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

\[ \begin{aligned} \dot{y_{0}} &= y_{0} y_{1} + y_{0} \left(- a + \frac{1}{b}\right) + y_{2} \\ \dot{y_{1}} &= - b y_{1} - y_{0}^{2} \\ \dot{y_{2}} &= - c y_{2} - y_{0} \end{aligned} \]

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)

Properties

Lyapunov spectrum

$+0.1059,\; +0.01083,\; -0.6443$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.181$

Divergence ∇·f

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

Equilibria

3 equilibria
0 stable · 3 unstable

Define it in TSDynamics

import tsdynamics as ts

sys = ts.systems.Finance()
traj = sys.integrate(final_time=100.0, dt=0.01)

exps = sys.lyapunov_spectrum()
ts.kaplan_yorke_dimension(exps)

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}
}