Primer on Complex Systems

Abstract

Complexity has become one of the buzzwords of modern science. It is almost impossible to browse through a recent issue of any scientific journal without running into terms such as complexity, complex behaviour, complex dynamics or complex systems mentioned in one way or another. But what do most scientists actually mean when they use these terms? What does it take for a system to become complex? Why is it important to know if it is complex or not?

Appendices

Appendix 1: Fixed Points of a Dynamical System

A dynamical system is usually defined by a system of N ordinary differential equations (or simply, ODEs) that, written in matrix form, becomes:

$$\displaystyle \begin{aligned} \dot{\mathbf{x}} = \mathbf{F}(\mathbf{x}), \end{aligned} $$

(1.12)

where F is an arbitrary N-dimensional non-linear function. The fixed points of the system are given by the solutions of F(x _fp) = 0, that correspond to all the time-independent solutions of the problem. At most, N different fixed points may exist. Each fixed-point can be either stable, unstable or neutral, depending on whether a small perturbation away from the fixed point is attracted back to it (i.e., stable), diverges away from it (i.e., unstable) or remains close to the fixed point without either being attracted or ejected from it (i.e., neutral).

To determine their type [39], one needs to linearize the system of ODEs in the neighbourhood of each fixed point, x _fp;k , where the index k = 1, 2, ⋯ , N _p , being N _p  ≤ N the number of fixed points of the system. To do this, one writes x = x _fp;k  + y and assumes that the norm of the perturbation |y| remains small. The behaviour of the perturbation, when advanced in time, will tell the type of the fixed point. A linear advance is sufficient. Thus, one proceeds by inserting x = x _fp;k  + y into Eq. 1.12, that yields the linear evolution equation of y:

$$\displaystyle \begin{aligned} \dot{\mathbf{y}} = \left. \frac{\partial\mathbf{F}}{\partial \mathbf{x}}\right|{}_{\mathbf{ x}= \mathbf{x}_{\mathrm{fp};k}} \cdot \mathbf{y}. \end{aligned} $$

(1.13)

The matrix formed by taking partial derivatives of F and evaluating them at the fixed point is called the jacobian matrix at the fixed point. The solution of this equation is given by the exponential of the jacobian,

$$\displaystyle \begin{aligned} \mathbf{y}(t) = \exp\left( \left. \frac{\partial\mathbf{F}}{\partial \mathbf{ x}}\right|{}_{\mathbf{x}= \mathbf{x}_{\mathrm{fp};k}} (t - t_0) \right) \cdot \mathbf{ y}(t_0). \end{aligned} $$

(1.14)

Therefore, it is the eigenvalues of the jacobian at the fixed point that define the type of the fixed point. If there is any eigenvalue with a positive real part, the perturbation will grow and the solution will move away from the fixed point, and the fixed point is unstable. If all eigenvalues have negative real parts, the amplitude of the perturbation will eventually vanish, and the solution will come back to the fixed point, which is stable. If, on the other hand, all eigenvalues are purely imaginary, the norm of the perturbation remains constant over time and, although the solution never gets back to the fixed point, it does not run away from it. The fixed point is then neutral.

Problems

1.1

Predator-Prey Model: Definition

Prove that, if a term like $\gamma n_l - \nu n_l^2$ had been used in Eq. 1.2 of the predator-prey model to account for lynx mating and overpopulation, a finite number of lynx might exist in the absence of hares, which would make no sense in the context of the model.

1.2

Lotka-Volterra Model: Periodic Orbits

• (a) Prove that the predator-prey model, if 𝜖 = 0, admits the conserved quantity:

  $$\displaystyle \begin{aligned} K(n_l, n_h):= \alpha \log(n_l) + \gamma \log(n_h) - \beta n_l - \delta n_h \end{aligned} $$

  (1.15)

• (b) Write a numerical code to plot the orbits K(n _l , n _h ) = K ₀, for arbitrary K ₀.

1.3

Predator-Prey Model: Fixed Points

With the help of the techniques discussed in Appendix 1 in this chapter,

• (a) prove that all the fixed points of the PP model are given by Eqs. 1.3–1.5.

• (b) show that the jacobians at the fixed points are given by:

  $$\displaystyle \begin{aligned} A: \left(\begin{array}{cc} \alpha & 0 \\ 0 & -\gamma\end{array}\right),~~~ B: \left(\begin{array}{cc} -\alpha & -\beta\alpha/\epsilon \\ 0 & \delta\alpha/\epsilon-\gamma\end{array}\right),~~ C: \left(\begin{array}{cc} -\epsilon\gamma/\delta & -\beta\gamma/\delta \\ \left(\delta\alpha - \epsilon \gamma\right)/\beta & 0 \end{array}\right) \end{aligned} $$

  (1.16)

• (c) find the eigenvalues of each jacobian.

• (d) classify the type of each fixed point.

1.4

Predator-Prey Model: Phase Space

Write a computer code that numerically integrates the dynamical system given by Eqs. 1.1–1.2, starting from appropriate initial conditions. Use the code to reproduce all the phase space plots shown in Fig. 1.5.

1.5

The Running Sandpile: Building of a Cellular Automata

Write a computer program that evolves in time the one-dimensional running sandpile described in Sect. 1.3.1. The output of the program should include at least the time record of the state of each cell of the sandpile (stable or unstable), its height, the total mass confined and the outflux coming out of the bottom of the pile. This program will be the basis of several exercises proposed in later chapters.

1.6

Diffusive Equation: Global Confinement Scaling

Assume a one-dimensional system of size L driven by a source density S, uniform both in time and space. If transport in the system takes place according to the diffusive equation (Eq. 1.7) with boundary conditions: n(L) = dn/dx(0) = 0, show that:

• (a) profiles are not stiff by finding the steady-state profile as a function of S and D;

• (b) the scaling of the global confinement time, τ _CF , defined as the ratio of the total mass contained to the total drive strength, is τ _CF  ∝ L ²;

• (c) there is no power degradation if D is independent of the external source S.