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The phase plane system is $\dot{x}=y$, $\dot{y}=-cy-\sin x$. Its equilibrium points are $(k\pi ,0)$ for every integer $k$. At such a point the Jacobian matrix and characteristic equation areFor even $k$, the equilibrium point is a stable focus for $0<c<2$, with eigenvalues $-c/2\pm i\sqrt{1-c^2/4}$; it is a stable node for $c>2$, with two distinct negative real eigenvalues. The excluded value $c=2$ gives the repeated critical case.

For odd $k$, the eigenvalues are $(-c\pm \sqrt{c^2+4})/2$. One is positive and one negative, so each is a saddle equilibrium for every $c>0$. These classifications follow from the linearization of a dynamical system, since all the relevant eigenvalues have nonzero real part for $c>0$. The even equilibria remain hyperbolic at $c=2$ too, although their linearization then has a repeated eigenvalue; that critical case is excluded from the requested classification.

Use $y$ for height above the horizontal wall. In the lubrication approximation, vertical momentum is hydrostatic and the interfacial stress balance with variable surface tension givesThe horizontal equation is $\mu u_{yy}=B$, with $u(0)=0$ and $\mu u_y(h)={\gamma }_x=-A{C}_x$. ThusIntegrating across the film and adding surface diffusion to surfactant advection gives the dimensional thin-film mass flux and insoluble surfactant flux:These include Marangoni stress, hydrostatic leveling, capillarity and surface diffusion with their signs fixed by the surface traction. In particular the derivative acts on the product of surface tension and curvature, not just on curvature.

Choose $h_{\ast }=(A{C}_0/(\rho g){)}^{1/2}$ and $u_{\ast }=A{C}_0{h}_{\ast }/(\mu L)$. The dimensionless definitions areTogether with the stated concentration and horizontal scales, these giveIn steady flow $Q,J$ are constants by liquid and surfactant conservation.

With capillarity and diffusion neglected, solve these two linear equations for the gradients, in the region $H>0,\Gamma >0$:For positive $Q,J$, the phase plane nullclines are $H=3Q\Gamma /(2J)$ for ${\Gamma }_X=0$ and $H=2Q\Gamma /J$ for $H_X=0$. Above both lines, trajectories go left and upward; between them they go left and downward; below both they go right and downward. There is no positive-quadrant equilibrium. The axes are singular boundaries of this positive-flux reduction, not regular equilibria.

The phase portrait shows these trajectories for one choice of positive fluxes; the two nullcline slopes rescale with $Q/J$.

Write $z=x+y$. When ${\mu }_1={\mu }_2=0$ the system gives $\dot{z}=y-z^3$ and $\dot{y}=-z^3$. Choosingmakes $V$ positive definite and radially unbounded, while its orbital derivative isThe set $\{\dot{V}=0\}$ is $z=0$. A trajectory remaining there must also have $\dot{z}=y=0$, so its largest invariant subset is the origin. The LaSalle invariance principle therefore proves that the origin is globally asymptotically stable. In the phase plane, $\dot{x}=0$ on $y=0$ and $\dot{y}=0$ on $x+y=0$; every nonstationary trajectory crosses the nested level curves of $V$ inward and tends to the origin.

At an equilibrium point, $y=0$ andHence the origin always exists, while $x=\pm \sqrt{{\mu }_1}$ exist for ${\mu }_1>0$. At $(x_{\ast },0)$ the Jacobian matrix isA stationary bifurcation occurs when a real eigenvalue passes through zero, hence onA Hopf bifurcation requires zero trace and positive determinant. At the origin this givesand on either nonzero branch it givesThus the Hopf locus consists of the negative ${\mu }_1$-axis and the ray ${\mu }_2=3{\mu }_1$ in the first quadrant; the stationary locus is the ${\mu }_2$-axis.

Now set ${\mu }_2=-1$, write $\mu ={\mu }_1$, and append $\dot{\mu }=0$. At $(x,y,\mu )=(0,0,0)$ the extended centre manifold for a parameter is tangent to the $(x,\mu )$-plane, so put $y=h(x,\mu )$. Its centre-manifold invariance equation isUnder the ordering $\mu =O(x^2)$, solving through cubic order givesThis is the pitchfork bifurcation normal form: $x=0$ is stable for $\mu <0$ and unstable for $\mu >0$, while the two stable branches $x=\pm \sqrt{\mu }$ emerge for $\mu >0$. The bifurcation is therefore a supercritical pitchfork bifurcation.