Periodic mean-field games: necessary resonance and an explicit family with global periodic branches
Problem ID: C73-38.
Research attempt; independent review pending.Declared family and target
The catalogue asks for a classification of genuine periodic MFG equilibria, their bifurcation points, and continuation of their branches. We study the spatially homogeneous coupling and quadratic Hamiltonian subfamily on the unit-volume torus ${\mathbb{T}}^d={\mathbb{R}}^d/\mathbb Z^d$:
\begin{equation}\label{wsref-0-6d-66-67}
\begin{cases}
\lambda-v_t-\nu\Delta v+\tfrac12|Dv|^2=f(m),\\
m_t-\nu\Delta m-{\operatorname{div}}(mDv)=0,\\
m>0,\quad\int_{{\mathbb{T}}^d}m(t,x)dx=1,
\end{cases}
\end{equation} where $\nu>0$, $f\in C^1((0,\infty))$, and $v,m$ are $C^{1,2}$ and $P$-periodic for some $P>0$. Fix $\int_0^P\int v=0$. The stationary branch is $(v,m,\lambda)=(0,1,f(1))$. Genuine periodicity means that $(Dv,m)$ is nonconstant in time.
We retain the nonlinear, period-independent exclusion region and complete linear mode calculation from Version 1. Version 2 proves a stronger necessary resonance theorem for arbitrary finite-period bifurcation sequences, with no differentiable-branch or nonzero-tangent assumption. All statements are proved below. Version 3 adds an explicit inverse-square coupling family, determines its complete periodic-existence parameter set, and constructs nonlinear branches with fully described endpoint behavior. The general necessary conditions do not imply nonlinear existence at every resonance or global continuation for arbitrary couplings. The original periodic existence examples are already known from [CN, Theorem 6.1 and Remark 6.4]; the published agenda is discussed in [CD, Section 3.3]. Version 1 records a check of those sources and their scope on 9 October 2026; the present version retains those references and does not claim an additional literature search. The computations here are not claimed as new bifurcation theory.
A nonlinear identity and a density-gradient estimate
Write $Q_P=({\mathbb{R}}/P\mathbb Z)\times{\mathbb{T}}^d$, with integration over one period, and let $q_1=4\pi^2$ be the first positive eigenvalue of $-\Delta$.
Lemma: Every solution of
(mfg) satisfies
\begin{align}
0&=\int_{Q_P}(f(m)-f(1))(m-1)
+\tfrac12\int_{Q_P}(m+1)|Dv|^2,\label{wsref-0-6d-6f-6e-6f-69-64}\\
\nu\int_{Q_P}\frac{|Dm|^2}{m}
&=-\int_{Q_P}Dm\cdot Dv.\label{wsref-0-65-6e-74-72-6f-70-79-69-64}
\end{align} In particular,
\begin{equation}\label{wsref-0-63-61-75-63-68-79}
\int_{Q_P}m|Dv|^2\geq\nu^2\int_{Q_P}\frac{|Dm|^2}{m}.
\end{equation}
Proof: Subtract the first stationary equation from the first equation of
(mfg), multiply by $\rho=m-1$, and integrate over $Q_P$. The constant multiplier $\lambda-f(1)$ disappears because $\int\rho=0$ at every time. Periodicity and the Fokker--Planck equation give \[
\int_{Q_P}(-v_t)\rho
=\int_{Q_P}v\rho_t
=-\nu\int_{Q_P}Dv\cdot D\rho-\int_{Q_P}m|Dv|^2.
\] The diffusion term $-\nu\Delta v$ cancels the first term on the right. Combining the remaining term with $|Dv|^2\rho/2$ proves
(monoid).
For (entropyid), multiply the Fokker--Planck equation by $1+\log m$ and integrate. Positivity and compactness make this a valid smooth multiplier. Its time derivative integrates to zero over a period; spatial integration by parts gives exactly (entropyid). Let $I=\int|Dm|^2/m$ and $J=\int m|Dv|^2$. Weighted Cauchy--Schwarz gives $\nu I\leq\sqrt{IJ}$. If $I=0$, (cauchy) is automatic; otherwise division by $\sqrt I$ and squaring prove it. ∎
Theorem (A nonlinear exclusion region): Let $0<\varepsilon<1$, $L\geq0$, and suppose \[
(f(s)-f(1))(s-1)\geq-L(s-1)^2
\quad(1-\varepsilon\leq s\leq1+\varepsilon).
