lacunary - Mathnotes

We use x˙ to represent the time derivative of x, that is, x˙=dx/dt.

Definition: First-order system (also: one-dimensional system) \@{first-order-system}

A first-order system is a system of the form

x˙=f(x),

where x=x(t) is a @real-valued function of time t, and f(x) is a smooth real-valued function of x.

We can interpret a @differential-equation of this form as a as a vector field. f(x) tells us in which direction and at what magnitude we move at any point on the real line (where f(x) is defined.) We can do this by plotting x˙ vs x.

You can see that when x is to the right of 0, the flow is to the right, and when x is to the left of 0 the flow is to the left.

At some points, there is no flow.

Definition: Stable \@{stable}

A fixed point where nearby points flow towards it is said to be stable.

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Definition: Unstable \@{unstable}

A fixed point where nearby points flow away from it is said to be unstable.

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Remark \@{remark-5}

The convention, when drawing a dynamical system as a vector field is to draw filled in circles for stable fixed points and empty circles for unstable fixed points.

Definition: Phase point \@{phase-point}

The starting point, x0, where we place a particle is called a phase point.

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Definition: Trajectory \@{trajectory}

The function describing the path taken by a particle starting at a phase point is called a trajectory and represents a solution to a @differential-equation with @initial-conditions x=x0.

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Definition: Phase Portrait \@{phase-portrait}

A drawing that shows the different trajectories taken from different phase points in a system is called a phase portrait.

We can also visualize trajectories by plotting x(t) vs t.

Here's both views together.

Definition: Globally stable \@{globally-stable}

A fixed point x∗ that is approached from any starting position on the real line (other than that at x∗ itself) is said to be globally stable.

Application: Population Growth

A very simple model of population growth is just exponential growth. You can model this as

N˙=rN,

where r>0. You can see by modeling this on the demo's above that population just goes to infinity with this. This is not realistic. A better model assumes there is a certain carrying capacity K, where if the population N exceeds K, growth actually becomes negative. This is modeled using the logistic equation

N˙=rN(1−NK).

Linear Stability Analysis

While it's nice to have a visual intuition for whether a fixed point is stable or not, sometimes it's also nice to know analytically.

Theorem \@{linear-stability-analysis}

Let x∗ be a fixed point of x˙=f(x). Then, if f′(x)≠0, if f′(x∗) is negative, then x∗ is a stable fixed point . If f′(x∗) is positive, then x∗ is an unstable fixed point.

Remark \@{linear-stability-analysis-remark}

This comes from letting u(t)=x(t)−x∗ be a small perturbation away from x∗, differentiating it, writing its taylor series, then noticing that f(x∗)=0 and terms greater than the linear term matter less than the linear term and writing

u˙=f′(x∗)u.

This is called the linearization about x∗.

It also only works if f′(x)≠0. If that's not the case, the best bet is to fall back to graphical analysis or to solve explicitly if possible.

Definition: Half-stable \@{half-stable}

A fixed point x∗ where f′(x)=0 and the stability depends on which side of x∗ x0 lies on is called half-stable. It is denoted with a half-filled circle.

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Existence and Uniqueness

Theorem \@{existence-and-uniqueness-of-ivp-solutions}

Consider the @initial-value-problem

x˙=f(x),x(0)=x0.

Suppose that f(x) and f′(x) are continuous on an open interval R of the x-axis, and suppose that x0 is a point in R. Then, the initial value problem has a solution x(t) on some time interval (−τ,τ) about t=0, and the solution is unique.