2.1 A 1D energy balance model

The fundamental mechanism of 1D EBMs is that the temperature $u=u(t,x)$, averaged in the zonal direction, evolves in time due to (i) the diffusion of energy between adjacent regions, (ii) the energy absorbed by the planet and (iii) the energy emitted by the planet. The 1D EBM we consider in this paper is a Seller-type EBM where the absorbed radiation depends on an additive parameter (Bastiaansen et al., 2022). We only add a change in the diffusion term in order to get a non-degenerate parabolic PDE. Given an initial condition $\stackrel{\mathrm{̃}}{u}$, the non-linear, parabolic, reaction–diffusion PDE governing the model is given by

where R_a and R_e represent the radiation absorbed and emitted by the planet per unit area, respectively. C_T is the heat capacity, and the differential term parameterizes the meridional heat transport. The boundary conditions impose no flux at the poles. We now provide further details regarding the parameterization of these terms. The values of the constants of the model can be found in Table 1.

Firstly, the absorbed radiation is assumed to have the following form:

where Q₀ is the solar radiation per unit area, and α is the albedo. The solar radiation is assumed to be

where ${\widehat{Q}}_{\mathrm{0}}$ is the mean solar radiation, and c_i represents constants. The albedo, which is the proportion of the incident light or radiation that is reflected by a surface, is parameterized by a smooth monotonically decreasing function with a peak derivative in a reference temperature u_ref close to the melting point of ice. Specifically,

where K>0 is a rate parameter and α₁>α₂ are respectively the ice albedo and the water albedo.

Second, the emitted radiation is modelled using the Stefan–Boltzmann law, in other words assuming that the Earth radiates as a black body. Under this assumption, the energy radiated is proportional to the fourth power of its temperature and it is given by

where ε₀ and σ₀ are respectively the emissivity and Boltzmann's constant. The additive parameter q describes, in a simplified way, the radiative forcing by CO₂, i.e. the effect of atmospheric CO₂ on the energy budget (IPCC, 2014). It is worth explaining (i) the additive structure of q and (ii) its independence on the spatial variable x. About the first point, denote by C the global CO₂ concentration in part per million (ppm) and assume, just for explanation purposes, a dependence of the outgoing radiation both on u and C. To avoid confusion, we denote this by ${\widehat{R}}_{\mathrm{e}}={\widehat{R}}_{\mathrm{e}}(u,C)$, the outgoing radiation depending on temperature and CO₂ concentration. If we linearize ${\widehat{R}}_{\mathrm{e}}$ with respect to temperature, we get

where $\widehat{u}$ is a reference temperature. In Myhre et al. (1998), three radiative transfer models are used in order to get that the dependence of the outgoing radiation with respect to changes in CO₂ is given by

where C₀ is a reference CO₂ concentration, and $A_{\mathrm{1}},A_{\mathrm{2}}>\mathrm{0}$ are explicit constants. In conclusion,

and thus the radiative forcing of CO₂ has an additive structure. About point (ii), we adopt the well-mixed hypothesis for CO₂. In other words, we assume that atmospheric CO₂ is globally homogeneous, thereby inducing a radiative forcing q independent of latitude (IPCC, 2001). This assumption overlooks the spatial pattern of CO₂ concentration, which affects many aspects of the climate system, such as the poleward heat transport (Huang et al., 2017). It was the state-of-the-art assumption 2 decades ago, although today it is common to keep in consideration the spatial distribution of radiative forcing (IPCC, 2001; Byrne and Goldblatt, 2014; Zhang et al., 2019).