\] If
\begin{equation}\label{wsref-0-74-68-72-65-73-68-6f-6c-64}
L<\nu^2q_1\frac{2+\varepsilon}{2(1+\varepsilon)^2},
\end{equation} then, for every $P>0$, any solution of
(mfg) satisfying $\|m-1\|_\infty\leq\varepsilon$ is the stationary solution $(v,m,\lambda)=(0,1,f(1))$. Consequently, if $f'(1)>-\nu^2q_1$, there is a neighborhood of $m=1$ in the uniform norm containing no nonstationary periodic solutions, uniformly over all periods.
Proof: The ratio $(m+1)/(2m)$ is at least $c_\varepsilon=(2+\varepsilon)/(2(1+\varepsilon))$. Lemma
Lemma, the bound $m\leq1+\varepsilon$, and spatial Poincaré inequality for the mean-zero function $m-1$ give \begin{align*}
\tfrac12\int(m+1)|Dv|^2
&\geq c_\varepsilon\int m|Dv|^2
\geq c_\varepsilon\nu^2\int\frac{|Dm|^2}{m}\\
&\geq\frac{c_\varepsilon\nu^2}{1+\varepsilon}\int|Dm|^2
\geq\nu^2q_1\frac{2+\varepsilon}{2(1+\varepsilon)^2}
\int|m-1|^2.
\end{align*} All integrals here are over $Q_P$. Substitution in
(monoid) yields \[
0\geq\left(\nu^2q_1\frac{2+\varepsilon}{2(1+\varepsilon)^2}-L\right)
\int|m-1|^2.
\] The coefficient is positive, so $m=1$. Identity
(monoid) then forces $Dv=0$. The HJB equation says $v_t=\lambda-f(1)$; periodicity makes this constant derivative zero, and normalization gives $v=0$.
For the final assertion choose $\max\{0,-f'(1)\}<L<\nu^2q_1$. Continuity of $f'$ ensures the secant inequality in a sufficiently small density neighborhood. The factor on the right of (threshold) tends to one as $\varepsilon\downarrow0$, so shrinking the neighborhood also makes (threshold) hold. No step depends on $P$. ∎
Corollary (The homogeneous monotone region): If $f$ is nondecreasing on $(0,\infty)$, then
(mfg) has no genuinely periodic solution of any amplitude or period; its only normalized periodic solution is $(0,1,f(1))$.
Proof: Both integrands in
(monoid) are nonnegative, so $Dv=0$. The Fokker--Planck equation reduces to $m_t=\nu\Delta m$. Multiplication by $m$ and integration over one period gives $\int|Dm|^2=0$. Unit mass then implies $m=1$, and the HJB equation and normalization determine $v$ and $\lambda$ as in the preceding proof. ∎
All linear periodic modes and what they imply
Proposition: Let $a=f'(1)$ and linearize
(mfg) about $(0,1,f(1))$. For a real spatial Laplace eigenfunction $\phi$ with $-\Delta\phi=q\phi$, $q>0$, write $w(t,x)=U(t)\phi(x)$ and $\rho(t,x)=R(t)\phi(x)$. The mode equations are
\begin{equation}\label{wsref-0-6d-61-74-72-69-78}
\frac{d}{dt}\begin{pmatrix}U\\R\end{pmatrix}
=A_q\begin{pmatrix}U\\R\end{pmatrix},\qquad
A_q=\begin{pmatrix}\nu q&-a\\-q&-\nu q\end{pmatrix},
\qquad A_q^2=q(\nu^2q+a)I.
\end{equation} A nonconstant $P$-periodic mode exists if and only if
\begin{equation}\label{wsref-0-72-65-73-6f-6e-61-6e-63-65}
a=-\nu^2q-\frac{(2\pi n/P)^2}{q}
\quad\hbox{for some integer }n\geq1.
\end{equation} If $a=-\nu^2q$, the periodic modes at that spatial eigenvalue are stationary and equal to the kernel of $A_q$. If $a> -\nu^2q$, there is no nonzero periodic mode at that eigenvalue. The spatially constant mode vanishes after mass and value normalization, including the variation of $\lambda$.
