Finance

Systems / ODEs / Population dynamics

The Ma-Chen macroeconomic system — interest rate, investment demand, and price index whose quadratic couplings yield a low-dimensional chaotic attractor.

continuous · ODE3 dimensionschaotic

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Definition

\[ \begin{aligned} \dot{x} &= x y + x \left(- a + \frac{1}{b}\right) + z \\ \dot{y} &= - b y - x^{2} \\ \dot{z} &= - c z - x \end{aligned} \]

Parameters

Symbol	Default	Role
a	0.001	savings amount (interest-rate damping)
b	0.2	per-unit investment cost (demand damping)
c	1.1	elasticity of demand (price-index damping)

State variables: x, y, z

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = - a - b - c + y + \frac{1}{b}$
state-dependent

Equilibria

3 equilibria
0 stable · 3 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.Finance()
traj = system.run(final_time=100.0, dt=0.01)
ts.plot(traj)

spectrum = ts.analysis.lyapunov_spectrum(system)
ts.analysis.kaplan_yorke_dimension(spectrum)

Reference

Cai & Huang (2007), Int. J. Nonlinear Sci. 3, 235-241

BibTeX

@misc{finance,
  title = {Finance system},
  note = {Cai & Huang (2007), Int. J. Nonlinear Sci. 3, 235-241}
}