The third component of the model is the term ∂_x(κ(x)u_x). It parameterizes the meridional heat transport, that is, the phenomenon resulting from the poleward transportation of heat by the Earth–atmosphere system due to the surplus of net radiation heating in the tropics and the deficit in the poleward regions. Usually, the diffusion function κ(x) is assumed null at the poles, i.e. with a form such as $\mathit{\kappa }\left(x\right)=D(\mathrm{1}-x^{\mathrm{2}})$, where D is a diffusion constant. This choice is based on the paradigm of mimicking the conduction of heat on a sphere; see North and Kim (2017) for a derivation. On the other hand, it leads to mathematical difficulties in the treatment of the singular PDE arising, in particular in the study of the corresponding variational problem, from which all our results follow. To avoid these difficulties, which at the moment remain an open problem to solve, we add as a simplifying assumption that κ is non-degenerate and given by

We choose δ=0.003, but its value is not important for the results of this work, and different choices can be made.

For the parabolic problem (Eq. 8), the global existence and uniqueness of the solution can be demonstrated, given a regular initial condition (Temam, 1997). Furthermore, if the initial condition is non-negative, the solution remains non-negative for any time t>0. This can be shown proving that $[\mathrm{0},+\mathrm{\infty })$ is an invariant region for Eq. (8), exploiting the fact that there exist $C_{\mathrm{1}},C_{\mathrm{2}}>\mathrm{0}$ such that $R(x,u;q)>C_{\mathrm{1}}>\mathrm{0}$ for all $x\in [-\mathrm{1},\mathrm{1}]$ and $u\in [\mathrm{0},C_{\mathrm{2}}]$ (Smoller, 2012).

We recall the formulation of stochastic EBMs using the theory of SPDEs (Da Prato and Zabczyk, 2014). Denote by Δ the Laplace operator with Neumann boundary conditions. Given an initial condition $\stackrel{\mathrm{̃}}{u}\in H$, we consider the following SPDE:

where ε>0, and (W_t)_t≥0 is a cylindrical Brownian motion on H. Under the minor cutoff modifications introduced in Sect. 3.1, it can be proved that the H-valued stochastic process (u_t)_t which solves in a mild sense (9) is unique and has continuous trajectories (Da Prato and Zabczyk, 2014). In addition to this there exists a unique Gibbs invariant measure

where ℛ is as in Eq. (6), and $\mathit{\mu }\sim \mathcal{N}(\mathrm{0},-\frac{{\mathit{\epsilon }}^{\mathrm{2}}}{\mathrm{2}\mathit{\kappa }}{\mathrm{\Delta }}^{-\mathrm{1}})$ is a symmetric Gaussian measure on H with covariance $\mathcal{Q}=-\frac{{\mathit{\epsilon }}^{\mathrm{2}}}{\mathrm{2}\mathit{\kappa }}{\mathrm{\Delta }}^{-\mathrm{1}}$ (Da Prato, 2004, 2006). The covariance operator $\mathcal{Q}:H\to H$ is the unique linear operator such that ${\int }_H\langle h_{\mathrm{1}},\mathit{\varphi }\rangle \langle h_{\mathrm{2}},\mathit{\varphi }\rangle \mathit{\mu }\left(\mathrm{d}\mathit{\varphi }\right)$ for each $h_{\mathrm{1}},h_{\mathrm{2}}\in H$, where $\langle \cdot ,\cdot \rangle$ denotes the scalar product in H. Further, it can be shown that 𝒬 is symmetric and positive-definite, and its eigenvalues (λ_k)_k∈ℤ satisfy ${\sum }_{k\in \mathbb{Z}}{\mathit{\lambda }}_k<\mathrm{\infty }$. In the following lines, given a symmetric, positive-definite operator 𝒬 such that ${\sum }_{k\in \mathbb{Z}}{\mathit{\lambda }}_k<+\mathrm{\infty }$, we are going to explain how to construct an H-valued random variable X with law 𝒩(0,𝒬). Indeed, consider a sequence (R_k)_k∈ℤ of independent and identically distributed ℝ-𝒩(0,1) random variables defined on a probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$. We can assume without loss of generality that the eigenvectors (e_k)_k∈ℤ associated with the eigenvalues (λ_k)_k∈ℤ form an orthonormal basis of H. Then, the H-valued random variable

is well defined, i.e. the series defining X converges in $L^{\mathrm{2}}(\mathrm{\Omega },\mathcal{F},\mathbb{P};H)$ and has law 𝒩(0,𝒬) (Da Prato, 2006, Proposition 2.18). Further, the convergence also holds ℙ almost surely in H (Da Prato, 2006, Proposition 2.13).

