Eigenvalues and the Trace-Determinant Plane of a Linear Map

Drag the point on the trace-determinant plane [1], and the corresponding eigenvalues are displayed around the unit circle in the complex plane.
The graphic on the left shows the discriminant parabola and the lines for unit eigenvalues. Where in the trace-determinant plane are the eigenvalues the same? Can you make both eigenvalues have modulus one?
Let be a matrix; then the trace is the sum of the eigenvalues and the determinant is the product of the eigenvalues. The inverse relation is found by using the quadratic formula on the characteristic polynomial.
The eigenvalues of the matrix determine how the orbit/flow behaves. Can you find the critical point at which a period-doubling bifurcation occurs in the map/flow?



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