Contents

- 1 How do you draw a phase portrait?
- 2 What does a phase portrait show?
- 3 How do you draw a phase plane?
- 4 Is a phase portrait a vector field?
- 5 What is the phase trajectory?
- 6 How do you make a phase portrait in Python?
- 7 What is a stable spiral?
- 8 How do you determine if a critical point is stable or unstable?
- 9 How do you calculate Isoclines?
- 10 Can trajectories cross?

## How do you draw a phase portrait?

To sketch the phase plane of such a system, at each point (x0,y0) in the xy- plane, we draw a vector starting at (x0,y0) in the direction f(x0,y0)i + g(x0,y0)j. Definition of nullcline. The x-nullcline is a set of points in the phase plane so that dx dt = 0.

## What does a phase portrait show?

Also, the direction of the vectors give the direction of the trajectory as t t increases so we can show the time dependence of the solution by adding in arrows to the trajectories. Doing this gives the following sketch. This sketch is called the phase portrait.

## How do you draw a phase plane?

To sketch the phase plane of such a system, at each point (x0,y0) in the xy- plane, we draw a vector starting at (x0,y0) in the direction f(x0,y0)i + g(x0,y0)j. Definition of nullcline. The x-nullcline is a set of points in the phase plane so that dx dt = 0.

## Is a phase portrait a vector field?

The phase portrait is a plot of a vector field which qualitatively shows how the solutions to these equations will go from a given starting point.

## What is the phase trajectory?

The trajectory of a point in a phase space, representing how the state of a dynamical system changes with time. A point w of a non-closed phase trajectory divides it into two parts — the positive and negative semi- trajectories.

## How do you make a phase portrait in Python?

- Other Visualization Packages in Python.
- Phase Portraits. Ingredients. Libraries. Example Model. Recipe. Step 1: Define the range over which to plot. Step 2: Calculate the state space derivatives at a point. Step 3: Calculate the state space derivatives across our sample space. Step 4: Plot the phase portrait.

## What is a stable spiral?

A fixed point for which the stability matrix has eigenvalues of the form (with ). SEE ALSO: Elliptic Fixed Point, Fixed Point, Hyperbolic Fixed Point, Stable Improper Node, Stable Node, Stable Star, Unstable Improper Node, Unstable Node, Unstable Spiral Point, Unstable Star.

## How do you determine if a critical point is stable or unstable?

An unstable critical point is one that is not stable. Informally, a point is stable if we start close to a critical point and follow a trajectory we will either go towards, or at least not get away from, this critical point.

## How do you calculate Isoclines?

In an equation of the form y’ = f(x, y), the isoclines are lines in the (x, y) plane obtained by setting f(x, y) equal to a constant. This gives a series of lines (for different constants) along which the solution curves have the same gradient.

## Can trajectories cross?

It’s said in elementary classical mechanics texts that the phase trajectories of an isolated system can ‘t cross. But clearly they can, for example for the pendulum, the trajectories look like this: The points where they cross correspond to unstable equilibrium.