Real Eigenvalues — Question 2

Question 2 – Solution

Strategy. Track stable and unstable coefficients separately; perpendicular distance measures departure from the stable line.

Step 1: Separate decay from growth. Eigenpairs are $3,(1,1)^T$ and $-1,(1,-1)^T$. For data $(p,q)$, $$X=\frac{p+q}{2}e^{3t}\binom 11+\frac{p-q}{2}e^{-t}\binom 1{-1}.$$ These independent vectors give the full family. A forward solution is bounded exactly when $p+q=0$, and then it tends to zero.

Step 2: Interpret the invariant lines. The unstable line $y=x$ points away from the origin; the stable line $y=-x$ points toward it. The opposite signs make the origin a saddle. Each modal coefficient preserves its sign unless identically zero. Thus off-line trajectories cannot cross either eigenline at finite time. Each equilibrium or nonzero half-line is a separate trajectory.

Step 3: Compute modal balance. The perturbed solution is $X=\varepsilon e^{3t}(1,1)^T+e^{-t}(1,-1)^T$. Equality of modal magnitudes requires $\varepsilon e^{3t}=e^{-t}$, hence $$\boxed{t_{\mathrm{balance}}=\tfrac 14\ln(1/\varepsilon).}$$ The eigenvectors have equal length, so the same time balances their Euclidean contributions. It is nonnegative under the stated assumption.

Step 4: Measure the delayed departure. Distance to $y=-x$ is $|x+y|/\sqrt 2=\sqrt 2\varepsilon e^{3t}$, strictly increasing from a value below one. Its first value one occurs at $$\boxed{t_{\mathrm{exit}}=\tfrac 13\ln\!\left(\frac 1{\sqrt 2\varepsilon}\right).}$$ The delay diverges as $\varepsilon\downarrow 0$, but every positive error has a growing unstable component. Only exact cancellation stays on the stable line. The figure illustrates $\varepsilon=1/10$.