Proof: The linear equations at positive spatial frequencies are $-w_t-\nu\Delta w=a\rho$ and $\rho_t-\nu\Delta\rho-\Delta w=0$. Substitution yields
(matrix); direct multiplication gives its square. If $q(\nu^2q+a)>0$, the eigenvalues are nonzero real numbers of opposite sign, so neither exponential over a positive period equals one. If that product is zero, $A_q\ne0$ and $A_q^2=0$, so $e^{PA_q}=I+PA_q$. Periodic initial vectors are exactly $\ker A_q$ and give stationary modes. If the product is negative, put $\omega_q=\sqrt{-q(\nu^2q+a)}>0$. Then \[
e^{tA_q}=I\cos(\omega_qt)+\frac{A_q}{\omega_q}\sin(\omega_qt).
\] Its eigenvalues over one period are $e^{\pm i\omega_qP}$. A nonzero real fixed vector exists precisely when $\omega_qP=2\pi n$ for a positive integer $n$, proving
(resonance). At resonance every initial vector in this two-dimensional mode is periodic. The spatial zero mode has $\rho_0=0$ by mass, $-w_0'+\delta\lambda=0$ by the HJB equation, and hence $\delta\lambda=0$ by periodicity; the normalization makes $w_0=0$. Expansion in a complete real Fourier eigenbasis gives the stated list of modes without assuming simple eigenvalues. ∎
Proposition (A necessary condition for a differentiable periodic branch): Let $f_\eta$ be smooth in $(\eta,m)$, and suppose a branch of periodic solutions that is $C^1$ as a curve in the rescaled $C^{1,2}$ function space, with parameters $(\eta_\epsilon,P_\epsilon)$, passes through $(0,1,f_{\eta_0}(1))$ at $\epsilon=0$. Take derivatives after scaling time to a common unit period, subtract the stationary branch in the multiplier, and assume the resulting first variation $(w,\rho)$ is nonconstant in time. Then some positive spatial eigenvalue $q$ and integer $n\geq1$ satisfy
(resonance) with $a=f'_{\eta_0}(1)$ and $P=P_0$.
Proof: Differentiating the rescaled equations is legitimate by the assumed $C^1$ convergence in $C^{1,2}$. Terms involving $P'_0$ multiply time derivatives of the stationary state and vanish. The derivative of $f_\eta(1)$ cancels against the subtracted stationary multiplier. The remaining equations are exactly the linear equations of Proposition
Proposition with period $P_0$ after returning to physical time. A time-dependent first variation has a time-dependent Fourier mode, to which that proposition applies. ∎
For example, $q=q_1=4\pi^2$ gives the basic oscillatory frequency $\omega_1=2\pi\sqrt{-4\pi^2\nu^2-a}$ when $a<-4\pi^2\nu^2$. At $\nu=1$, its fundamental period $1/\sqrt{-4\pi^2-a}$ matches the normalization in [CN, Theorem 6.1]. That theorem supplies nonlinear existence in its specified regime; a calculation of imaginary eigenvalues by itself does not.
A resonance test without a differentiable branch
The last proposition of Version 1 required a differentiable branch with nonstationary first variation. A branch may have a zero first variation, or a stationary tangent, and a bifurcating sequence need not lie on any chosen differentiable curve. We now remove these restrictions for the homogeneous quadratic family at a finite limiting period. The proof uses time differences of the actual nonlinear solution, so stationary linear degeneracies do not enter its conclusion.
Fix $\eta_0$ and $P_0>0$. Let $f_\eta(s)$ be $C^2$ jointly on a neighborhood of $(\eta_0,1)$ and write $a_\eta=\partial_s f_\eta(1)$. Spatial Fourier frequencies are $\xi=2\pi k$, $k\in\mathbb Z^d$; write $q=|\xi|^2$ and $\omega_n=2\pi n/P$. All $L^2$ norms below use normalized measure $P^{-1}dt\,dx$ on $Q_P$. The zero temporal mean condition for a function $r$ means $\int_0^P r(t,x)dt=0$ for every $x$, or its $L^2$ interpretation. The corresponding condition for a spatial gradient is imposed on that gradient, not on an arbitrary spatially constant part of its primitive.
