical Cyclones
The dynamics of warm-core rotating vortices that form over tropical oceans and can reach devastating intensities.
Life sciences
Quant ecology
Food Web Dynamics
Food webs describe energy flow and species interactions across trophic levels, combining network theory with nonlinear population dynamics.
Island Biogeography
MacArthur and Wilson's equilibrium theory explains species richness on islands through the balance of immigration and extinction, predicting the species-area relationship.
Logistic Population Growth
The logistic equation models density-dependent population growth toward a carrying capacity, with rich extensions including chaos, Allee effects, and generalized forms.
Lotka-Volterra Equations
A foundational system of nonlinear ODEs describing predator-prey population dynamics and their oscillatory coexistence.
Metapopulation Dynamics
The Levins metapopulation model and its extensions describe how species persist as networks of local populations connected by dispersal.
SIR Epidemic Model
The SIR compartmental model partitions a population into Susceptible, Infected, and Recovered classes to describe epidemic spread through a basic system of ODEs.
Physical sciences
Quantum computing
Engineering & CS
Robotics
Motion Planning
Algorithms for finding collision-free, dynamically feasible, and optimally smooth robot trajectories from start to goal.
Optimal Control in Robotics
Optimal control finds inputs that minimize a cost functional over a trajectory, from LQR for linear systems to MPC and iLQR for nonlinear robots.
PID Control
The proportional-integral-derivative controller — the most widely deployed feedback control law in engineering.
Engineering & CS
Signal processing