As mentioned in the introduction, this measure is concentrated on minimum points of the functional F_q. A heuristic explanation of this fact can be found in Sect. 3.1.

The stationary problem associated with the 1D EBM is given by the elliptic equation for u=u(x):

These solutions can be either stable or unstable, depending on the long-term behaviour of their infinitesimal perturbations. As pointed out in Bastiaansen et al. (2022), if the reaction–diffusion equation for $u=u(t,x)$ was space-homogeneous, i.e. of the form

then the stable steady-state solutions would correspond to functions that are constant in space and time, with values given as the roots of

A rigorous result in this direction has been shown in Gaspar and Guaraco (2018). Indeed, for a fixed double-well symmetric potential, it has been proved that (i) if κ is large enough, the only steady-state solutions of Eq. (12) are the constants where the potential is critical, and (ii) the number of unstable steady-state solutions to Eq. (12) can be made arbitrarily large as κ→0. Introducing a spatial dependence in $R=R(x,u)$ leads to a space-heterogeneous model. Depending on the space heterogeneity, it can exhibit any number of both stable and unstable steady-state solutions (Bastiaansen et al., 2022). The variational approach to the study of steady-state solutions provides a tool for characterizing the stable ones, which are the local minimum points of a functional.

In the following paragraph, we describe the properties of the solutions of Eq. (11). As the parameter q changes, numerical simulations for Eq. (11) suggest the existence of either one or three steady-state solutions. That is, there exists q₁<q₂ s.t. Eq. (8) has one steady-state solution if q<q₁ or q>q₂, and the steady-state solutions are three if $q_{\mathrm{1}}\le q\le q_{\mathrm{2}}$. In the latter case, we denote the solutions by $u_{\mathrm{S}}\le u_{\mathrm{M}}\le u_{\mathrm{W}}$, corresponding respectively to the snowball climate, a middle (or intermediate) climate and the warm climate. As an analogy, we denote by u_S the unique steady-state solution for q<q₁ and by u_W the unique one for q>q₂. Figure 2a shows the bifurcation diagram of the model in the $(q,\stackrel{\mathrm{‾}}{u})$ plane, where $\stackrel{\mathrm{‾}}{u}={\int }_{-\mathrm{1}}^{\mathrm{1}}u\left(x\right)\mathrm{d}x$ denotes the average temperature. Figure 2b depicts the three steady-state solutions for $q=\mathrm{25}\in (q_{\mathrm{1}},q_{\mathrm{2}})$.

Figure 2(a) Bifurcation diagram of the steady-state solutions in the $(q,\stackrel{\mathrm{‾}}{u})$ plane, with $\stackrel{\mathrm{‾}}{u}={\int }_{-\mathrm{1}}^{\mathrm{1}}u\left(x\right)\mathrm{d}x$. Solid lines denote stable solutions u_S and u_W, while dashed lines denote the unstable solution u_M. (b) Steady-state solutions of the EBM for q=25. In every point x of the space domain, the three steady-state solutions satisfy $u_{\mathrm{S}}\left(x\right)<u_{\mathrm{M}}\left(x\right)<u_{\mathrm{W}}\left(x\right)$, with maximum temperature attained at the Equator and minimum temperature attained at the poles.

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A stability analysis can be conducted to determine the stability of the steady-state solutions. The results show that u_S and u_W are stable, while the middle climate, u_M, is unstable. Furthermore, it is worth noting that special values $q=q_{\mathrm{1}},q_{\mathrm{2}}$ correspond to bifurcation points of saddle-node type, where the unstable solution u_M collides with either u_W (for q=q₁) or u_S (for q=q₂) and then disappears. These numerical findings regarding the number and stability of the steady-state solutions will be supported and validated using rigorous arguments, as in the next section.