Lemma (A uniform multiplier away from oscillatory resonance): Suppose \begin{equation}\label{wsref-0-6e-6f-6e-72-65-73-6f-6e-61-6e-63-65}
a_{\eta_0}\ne-\nu^2q-\frac{(2\pi n/P_0)^2}{q}
\quad\text{for every }q=4\pi^2|k|^2>0
\text{ and every integer }n\geq1.
\end{equation} There are neighborhoods $J$ of $\eta_0$ and $I$ of $P_0$ and a constant $K<\infty$ such that, for $\eta\in J$, $P\in I$, every periodic solution of \begin{equation}\label{wsref-0-66-6f-72-63-65-64-6c-69-6e-65-61-72}
-w_t-\nu\Delta w-a_\eta r=h+\ell(t),\qquad
r_t-\nu\Delta r-\Delta w={\operatorname{div}} J_0
\end{equation} with $\int_{{\mathbb{T}}^d}r(t)=0$ and zero temporal means for $Dw,r$ obeys \begin{equation}\label{wsref-0-6d-75-6c-74-62-6f-75-6e-64}
\left(\|Dw\|_2^2+\|r\|_2^2\right)^{1/2}
\leq K\left(\|h\|_2^2+\|J_0\|_2^2\right)^{1/2}.
\end{equation} Here $J_0$ is a vector field and $\ell(t)$ is any spatially constant function. The notation $J_0$ in this lemma is unrelated to the parameter neighborhood $J$. Square-integrable right-hand sides suffice when the equations hold distributionally and the left-hand quantities are defined. No assumption excludes stationary degeneracies $a_{\eta_0}=-\nu^2q$.
Proof: The zero spatial frequency contributes nothing to $Dw$ or $r$. The zero temporal frequencies of these quantities also vanish by assumption. At every remaining frequency, write $a=a_\eta$, $\omega=\omega_n$ and let hats denote Fourier coefficients. The system is \[
\begin{pmatrix}\nu q-i\omega&-a\\q&\nu q+i\omega\end{pmatrix}
\binom{\widehat w}{\widehat r}
=\binom{\widehat h}{i\xi\cdot\widehat J_0}.
\] Its determinant and inverse formulas are \begin{align*}
D_{q,n}(a,P)&=\nu^2q^2+\omega_n^2+a q,\\
\widehat w&=\frac{(\nu q+i\omega)\widehat h
+a\,i\xi\cdot\widehat J_0}{D_{q,n}},\\
\widehat r&=\frac{-q\widehat h
+(\nu q-i\omega)i\xi\cdot\widehat J_0}{D_{q,n}}.
\end{align*} Condition
(nonresonance) makes these determinants nonzero at $(a_{\eta_0},P_0)$ for all $q>0$ and $n\ne0$. We justify uniform boundedness over the entire frequency set, including perturbations of the period. First restrict to compact initial neighborhoods where $|a|\leq A$ and $0<P_-\leq P\leq P_+<\infty$. If $q\geq2A/\nu^2$, then \[
D_{q,n}\geq\tfrac12\nu^2q^2+\omega_n^2.
\] There are only finitely many spatial frequencies below that threshold. For any such $q$, when $|n|$ is sufficiently large uniformly in $P\in[P_-,P_+]$, one has $\omega_n^2\geq2Aq$, and hence $D_{q,n}\geq\nu^2q^2+\tfrac12\omega_n^2$. Thus outside a finite set of pairs of frequencies, \[
D_{q,n}\geq c_0(q^2+\omega_n^2)>0
\] for a constant $c_0$ depending only on $\nu$ and the initial neighborhoods. On the finite remaining set, nonresonance and continuity allow the neighborhoods to be reduced so that all $|D_{q,n}|$ are bounded away from zero.
Using $|\xi\cdot\widehat J_0|\leq\sqrt q|\widehat J_0|$, the four scalar bounds for the inverse mapping into $(\sqrt q|\widehat w|,|\widehat r|)$ have coefficients \[
\frac{\sqrt q\sqrt{\nu^2q^2+\omega^2}}{|D_{q,n}|},\quad
\frac{|a|q}{|D_{q,n}|},\quad
\frac{q}{|D_{q,n}|},\quad
\frac{\sqrt q\sqrt{\nu^2q^2+\omega^2}}{|D_{q,n}|}.
