Generative eBooks

Bifurcations: a first glance

Generative eBooks — 2016-06-08T12:52:22Z

We've just seen that solutions of one-dimensional systems behave rather trivially. But dependence on parameters adds spice because qualitative changes in the behavior of solutions can occur. These changes are called bifurcations.
This chapter aims at introducing a few examples and not developing the full theory. We'll explore additional types of bifurcations when we step up to two- and three-dimensional systems.
The basic setting is a one-dimensional differential equation (…)

- Part I : One-dimensional systems: flows on the real line

Interlude: Impossibility of periodic oscillations

Generative eBooks — 2016-06-08T12:51:48Z

Being constrained to move on the real line forces solutions to increase or decrease monotonically, or remain constant (fixed point). More precisely, non-constant solutions either approach a fixed point, or diverge to $\pm\infty$ monotonically. To state it more geometrically, a particle flowing on the real line according to $\dotx=f(x)$ never reverses direction. Hence there are no periodic solutions to such equations.
Let's prove analytically the absence of periodic solutions. Suppose on (…)

- Part I : One-dimensional systems: flows on the real line

Fixed points & their stability

Generative eBooks — 2016-06-08T12:51:19Z

For an equation $\dotx=f(x)$, recall that a fixed point $\barx$ is a point such that $f(\barx)=0$; it corresponds to a constant or stationary solution $x(t)=\barx$ that is obviously defined for all $t\in\mathbbR$.
Consider again the logistic equation $\dotx=bx-dx^2$ for $x\geq 0$; it has two fixed points, namely $0$ and $b/d$. Intuitively, $0$ is an unstable fixed point since it ‘repulses' solutions that start from $x_0\in (0,b/d)$, while $b/d$ is a stable fixed point since it ‘attracts' (…)

- Part I : One-dimensional systems: flows on the real line

Existence, uniqueness and lifetime of solutions

Generative eBooks — 2016-06-08T12:50:50Z

If we are given a system $\dotx=f(x)$ and an initial state $x_0$, does there exist a solution $x(t)$ to this equation such that $x(0)=x_0$? It turns out that if $f$ is a continuous function, then existence is guaranteed. The example $$ f(x)= \begincases 1 & \textif\quad x< 0\ 1 & \textif\quad x\geq 0 \endcases $$ shows that when $f$ is discontinuous, things can go wrong: there is no solution that satisfies $x(0)=0$.
The next natural question is about uniqueness. What can go (…)

- Part I : One-dimensional systems: flows on the real line

Prelude: graphical study of one-dimensional differential equations

Generative eBooks — 2016-06-08T12:50:19Z

The graphical study of the logistic equation we did in a previous chapter paves the way for the general case. For if $\dotx=f(x)$, plot $\dotx$ versus $x$, that is, the graph of $f$. Intersection points with the $x$-axis are fixed points. Between two fixed points, the graph is either over or under the $x$-axis: in the former case $\dotx>0$, which means that $x(t)$ increases and can be represented by drawing arrows on the $x$-axis that point to the right; in the latter case $\dotx‘Allee (…)

- Part I : One-dimensional systems: flows on the real line