Networks

On Spatial networks

In particular I looked at networks formed by the Physarum polycephalum, when connecting food sources. I used a mathematical model of these networks and looked at their features. They turn out to perform rather well under metrics of efficiency and robustness. They also display typical features of Spatial networks, in particular Planar networks.

See Physarum machines and physarum solver, and project in Overleaf. See code in Dropbox.

Complex systems

On Percolation

In particular, on the Relations between the stability of Boolean networks and percolation

Nonlinear systems

On the Duffing oscillator

The effects of small damping, nonlinearity and forcing on a harmonic oscillator:

$\ddot{x} + \beta \dot{x} + x + \delta x^3 = \Gamma \cos{\omega t}$

• The simple harmonic oscillator (forced and damped, in general)
• Duffing oscillator
  • Free (unforced) Duffing oscillator
    • Free undamped Duffing oscillator
    • Free damped Duffing oscillator
  • Forced damped Duffing oscillator.

There are potentially $8$ qualitatively different forms of the equation, depending of which combination of the $3$ parameters considered are non-zero.

The Duffing Equation: Nonlinear Oscillators and their Behaviour

References on the Duffing oscillator