\] Each is uniformly bounded on the finite set. On its complement the determinant estimate proves the same fact: for example the first is at most a fixed constant times $\sqrt q/(q^2+\omega^2)^{1/2}\leq C/\sqrt q$, and $q\geq4\pi^2$. The other two coefficients are bounded by constants times $q/(q^2+\omega^2)\leq1/q$. Choose $K$ larger than the resulting uniform matrix norm. Apply the coefficient bound, sum its squares over all frequencies, and use Parseval's identity. This proves (multbound). Any frequencies of $h,J_0$ outside the relevant set only increase its right side. The function $\ell(t)$ has zero coefficient at every positive spatial frequency and therefore has no effect on this argument. ∎
Theorem (No small genuine periodic solutions off resonance): Under
(nonresonance), there are neighborhoods $J,I$ as in Lemma
Lemma (A uniform multiplier away from oscillatory resonance) and $\varepsilon_0>0$ such that, for every $\eta\in J$ and $P\in I$, every $C^{1,2}$ periodic solution of
(mfg) with $f=f_\eta$ and \[
\|m-1\|_\infty+\|Dv\|_\infty\leq\varepsilon_0
\] is stationary in time. It may be a nonuniform stationary solution; the theorem does not exclude a stationary spatial bifurcation. Consequently every genuine periodic bifurcation point $(\eta_0,P_0)$ from the homogeneous branch, in the catalogue's finite-period sense, satisfies
(resonance) for some $q>0$ and $n\geq1$. This conclusion applies to arbitrary converging sequences and needs neither a differentiable branch nor a condition on its first variation.
Proof: Put $r=m-1$, $g=Dv$, and $a=a_\eta$. Define the nonlinear remainders \[
H_\eta(r,g)=f_\eta(1+r)-f_\eta(1)-a r-\tfrac12|g|^2,
\qquad J(r,g)=r g.
\] The equations can be written \[
-v_t-\nu\Delta v-a r=H_\eta(r,g)+f_\eta(1)-\lambda,
\qquad r_t-\nu\Delta r-\Delta v={\operatorname{div}} J(r,g).
\] For a temporal shift $s$ let $\delta_s r(t)=r(t+s)-r(t)$, with times taken modulo $P$, and similarly define $\delta_s v,g,H,J$. The difference fields solve
(forcedlinear) with $w=\delta_s v$, density $\delta_s r$, and forcings $\delta_s H$, $\delta_s J$; the scalar constants cancel. They have zero temporal mean because integration over a full period is invariant under the shift. Also $\int\delta_s r(t)=0$ because both densities have mass one. Thus Lemma
Lemma (A uniform multiplier away from oscillatory resonance) applies.
By joint continuity of $\partial_s f_\eta$, after shrinking $J$ if necessary there is a modulus $\omega_f(\varepsilon)\to0$ such that \[
|\partial_s f_\eta(1+z)-a_\eta|
\leq\omega_f(\varepsilon)\quad
(\eta\in J,\ |z|\leq\varepsilon).
\] If $\|r\|_\infty,\|g\|_\infty\leq\varepsilon$, the mean-value formula and difference of squares give pointwise \begin{align*}
|\delta_s H|&\leq\omega_f(\varepsilon)|\delta_s r|
+\varepsilon|\delta_s g|,\\
|\delta_s J|&\leq\varepsilon|\delta_s r|
+\varepsilon|\delta_s g|.
\end{align*} In the second line use $r(t+s)g(t+s)-r(t)g(t)
=r(t+s)\delta_s g+g(t)\delta_s r$. Minkowski's inequality and $\|z\|_2$ for each component bounded by the combined two-component norm show \[
\left(\|\delta_s H\|_2^2+\|\delta_s J\|_2^2\right)^{1/2}
\leq[\omega_f(\varepsilon)+3\varepsilon]
\left(\|\delta_s g\|_2^2+\|\delta_s r\|_2^2\right)^{1/2}.
