Bifurcations & XPPAUT. Outline. Why to study the phase space? Bifurcations / AUTO Morris-Lecar. A Geometric Way of Thinking. Exact solution:. N. t. K/2. Logistic Differential Equation. K. K. y. x. Graphical/Topological Analysis. When do we understand a dynamical system?
Study with AUTO (see later) the forcast for lovers governed by the general linear system:
Consider combinations of different types of lovers, e.g.
We begin with the classic Lotka-Volterra model of competion between two species competing for the same (limited) food supply.
Principle of Competitive Exclusion:
Two species competing for the same limited resource typically cannot coexist.
Study the phase space of the Rabbit vs. Sheep problem for different parameter. Try to compute the bifurcation diagram (see later in this lecture!) with respect to some parameter.
Suppose is the phase of the firefly‘s flashing.
is the instant when the flash is emitted.
is its eigen-frequency.
If the stimulus with frequency is ahead in the cycle, then we assume that the firefly speeds up. Conversely, the firefly slows down if it‘s flashing is too early. A simple model is:
can be simplyfied by introducing relative phases:
We obtain the non-dimensionalised equation:
Repeat the Bifurcation analysis for all prototypical cases mentioned above!