Table 1Parameters and constants appearing in the Seller EBM (8).

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3.1 Potential functional and its minimizer

In this section, we (i) provide an intuitive motivation for why the invariant measure for the stochastic EBM concentrates on minimum points of the functional F_q, (ii) prove the existence of global minimum points for F_q using the direct method, and (iii) present sufficient conditions on the viscosity κ and the space-averaged potential $\stackrel{\mathrm{‾}}{\mathcal{R}}\left(u\right)=\frac{\mathrm{1}}{\mathrm{2}}{\int }_{-\mathrm{1}}^{\mathrm{1}}\mathcal{R}(x,u)\mathrm{d}x$, with ${\partial }_u\mathcal{R}=-R=R_{\mathrm{e}}-R_{\mathrm{a}}$, to ensure that the 1D EBM has at least three steady-state solutions.

Firstly, consider the stochastic EBM (9). Assume that for a negative value of u, where the model has no physical meaning, the Stefan–Boltzmann law is extended as

And β is smoothly extended to $\stackrel{\mathrm{̃}}{\mathit{\beta }}$ by setting it to zero outside the physically relevant range, as described in the Supplement. Then, Eq. (9) possesses a unique Gibbs invariant probability measure given by

where $(u{)}_{+}=max\mathit{\{}u,\mathrm{0}\mathit{\}}$ is the positive part, $\mathcal{N}(\mathrm{0},-\frac{{\mathit{\epsilon }}^{\mathrm{2}}}{\mathrm{2}}{\mathrm{\Delta }}^{-\mathrm{1}})$ denotes a symmetric Gaussian measure with covariance operator $\mathcal{Q}=-\frac{{\mathit{\epsilon }}^{\mathrm{2}}}{\mathrm{2}\mathit{\kappa }}{\mathrm{\Delta }}^{-\mathrm{1}}$ over the Hilbert space $H=L^{\mathrm{2}}(-\mathrm{1},\mathrm{1})$ and Z is the normalization constant. See Da Prato (2004) for a rigorous derivation of the invariant measure for a reaction–diffusion model with a polynomial homogeneous reaction term. We move to explain in what sense ν is concentrated around minimum points of F_q. In fact, for u∈H the Gaussian measure μ is formally given by

where ${\mathcal{Q}}^{-\mathrm{1}}=-\frac{\mathrm{2}\mathit{\kappa }}{{\mathit{\epsilon }}^{\mathrm{2}}}\mathrm{\Delta }.$ Here, Z₁ is a normalization constant, $\langle \cdot ,\cdot \rangle$ denotes the scalar product in H and du is a formal notation for the Lebesgue measure on H. If we perform an integration by parts, we get

Plugging the previous identity into Eq. (10), we obtain

From this heuristic formula, we see that points u such that F_q(u) is not a global minimum have exponentially smaller density than the minimum points. Indeed, if u₁ is a global minimum point and u≠u₁, then the mass given by ν in a small neighbourhood around u is exponentially smaller than the mass given to a neighbourhood of the same size around u₁; in particular, the ratio between the two masses is given by $\exp \left(-\frac{\mathrm{2}}{{\mathit{\epsilon }}^{\mathrm{2}}}\left(F_q\left(u\right)-F_q\left(u_{\mathrm{1}}\right)\right)\right)$. The previous derivation is formal, because the Lebesgue measure cannot be defined on an infinite-dimensional Hilbert space. For a more rigorous explanation, see Sect. S2 in the Supplement.