\] Choose $0<\varepsilon_0<1$ so small that $K[\omega_f(\varepsilon_0)+3\varepsilon_0]<1$. Combining this inequality with (multbound) forces $\delta_s g=\delta_s r=0$ in $L^2$ for every $s$. The fields are continuous, so their differences vanish pointwise. Hence $Dv$ and $m$ are independent of time. It follows that $v(t,x)=V(x)+c(t)$. The HJB equation makes $c'(t)$ a constant, since its other terms are independent of time. Periodicity makes that constant zero, so $v$ itself is stationary.
For the final assertion, assume a genuine bifurcation sequence converges in the stated rescaled $C^{1,2}$ topology with $P_n\to P_0>0$. Its parameters eventually lie in $J\times I$, and its density and gradient eventually satisfy the preceding smallness condition. Under nonresonance it would therefore consist of stationary solutions, a contradiction. This proves the claimed necessary resonance. ∎
The finite-period assumption matters: this estimate is uniform only for periods in a neighborhood of $P_0>0$. It makes no claim about branches with $P_n\to\infty$ or $P_n\to0$. A nontrivial spatial stationary kernel cannot obstruct the proof because taking temporal differences removes precisely the temporal zero modes.
An explicit family: existence, bifurcation and continuation
The preceding sections retain the necessary conditions from Version 2. They do not supply a nonlinear branch. We now supply one in closed form, and determine the entire parameter set admitting genuine periodic solutions for the declared family
\begin{equation}\label{wsref-0-69-6e-76-65-72-73-65-66-61-6d-69-6c-79}
f_\eta(m)=\frac{\eta}{2m^2},\qquad \eta\in{\mathbb{R}},\quad m>0,
\qquad H(p)=\tfrac12p^2,
\end{equation} on ${\mathbb{T}}={\mathbb{R}}/{\mathbb{Z}}$. This is a smooth family on exactly the positive-density domain in the catalogue. Derivatives need not stay bounded as $m\downarrow0$; we make no such assertion. For each fixed positive solution below, its density has a strictly positive minimum. These are explicit examples and an explicit branch analysis, not a novelty claim or a classification of all periodic solutions for positive $\eta$.
Lemma (A rotating-profile identity): Fix an integer $j\geq1$, put $k=2\pi j$, and let $0<r<1$, $c\in{\mathbb{R}}$. Write $q_r=\sqrt{1-r^2}$ and define periodic functions \[
M(z)=1+r\cos(kz),\qquad
P(z)=-\nu\frac{M'(z)}{M(z)}-c+\frac{c q_r}{M(z)}.
\] Then $M>0$, $\int_{\mathbb{T}} M=1$, $\int_{\mathbb{T}} P=0$, and there is a unique one-periodic smooth $U$ with $U'=P$ and $\int_{\mathbb{T}} U=0$. In fact $M,P,U$ all have spatial period $1/j$. They satisfy \begin{align}
\nu M''+(MP)'&=-cM',\label{wsref-0-72-6f-74-61-74-69-6e-67-46-50}\\
cP-\nu P'+\tfrac12P^2
&=-\frac{\nu^2k^2+c^2}{2}
+\frac{(1-r^2)(\nu^2k^2+c^2)}{2M^2}.\label{wsref-0-72-6f-74-61-74-69-6e-67-48-4a-42}
\end{align}
Proof: Positivity and unit mass follow directly from $0<r<1$. The logarithmic derivative $M'/M$ has zero integral over a spatial period. The change of variable $\theta=kz$ reduces the remaining integral to \[
\int_{\mathbb{T}}\frac{dz}{1+r\cos(kz)}=\frac1{\sqrt{1-r^2}}.
\] For completeness, over one $2\pi$ period put $y=\tan(\theta/2)$: \[
\frac1{2\pi}\int_{-\pi}^{\pi}\frac{d\theta}{1+r\cos\theta}
=\frac1\pi\int_{-\infty}^{\infty}
\frac{dy}{(1+r)+(1-r)y^2}
=\frac1{\sqrt{1-r^2}}.
\] It follows that $\int P=-c+cq_r/q_r=0$. The same zero integral holds over each interval of length $1/j$, since all the summands have that period. Thus the normalized periodic primitive exists, is unique, and has spatial period $1/j$.
