Question

Without doing further calculations, sketch bifurcation diagrams showing how the effect of adding small terms could lead to:

(i) no bifurcations;
(ii) 2 saddle-node bifurcations;
(iii) 4 saddle-node bifurcation

Answer 1

Bifurcation diagrams show how the behavior of a dynamical system changes as a parameter is varied. Saddle-node bifurcations occur when fixed points in the system collide and disappear. We can sketch bifurcation diagrams by using different equations to represent the behavior of the system.

Step-by-step explanation:

A bifurcation diagram shows how the behavior of a dynamical system changes as a parameter is varied. In the case of saddle-node bifurcations, they occur when two fixed points in the system collide and disappear as the parameter is changed. To sketch bifurcation diagrams, we can use simple equations to represent the behavior of the system. Without doing further calculations, we can illustrate the effect of adding small terms on the bifurcation diagrams:

(i) To show no bifurcations, we can use a simple linear equation, such as f(x) = x, where there are no changes in the fixed points as the parameter is varied.

(ii) To show 2 saddle-node bifurcations, we can use a quadratic equation, such as f(x) = x^2 - a, where the fixed points collide and disappear at two different parameter values.

(iii) To show 4 saddle-node bifurcations, we can use a quartic equation, such as f(x) = x^4 - ax^2, where the fixed points collide and disappear at four different parameter values.