Chapter 6.3, Problem 16E

Interpretation Introduction

Interpretation:

To show that for the given system, there is a trajectory connecting two saddle points. To sketch the phase portrait for the parameters equals to zero, less than zero, and greater than zero.

Concept Introduction:

A saddle point can be defined as a point on a graph that is neither a local minimum nor a local maximum. At the saddle point, the derivatives of x and y are both equal to zero.

Trajectory connecting two saddle points is called a saddle connection.

For non-zero value of parameter, phase portrait possesses different topological character in which saddle points are not connected by saddle connection.

A phase portrait’s topology can be structurally changed by an arbitrarily small perturbation parameter; such phase portraits are not structurally stable.