Solution. As a system, \(\theta ' = v\), \(v' = -\sin \theta - 0.5v\), with equilibria \((k\pi , 0)\). The Jacobian is \(\begin {bmatrix} 0 & 1\\ -\cos \theta & -0.5\end {bmatrix}\). At \(\theta = 2k\pi \): \(\tau = -0.5\), \(\Delta = 1\), \(\tau ^2 < 4\Delta \), a stable spiral (the pendulum hanging down). At \(\theta = (2k + 1)\pi \): \(\Delta = -1\), a saddle (the pendulum balanced upright). All are hyperbolic, so the linear classification is valid locally.

Solution. The eigenvalues are \(-1\) (eigenvector \(e_1\)) and \(1\) (eigenvector \(e_2\)). The \(y\)-axis is invariant, so \(W^u = \{x = 0\}\). For \(W^s\) try \(y = cx^2\): \(y' = 2cxx' = -2cx^2\), and the equation requires \(y' = cx^2 + x^2\), so \(c = -\frac 13\), and \(W^s\) is the parabola \(y = -x^2/3\).

Solution. Separating, \(x(t) = x_0/\sqrt {1 - 2ax_0^2t}\). If \(a < 0\) this tends to \(0\) like \(t^{-1/2}\), slower than any exponential; if \(a > 0\) it blows up at \(t = 1/(2ax_0^2)\). The linearisation \(x' = 0\) cannot distinguish them, which is why Theorem 10.2.2 requires hyperbolicity.

Solution. The equations become \(x' = (a - e)x - bxy\), \(y' = -(c + e)y + dxy\), with the same form, so by Theorem 10.5.1(iii) \(\bar x = (c + e)/d\) and \(\bar y = (a - e)/b\). Moderate fishing raises the average prey population and lowers the average predator population. Conversely a pesticide that kills both pests and their natural predators can increase the average number of pests.

Solution. \(I' = I(\beta S - \gamma )\), positive if and only if \(S > \gamma /\beta \). For the conserved quantity, \(\frac {d}{dt}\bigl (S + I - \frac \gamma \beta \log S\bigr ) = -\gamma I + \frac \gamma \beta \beta I = 0\). Initially \(S \approx S_0\), so the epidemic grows if and only if \(R_0 = \beta S_0/\gamma > 1\), the basic reproduction number. The conserved quantity then gives the final size: as \(I \to 0\), \(S_\infty \) solves \(S_\infty - \frac \gamma \beta \log S_\infty = S_0 + I_0 - \frac \gamma \beta \log S_0\).

Solution. The equilibria are \((0, 0)\), \((3, 0)\) and \((0, 2)\); the nullclines \(x + y = 3\) and \(x + y = 2\) are parallel, so there is no coexistence point. At \((3, 0)\), \(J = \begin {bmatrix} -3 & -3\\ 0 & -1\end {bmatrix}\), a stable node; at \((0, 2)\), \(J = \begin {bmatrix} 1 & 0\\ -2 & -2\end {bmatrix}\), a saddle. Moreover \((\log x - \log y)' = (3 - x - y) - (2 - x - y) = 1\), so \(x/y = (x_0/y_0)e^t \to \infty \): whatever the initial values, the ratio of \(x\) to \(y\) grows without bound and \(y \to 0\), while \(x \to 3\).