Spurious minima obstruct global synchrony

Fix an angular lift $a$ and distinct vertices $x,y$. By [F5], cosine has a local expansion $\cos (a_x-a_y+s)={\sum }_{k\ge 0}{b}_k{s}^k$ for $\mathrm{\mid }s\mathrm{\mid }<r$, with $r>0$. Choose $0<R<r$. Convergence at $s=R$ makes $\mathrm{\mid }{b}_k\mathrm{\mid }{R}^k$ bounded by some $M>0$, so ${\sum }_{k\ge 0}\mathrm{\mid }{b}_k\mathrm{\mid }{q}^k\le M\mathrm{/}(1-q\mathrm{/}R)$ for $0\le q<R$. For increments $h$, expand each finite polynomial $(h_x-h_y{)}^k$ by distributivity. The sum of the absolute values of its expanded terms is at most $(\mathrm{\mid }{h}_x\mathrm{\mid }+\mathrm{\mid }{h}_y\mathrm{\mid }{)}^k$. Thus substituting $s=h_x-h_y$ produces an absolutely convergent multivariable power series whenever $\mathrm{\mid }{h}_x\mathrm{\mid }+\mathrm{\mid }{h}_y\mathrm{\mid }<R$, representing $\cos (a_x+h_x-a_y-h_y)$. A diagonal term has $x=y$ and is constant. There are finitely many ordered vertex pairs, so choose a common positive polyradius, inside the angular chart, on which all these expansions converge absolutely. Adding their finitely many series with the coefficients in [F2] preserves absolute convergence and gives the energy there. This proves real analyticity in the several-variable sense of [F5] at every phase state.

For the unqualified assertion take $V=\{1,2\}$ and $E=\mathrm{\varnothing }$, with common intrinsic frequency zero. The adjacency matrix is zero, so the energy is identically zero and the rotating-frame vector field is identically zero. Every state is therefore a local minimum. The state $z=(0,\pi )$ is spurious, and every trajectory is constant. Its common-rotation orbit is $C=\{(c,c+\pi ):c\in S^1\}$. Thus $B(C)=C$: a constant trajectory has distance tending to zero from the closed orbit exactly when its initial state lies in that orbit.

Since $z$ is not fully synchronized, some two coordinates differ. Equality of those coordinates defines a closed subset of the torus, by continuity of subtraction modulo $2\pi$; hence a sufficiently small closed metric ball about $z$ contains no fully synchronized state. Shrink its radius $\epsilon >0$ so that the closed ball lies inside a Euclidean angular chart, inside a neighborhood on which ${\mathcal{E}}_G\ge {\mathcal{E}}_G(z)$, and inside the gradient-inequality neighborhood in [F3].

Every neighborhood of any point of $C$ contains a state obtained by changing only its second coordinate by a sufficiently small nonzero phase. This changes ${\theta }_2-{\theta }_1$ from $\pi$, so the perturbed state lies outside $C$. Therefore $B(C)$ has empty interior. In the rotation quotient the flow is the identity on the relative-phase circle, and the basin of the class with relative phase $\pi$ is a singleton, again with empty interior. This counterexample already respects the global-rotation equivalence of footnote 2; it does not rest on confusing a point with its orbit.

By analytic Lyapunov stability choose $\delta >0$ so that every initial state in $B_{\delta }(z)$ stays within distance $\epsilon \mathrm{/}2$ of $z$ for all nonnegative time, decreasing $\delta$ if necessary. Every such trajectory converges to a critical point $y$ by compact analytic gradient convergence. Its limit lies in the closed $\epsilon \mathrm{/}2$ ball. The positive-exponent inequality forces ${\mathcal{E}}_G(y)={\mathcal{E}}_G(z)$. Since a neighborhood of $y$ is contained in the $\epsilon$ ball where energy is at least that value, $y$ is itself a local minimum. It is spurious by step 2.1.

Thus the nonempty open set $B_{\delta }(z)$ is contained in the union of the point basins of spurious minima. It has positive normalized phase volume, and every point in it fails to converge to a fully synchronized state. Consequently no null exceptional set can contain all these failures, so the graph is not globally synchronizing. No measurability assertion about an arbitrary uncountable union is needed.

If the common-rotation orbit $C$ of $z$ is isolated among critical points, choose the ball in step 2.1 also inside a neighborhood $W$ for which every critical point in $W$ belongs to $C$. Step 3.1 then gives $y\in C$. Hence $B_{\delta }(z)\subseteq B(C)$. This establishes the asserted open subset of the orbit basin under the extra isolation hypothesis, without inferring that hypothesis from non-strict local minimality.

Finally, the analytic potential $U(x,y)=x^3+y^3$ has gradient $(3x^2,3y^2)$ and zero Hessian at the origin. For every $(x,y)\in {\mathbb{R}}^2$ and every $h>0$, $U(x,y)-U(x-h,y)=h(3(x-h\mathrm{/}2{)}^2+h^2\mathrm{/}4)>0$. Taking arbitrarily small $h$ proves that $U$ has no local minimum anywhere. Its polynomial formula is a finite power series, hence real analytic in the sense of [F5]. Its downward gradient trajectories from $x_0,y_0>0$ are $x(t)=x_0\mathrm{/}(1+3x_{0}t)$ and $y(t)=y_0\mathrm{/}(1+3y_{0}t)$, as direct differentiation verifies. They converge to the origin from an open quadrant. This illustrates the stated distinction for analytic potentials; it establishes no graph example or failure of a graph theorem. Together with steps 4.1, 4.2 and 2.2, this proves every mathematical assertion above.