IB Mathematics Applications & Interpretation Question Type 5: Interpreting Key Informative Points Using Phase Portraits Exercises

Sketch the phase portrait of the system $$\{\begin{array}{l}{x}^{\prime }=y\\ y^{\prime }=-4x-y\end{array}$$ and identify the type and stability of the origin.

The system $$\{\begin{array}{l}{x}^{\prime }=4x-y\\ y^{\prime }=2x+y\end{array}$$ has an equilibrium at the origin. Determine the eigenvalues and eigenvectors of the linearization, then classify the equilibrium.

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.

Given the nonlinear system $$\{\begin{array}{l}{x}^{\prime }=x(1-x)-xy\\ y^{\prime }=y(-2+x)\end{array}$$ draw the nullclines, find all equilibrium points, and determine the stability of each using linearization.

For the system $$\{\begin{array}{l}{x}^{\prime }=x-y^2\\ y^{\prime }=-y+x^2\end{array}$$ determine the type of equilibrium at the origin $(0,0)$ using linearization and comment on the effect of the nonlinear terms.

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.

Consider a two-dimensional autonomous system whose equilibrium at the origin is a non-hyperbolic node with eigenvalues ${\lambda }_1<0$ and ${\lambda }_2=0$.

Describe the qualitative behavior of the phase portrait in the neighborhood of the origin and provide a sketch showing representative trajectories.

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

$$J=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$$

Determine the type and stability of the origin.

A nonlinear predator–prey model is given by the following system of differential equations: $$\{\begin{array}{l}{x}^{\prime }=x(2-x-y)\\ y^{\prime }=y(-1+x)\end{array}$$ Locate the nontrivial equilibrium and classify it by computing the Jacobian and its eigenvalues.