Next, we discuss the properties of the functional $F_q:H^{\mathrm{1},\mathrm{2}}(-\mathrm{1},\mathrm{1})\cap \left\{u\ge \mathrm{0}\right\}\to \mathbb{R}$ given by

where B is a primitive of the co-albedo $\mathit{\beta }\left(u\right)=\mathrm{1}-\mathit{\alpha }\left(u\right)$, and $H^{\mathrm{1}}=H^{\mathrm{1},\mathrm{2}}(-\mathrm{1},\mathrm{1})$ denotes the Sobolev space of order 1 and exponent 2, i.e. the function space where a function u and its derivative u^′ (in a weak sense) are both square integrable over $[-\mathrm{1},\mathrm{1}]$. See Brezis (2011) for more details about Sobolev spaces. The functional F_q, depending on the parameter q, is known in the literature as a potential functional or Lyapunov function (North et al., 1979; North and Kim, 2017). The study of the functional F_q gives useful information thanks to its links with the invariant measure for the stochastic 1D EBM, as we have seen, and the stable steady-state solutions for the deterministic 1D EBM which emerge as necessary conditions for the stationarity of F_q. Going deeper with the former point, the first variation of F_q in the point u in direction h is given by

where in the last identity we have used integration by parts. Since h is arbitrary, u is a stationary point for the functional F_q if and only if it is a steady-state solution for the EBM. In particular, local extremum points for F_q correspond to steady-state solutions of the EBM. Any local minimizer of F_q represents a locally attractive solution of the deterministic 1D EBM. In view of our interpretation of F_q in terms of the invariant measure, however, global minimizers play a special role since if present and unique they are exponentially more likely than any other state (including minimizers that are just local). The following result establishes the existence of a global minimum point for F_q.

A rigorous proof of the previous result can be found in Sect. S3. The proof relies on standard arguments from the direct method of calculus of variation, exploiting the fact that the outgoing radiation in the EBM model prevents the temperature from being too high.

Concerning the existence of two local minimum points, let us describe a sufficient condition. Consider the potential function $\stackrel{\mathrm{‾}}{\mathcal{R}}:\mathbb{R}\to \mathbb{R}$ coming from the space-averaged model

If the viscosity κ>0 is sufficiently large and the function $\stackrel{\mathrm{‾}}{\mathcal{R}}$ has a double-well shape with sufficiently deep minimum values attained at the minimum points, then we are able to prove the existence of two minimum points for F_q. Further, it is possible to prove that the functional F_q satisfies a compactness condition known as the Palais–Smale condition. This property and the mountain pass theorem give the possibility to deduce the existence of a third steady-state solution. Next, we characterize a situation in which there are three steady-state solutions, two of which are local minimizers (Jabri, 2003). This is summarized in the following result.

• i.

  ${\stackrel{\mathrm{‾}}{\mathcal{R}}}^{\prime \prime }\left(u_i\right)>f\left(\stackrel{\mathrm{‾}}{\mathit{\epsilon }}\right)$, for $i=\mathrm{1},\mathrm{2}$;

• ii.

  $\mathit{\kappa }>g\left(\stackrel{\mathrm{‾}}{\mathit{\epsilon }}\right)$;

• iii.

  $\stackrel{\mathrm{‾}}{\mathit{\epsilon }}\le \mathit{\omega }$;

then ${\stackrel{\mathrm{̃}}{F}}_q$ has two local minimum points, ${\stackrel{\mathrm{̃}}{u}}_{\mathrm{1}},{\stackrel{\mathrm{̃}}{u}}_{\mathrm{2}}$, such that

• a.

  $B_{H^{\mathrm{1}}}(u_{\mathrm{1}},\stackrel{\mathrm{‾}}{\mathit{\epsilon }})\cap B_{H^{\mathrm{1}}}(u_{\mathrm{2}},\stackrel{\mathrm{‾}}{\mathit{\epsilon }})=\mathrm{\varnothing }$;

• b.

  ${\stackrel{\mathrm{̃}}{u}}_i\in B_{H^{\mathrm{1}}}(u_i,\stackrel{\mathrm{‾}}{\mathit{\epsilon }}),$ for $i=\mathrm{1},\mathrm{2}$.

• c.

