Question Type 5: Interpreting key informative points using phase portraits Exercises

Consider the system with Jacobian at the equilibrium $(0,0)$ given by

Determine the type and stability of the origin.

A phase portrait shows a saddle at the origin with unstable manifold along the $x$-axis and stable manifold along the $y$-axis. Propose a linear system with these properties and justify your choice.

A phase portrait exhibits spiral trajectories approaching an equilibrium inwards, making two rotations before settling. What does this indicate about the eigenvalues’ real and imaginary parts? Explain.

Sketch the phase portrait of the system

For the system

Consider a two-dimensional system whose equilibrium at the origin is a non-hyperbolic node (one zero eigenvalue, one negative). Describe the phase portrait qualitatively near the origin.

Consider the linear system

System:

The system

A nonlinear predator–prey model is given by

Given the nonlinear system