Multiplication by $M$ gives $MP=-\nu M'-cM+cq_r$; differentiation proves (rotatingFP). To check the HJB identity without suppressing its cancellation, put $C=cq_r$ and $D=-\nu M'/M$. Then $P=D-c+C/M$, so \[
cP+\tfrac12P^2=\tfrac12(D+C/M)^2-\tfrac12c^2,
\qquad
-\nu P'=\nu^2\frac{M''}{M}
-\nu^2\frac{(M')^2}{M^2}
+\nu C\frac{M'}{M^2}.
\] The mixed term $DC/M=-\nu C M'/M^2$ cancels the last term. The resulting sum is \[
\nu^2\frac{M''}{M}-\frac{\nu^2(M')^2}{2M^2}
+\frac{C^2}{2M^2}-\frac{c^2}{2}.
\] Use $M''=-k^2(M-1)$, $(M')^2=k^2\{r^2-(M-1)^2\}$ and $C^2=c^2(1-r^2)$. Putting terms over $2M^2$ gives (rotatingHJB). ∎
Theorem (Exact classification of the existence parameter set): For the smooth family
(inversefamily), the set of parameters admitting at least one genuine periodic solution is exactly \[
\mathcal P_f=(0,\infty).
\] More precisely, for every integer $j\geq1$ and every $\eta>0$, choose $0<r<1$ such that
\begin{equation}\label{wsref-0-61-6d-70-6c-69-74-75-64-65-63-6f-6e-64-69-74-69-6f-6e}
\eta>(1-r^2)\nu^2(2\pi j)^2,
\end{equation} put $k=2\pi j$, and choose either sign in
\begin{equation}\label{wsref-0-62-72-61-6e-63-68-70-61-72-61-6d-65-74-65-72-73}
c=\pm\sqrt{\frac{\eta}{1-r^2}-\nu^2 k^2},\qquad
\lambda=\frac{\eta}{2(1-r^2)}.
\end{equation} With the profiles of Lemma
Lemma (A rotating-profile identity), \[
m(t,x)=M(x-ct),\qquad v(t,x)=U(x-ct)
\] solve
(mfg) with coupling
(inversefamily). They have genuine temporal fundamental period
\begin{equation}\label{wsref-0-66-75-6e-64-61-6d-65-6e-74-61-6c-70-65-72-69-6f-64}
P_{\mathrm{time}}=\frac1{j|c|}.
\end{equation} They obey the required mass and value normalization.
Proof: Condition
(amplitudecondition) makes $c$ real and nonzero. Lemma
Lemma (A rotating-profile identity) verifies the Fokker--Planck equation after substitution of $m_t=-cM'$. The HJB left side is $\lambda+cP-\nu P'+P^2/2$. Equations
(rotatingHJB) and
(branchparameters) make its constant term vanish and its remaining coefficient equal to $\eta/(2M^2)$. Thus it is exactly $f_\eta(M)$, not an approximation. Each profile is smooth with positive minimum density $1-r$, mass one, and spatially centered $U$, so time integration also gives the stipulated zero space--time mean of $v$.
Since $r>0$, the density $1+r\cos(kx-kct)$ is not constant in time. Its least positive temporal period is $2\pi/(k|c|)=1/(j|c|)$. The value has the same period because $U$ has spatial period $1/j$. Thus the pair's fundamental period is the displayed value. For every $\eta>0$, one can choose $r<1$ sufficiently close to one so that (amplitudecondition) holds, proving $(0,\infty)\subseteq\mathcal P_f$. If $\eta\leq0$, then $f_\eta'(m)=-\eta/m^3\geq0$ for $m>0$. The homogeneous monotone corollary proved above excludes every genuine periodic solution, of any amplitude or period. This proves the reverse inclusion and the asserted exact parameter set. ∎
Theorem (Finite-period onset and the endpoints of the explicit branches): Fix $\eta>0$ and an integer $j\geq1$, and write $a_j=\nu^2(2\pi j)^2$. Both signs in
(branchparameters) give smooth branches on \[
r_*(\eta)<r<1,\qquad
r_*(\eta)=\sqrt{\max\{0,1-\eta/a_j\}}.
\] Their endpoint behavior is as follows.
- If $\eta>a_j$, the lower endpoint is a genuine finite-period bifurcation from $(v,m,\lambda)=(0,1,\eta/2)$. The limiting fundamental period is $P_0=1/(j\sqrt{\eta-a_j})$, and convergence holds in the catalogue's rescaled $C^{1,2}$ topology.