  If $\mathrm{|}\mathrm{|}u-u_{\mathrm{1}}\mathrm{|}{\mathrm{|}}_{H^{\mathrm{1}}}=\stackrel{\mathrm{‾}}{\mathit{\epsilon }}$, then $F_q\left(u\right)\ge F_q\left(u_{\mathrm{1}}\right)+\mathit{\delta }$, with $\mathit{\delta }=\mathit{\delta }\left(\stackrel{\mathrm{‾}}{\mathit{\epsilon }}\right)>\mathrm{0}.$

Note how the previous result can be also interpreted as giving sufficient conditions for the convergence of the stable solutions of a space-inhomogeneous EBM to the stable solution of the corresponding space-averaged model, as the diffusion becomes large.

3.2 Value function and uniqueness for the functional minimizer

The key element of this section is the value function, which is given by

From Sect. 3.1, we know that the previous infimum is indeed a minimum, and so V(q) can be interpreted as the minimum possible value attained by the potential functional over the possible temperature profiles u. Since a minimum point for F_q is also a stationary point for the functional, the value function can be evaluated numerically by computing the minimum of the three steady-state solutions u_S,u_M and u_W. Following this strategy, Fig. 3 shows $q\mapsto F_q\left(u_{*}\right)$, with $u_{*}\in \left\{u_{\mathrm{S}},u_{\mathrm{M}},u_{\mathrm{W}}\right\}$. Particularly, there exists a point q₃ such that u_S is the global minimum point of F_q for q<q₃, while u_W is the global minimum point for q>q₃. Further, for q=q₃ the function F_q has two different global minimum points u_S and u_W, and q=q₃ corresponds a non-differentiability point for V. In addition to this, the value function appears to be concave, thus with a decreasing derivative, where it exists.

Figure 3Potential functional F_q evaluated in the three steady-state solutions u_S,u_M and u_W. For q<q₃, u_S is the global minimum point, while u_W is a local minimum point. On the other hand, for q>q₃ the opposite happens. Solid lines correspond to values of the functional attained on stable solutions; dashed lines are for values corresponding to unstable ones.

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Summarizing, the numerical evaluations of V(q) suggest the following result, a rigorous proof of which is included in the Supplement.

• i.

  V is Lipschitz continuous.

• ii.

  q is a non-differentiable point for V if and only if there is more than one minimizer for F_q.

• iii.

  V is concave and V^′ is non-increasing.

We also see numerically that u_M is actually never a global minimizer for the specific functional F_q considered here, but we do no have rigorous proof of this fact. Let us briefly discuss the proofs of the previous points. The proof of (i) follows from the facts that the sup norm of the minimizer u₀ can be bounded uniformly in q and that, given a family {g_i}_i∈I of L_i Lipschitz functions g_i, the $\underset{i\in I}{inf}{g}_i$ is Lipschitz continuous if the constants L_i can be bounded uniformly. In our case, given u∈H¹ is non-negative, we have

where M>0 is the constant appearing in Eq. (15). On the other hand, the proof of point (ii) is less straightforward, although being very similar to the one for the existence of a solution for the variational problem. The proof of point (iii) makes use of the concept of semiconcavity, a generalization of that of concavity, which is fundamental in optimal control (Cannarsa and Sinestrari, 2004). The main reason for the concavity of V though is that V is an infimum over functions that are affine in q. Hence, the fact that q is additive is essential for this result. More details can be found in Sects. S4 and S5 in the Supplement.

3.3 Value function graph and bifurcation diagram

An additional property of the value function can be observed when comparing the bifurcation diagram (Fig. 4a) and the graph of the value function (Fig. 3).

Corollary 4. If V is differentiable, then $V^{\prime }\left(q\right)=-{\int }_{-\mathrm{1}}^{\mathrm{1}}{u}_{\mathrm{0}}\left(x\right)\mathrm{d}x$, where u₀ is the only minimizer for F_q.

