Nonlinear Systems

Abstract

Nonlinear problems are of interest to physicists, mathematicians and also engineers. Nonlinear equations are difficult to solve and give rise to interesting phenomena like indeterministic behavior, multistability or formation of patterns in time and space. In the following we discuss recurrence relations like an iterated function, systems of ordinary differential equations like population dynamics models or partial differential equations like the reaction diffusion equation. We discuss fixed points of the logistic mapping and analyze their stability. A bifurcation diagram visualizes the appearance of period doubling and chaotic behavior as a function of a control parameter. The Ljapunov exponent helps to distinguish stable fixed points and periods from chaotic regions. For continuous-time models, the iterated function is replaced by a system of differential equations. For stable equilibria all eigenvalues of the Jacobian matrix must have a negative real part. We discuss the Lotka–Volterra model, which is the simplest model of predator-prey interactions and the Holling-Tanner model, which incorporates functional response. Finally we allow for spatial inhomogeneity and include diffusive terms to obtain reaction-diffusion systems, which show the phenomena of traveling waves and pattern formation. Computer experiments study orbits and bifurcation diagram of the logistic map, periodic oscillations of the Lotka–Volterra model, oscillations and limit cycles of the Holling-Tanner model and finally pattern formation in the diffusive Lotka–Volterra model.

Problems

Problem 22.1: Orbits of the Iterated Logistic Map

This computer example draws orbits (Fig. 22.5) of the logistic map

$$\begin{aligned} x_{n+1}=r_{0}\cdot x_{n}\cdot (1-x_{n}). \end{aligned}$$

(22.102)

You can select the initial value $x_{0}$ and the variable r.

Problem 22.2: Bifurcation Diagram of the Logistic Map

This computer example generates a bifurcation diagram of the logistic map (Fig. 22.6). You can select the range of r.

Problem 22.3: Lotka–Volterra Model

Equation (22.47) are solved with the improved Euler method (Fig. 22.8). The predictor step uses an explicit Euler step to calculate the values at $t+\Delta t/2$

$$\begin{aligned} H_{pr}(t+\frac{\Delta t}{2})= & {} H(t)+\left( r_{H}H(t)-aH(t)P(t)\right) \frac{\Delta t}{2}\end{aligned}$$

(22.103)

$$\begin{aligned} P_{pr}(t+\frac{\Delta t}{2})= & {} P(t)+\left( bH(t)P(t)-m_{p}P(t)\right) \frac{\Delta t}{2} \end{aligned}$$

(22.104)

and the corrector step advances time by $\Delta t$

$$\begin{aligned} H(t+\Delta t)&= H(t)+\left( r_{H}H_{pr}(t+\frac{\Delta t}{2})-aH_{pr}(t+\frac{\Delta t}{2})P_{pr}(t+\frac{\Delta t}{2})\right) \Delta t\end{aligned}$$

(22.105)

$$\begin{aligned} P(t+\Delta t)&= P(t)+\left( bH_{pr}(t+\frac{\Delta t}{2})P_{pr}(t+\frac{\Delta t}{2})-m_{p}P_{pr}(t+\frac{\Delta t}{2})\right) \Delta t . \end{aligned}$$

(22.106)

Problem 22.4: Holling-Tanner Model

The equations of the Holling-Tanner model (22.64), (22.65) are solved with the improved Euler method (see Fig. 22.11). The predictor step uses an explicit Euler step to calculate the values at $t+\Delta t/2$:

$$\begin{aligned} H_{pr}(t+\frac{\Delta t}{2})= & {} H(t)+f(H(t), P(t))\frac{\Delta t}{2}\end{aligned}$$

(22.107)

$$\begin{aligned} P_{pr}(t+\frac{\Delta t}{2})= & {} P(t)+g(H(t), P(t))\frac{\Delta t}{2} \end{aligned}$$

(22.108)

and the corrector step advances time by $\Delta t$:

$$\begin{aligned} H(t+\Delta t)= & {} H(t)+f(H_{pr}(t+\frac{\Delta t}{2}), P_{pr}(t+\frac{\Delta t}{2}))\Delta t\end{aligned}$$

(22.109)

$$\begin{aligned} P(t+\Delta t)= & {} P(t)+g(H_{pr}(t+\frac{\Delta t}{2}), P_{pr}(t+\frac{\Delta t}{2}))\Delta t . \end{aligned}$$

(22.110)

Problem 22.5: Diffusive Lotka–Volterra Model

The Lotka–Volterra model with diffusion (22.95) is solved in 2 dimensions with an implicit method (21.2.2) for the diffusive motion (Figs. 22.12 and 22.14). The split operator approximation (21.3) is used to treat diffusion in x and y direction independently. The equations

$$ \left( \begin{array}{c} H(t+\Delta t)\\ P(t+\Delta t) \end{array}\right) =\left( \begin{array}{c} A^{-1}H(t)\\ A^{-1}P(t) \end{array}\right) +\left( \begin{array}{c} A^{-1}f(H(t), P(t))\Delta t\\ A^{-1}g(H(t), P(t))\Delta t \end{array}\right) $$

$$\begin{aligned} \approx \left( \begin{array}{c} A_{x}^{-1}A_{y}^{-1}\left[ H(t)+f(H(t), P(t))\Delta t\right] \\ A_{x}^{-1}A_{y}^{-1}\left[ P(t)+g(H(t), P(t))\Delta t\right] \end{array}\right) \end{aligned}$$

(22.111)

are equivalent to the following systems of linear equations with tridiagonal matrix (5.3):

$$\begin{aligned} A_{y}U=H(t)+f(H(t), P(t))\Delta t \end{aligned}$$

(22.112)

$$\begin{aligned} U=A_{x}H(t+\Delta t) \end{aligned}$$

(22.113)

$$\begin{aligned} A_{y}V=P(t)+g(H(t), P(t))\Delta t \end{aligned}$$

(22.114)

$$\begin{aligned} V=A_{x}P(t+\Delta t) . \end{aligned}$$

(22.115)

Periodic boundary conditions are implemented with the method described in Sect. 5.4.