- If $0<\eta<a_j$, then as $r\downarrow r_*(\eta)>0$, $c\to0$ and $P_{\mathrm{time}}\to\infty$. The spatial profiles converge smoothly to a nonuniform stationary solution with $M_*=1+r_*\cos(kx)$, $U_*'=-\nu M_*'/M_*$ and $\lambda_*=a_j/2$.
- If $\eta=a_j$, the lower endpoint is the homogeneous stationary profile and $P_{\mathrm{time}}\to\infty$.
- At the upper endpoint $r\uparrow1$, for every $\eta>0$, $\min m=1-r\to0$, $|c|\to\infty$, $\lambda\to\infty$ and $P_{\mathrm{time}}\to0$. Also $\|Dv\|_\infty\to\infty$.
These are continuation statements for this explicitly parameterized sinusoidal rotating family. They do not assert that all other branches have these endpoints, or that the branch is dynamically stable.
Proof: The open interval is exactly the strict positivity condition for the radicand in
(branchparameters); all its formulas are smooth there. For $\eta>a_j$, $c(r)$ extends smoothly to $r=0$, with nonzero limit $c_0=\pm\sqrt{\eta-a_j}$. The functions $M$ and $P$ tend smoothly to $1$ and $0$ respectively, and their normalized primitive tends to zero. Under the time rescaling $t=P_{\mathrm{time}}\tau$, \[
k(x-ct)=kx-2\pi\operatorname{sgn}(c)\tau.
\] This fixed rescaled phase and smooth dependence of all profile coefficients show convergence in $C^{1,2}$, indeed in every finite smooth norm, to the homogeneous profile. The multiplier tends to $\eta/2$. Every member with $r>0$ is genuine, so this is precisely a bifurcation sequence in the definition, with fixed parameter $\eta$. Its resonance agrees with Version 2: for $q=k^2$ and $\omega=2\pi/P_0=k|c_0|$, \[
f_\eta'(1)=-\eta=-\nu^2q-\omega^2/q.
\] The result is a nonlinear realization of this resonance for the declared family, not merely a linear frequency calculation.
When $0<\eta<a_j$, $r_*>0$, $1-r_*^2=\eta/a_j$ and the radicand tends to zero as $r\downarrow r_*$. The period therefore diverges. Since $r_*<1$, all spatial denominators stay bounded away from zero; the profiles converge in every finite spatial smooth norm to the stated ones. The identities of Lemma Lemma (A rotating-profile identity) still apply at $c=0$ and verify both stationary equations, with $\lambda_*=a_j/2$. At equality $\eta=a_j$, the same computation gives $r_*=0$, $c\to0$ and the homogeneous limit, again with divergent period. These infinite-period limits are different from the finite-period bifurcations covered by the preceding item and by Version 2.
Finally (branchparameters) proves all upper-end assertions except the gradient divergence directly. At a minimum of $M$, $M=1-r$ and $M'=0$, so \[
|P|=|c|\left(\sqrt{\frac{1+r}{1-r}}-1\right)\longrightarrow\infty.
\] Thus the gradient also leaves every bounded class. These explicit formulas continue the branch throughout its stated interval without an appeal to an unproved global bifurcation theorem. ∎
Remaining gap
The retained exclusion and resonance theorems constrain arbitrary small periodic solutions in the homogeneous quadratic family. Version 3 goes beyond necessary conditions: for $f_\eta(m)=\eta/(2m^2)$ it proves the exact existence set $\mathcal P_f=(0,\infty)$ and supplies explicit branches with finite-period bifurcation, infinite-period stationary limits, and loss of a positive density bound at their other endpoint. This does not classify every periodic solution even within that family, establish orbital stability, or give a general continuation theory for spatially dependent couplings and general Hamiltonians. Those broader classification questions in the catalogue remain unresolved.
Completion Estimate
55%
This is a heuristic assessment of the full catalogue target. The version now proves an exact parameter classification and actual nonlinear continuation for one declared smooth coupling family, in addition to the retained necessary-condition theory. Classification of all periodic solutions, general couplings and Hamiltonians, and stability remain open in this attempt.
References