In other words, the part of the bifurcation diagram that corresponds to the global minimizer, represented by the subgraph $(q,{\int }_{-\mathrm{1}}^{\mathrm{1}}{u}_{\mathrm{0}}(x),\mathrm{d}x)$, can be determined based on the knowledge of V^′, and vice versa. Figure 4 compares Figs. 2a and 3, highlighting in magenta the corresponding parts of the two graphs. From the mathematical point of view, the previous result is a consequence of the proof of Theorem 3.

Figure 4Comparison between the value function graph (left) and bifurcation diagram (right) for the 1D EBM. The magenta-shaded area highlights the parts of the plots which are in one-to-one correspondence. (a) Functional F_q evaluated on steady-state solutions, as in Fig. 3. (b) Bifurcation diagram, as in Fig. 2a.

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It is worth pointing out that by combining Theorem 3 and Corollary 4, a valuable property emerges, i.e. the global mean temperature of the functional minimizer is non-decreasing with respect to q. In other words, as the concentration of CO₂ rises, the global mean temperature increases. Additionally, through this monotonicity and Froda's theorem (Rudin, 1976, Theorem 4.30), we also establish that the global mean temperature is continuous, except for, at most, a countable number of upward jumps.

In the second part of this section, we demonstrate the applicability of Corollary 4 to other reaction–diffusion equations. We use as an example a spatially heterogeneous Allen–Cahn equation (ACE), already considered in Bastiaansen et al. (2022). For an initial condition $\stackrel{\mathrm{̃}}{u}$, this model is given by

The associated elliptic problem for u=u(x) is

In this case, the potential functional takes the form

and all the properties discussed in Sect. 3.1 and 3.2 can be extended to this equation. Specifically, Theorems 1, 2 and 3 hold. But in this case, the structure of the bifurcation diagram is more complex, even if symmetric with respect to q=0. Indeed, through numerical experiments, it is possible to deduce the existence of $\mathrm{0}<q_{\mathrm{4}}<q_{\mathrm{5}}$ such that (a) for |q|>q₅ or |q|<q₄, there exists a single steady-state solution, which is stable, (b) for $q_{\mathrm{4}}<\mathrm{|}q\mathrm{|}<q_{\mathrm{5}}$ there are three steady-state solutions, two of which are stable while the third is unstable. Further, $q=q_{\mathrm{4}},q_{\mathrm{5}}$ are bifurcation points of the saddle-node type. We denote by u₁ the steady-state solution for $q<-q_{\mathrm{5}}$, by u₂ and u₃ the steady-state solutions appearing at the bifurcation point $q=-q_{\mathrm{5}}$ and existing for $-q_{\mathrm{5}}<q<-q_{\mathrm{4}}$ in addition to u₁, and by u₄ and u₅ the steady-state solutions appearing at q=q₄ and existing for $q_{\mathrm{4}}<q<q_{\mathrm{5}}$ in addition to u₃. Regarding the potential functional J_q, in this case there exists $q_{\mathrm{6}}\in (q_{\mathrm{4}},q_{\mathrm{5}})$ such that u₁ is the global minimum point for the functional for $q<-q_{\mathrm{6}}$, and u₃ is the global minimum point for $-q_{\mathrm{6}}<q<q_{\mathrm{6}}$, while u₅ becomes the global minimum point for q>q₆. A picture for the bifurcation diagram just described and the value function is shown in Fig. 5. Note that $q=\pm q_{\mathrm{6}}$ represents the only values of the parameter q for which the value function is not differentiable and also the only points in which the global minimizer of the variational problem is not unique.

Figure 5Comparison between the value function and the bifurcation diagram for the non-homogeneous ACE. The magenta-shaded area highlights the parts of the plots which are in one-to-one correspondence. (a) Potential functional evaluated on the steady-state solutions: u₁ is the global minimum point for $q<-q_{\mathrm{6}}$, u₃ is the global minimum point for $-q_{\mathrm{6}}<q<q_{\mathrm{6}}$, u₅ is the global minimum point for q>q₆. Note that $q=\pm q_{\mathrm{6}}$ are the non-differentiability point for the value function, corresponding to non-uniqueness of the minimizer. (b) Bifurcation diagram.

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