Papers

We study the relative periodic orbits in the Gutzwiller-type anisotropic Kepler problem, which is a generalized model derived from the classical Gutzwiller anisotropic Kepler problem. By reducing the rotational symmetry of the $z$-axis, we obtain a reduced system with two degrees of freedom and its energy surface forms a compact and regular three-sphere in a certain parameter range. Combining the estimation of the Seifert rotation number of the planar Kepler orbit and the CHHL formula introduced in \cite{CHHL23}, we prove that this system admits infinitely many periodic orbits on every compact regular energy surface. This model can be applied to the $(1+2n)$-body problem to find infinitely many relative periodic orbits with hip-hop symmetry as well as relative periodic orbits in the $n$-pyramidal problem.

We formulate electrostatics in \rthree as the simplest instance of the theory of stresses: the virtual velocity field of continuum mechanics is simply replaced by a scalar potential field. A charge distribution is then a linear functional on potential fields whose value is the potential energy, and it is represented by a volume charge density and by surface charge densities on the boundaries of subbodies. Cauchy's theorem gives a vector field $\boldsymbol P$, the analog of the traction stress, and a scalar field $r$, so that the energy is $\int(r\varphi+\boldsymbol P\cdot\nabla\varphi)dV$. The pair $(r,\boldsymbol P)$, the analog of a variational stress, splits every charge distribution uniquely into free charge, with density $r$, and a dipole distribution, with density $\boldsymbol P$, which carries the bound charges $-\nabla\cdot\boldsymbol P$ and $\boldsymbol P\cdot\boldsymbol{\nu}$; the fields $r$ and $\boldsymbol P$ are the densities of charge and of dipole moment. We do not impose gauge invariance of the energy. Instead, we show that it is the analog of the balance of forces on every subbody, and that it singles out the neutral charge distributions. For charges embedded in matter, the forces exerted by the electric field are derived. They are shown to be gauge invariant for every charge distribution, and they include an asymmetric active stress $\boldsymbol E\boldsymbol\otimes\boldsymbol P$ whose skew-symmetric part is the couple density $\boldsymbol P\times\boldsymbol E$.

We construct a finite discrete realization of the Smorodinsky–Winternitz I superintegrable system on a triangular region of the two-dimensional square lattice. The construction is based on a pair of commuting number operators with finite spectrum together with an associated ladder-operator structure. We show that the resulting model is maximally superintegrable and that its symmetry algebra admits a Hahn-algebra presentation. Its spectral problem is solved exactly in terms of the bivariate dual Hahn polynomials of Tratnik type. Finally, we show that the continuum limit recovers the continuous Smorodinsky–Winternitz I system together with its ladder operators, eigenfunctions and symmetry algebra, thereby establishing the present construction as a genuine finite discrete realization of the continuous model.

We consider distributed distance-based formation control with prescribed transient performance for multi-agent systems modeled by unknown nonlinear dynamics of relative degree greater than one. We introduce a virtual leader whose trajectory is tracked with prescribed transient behavior by a designated subset of agents. The undirected communication graph is either a tree graph or a minimally and infinitesimally rigid graph. In the latter case, we further consider the objective of centroid tracking. In each setting, a distributed and model-free control law is developed, and we establish the satisfaction of the prescribed funnel constraints on the formation and tracking errors, and boundedness of all closed-loop signals. Numerical simulations illustrate the effectiveness of the proposed control laws.

The classical Strang projector-splitting integrator is widely observed to converge with order two, whereas the error analysis that remains uniform as the smallest singular value retained in the low-rank approximation tends to zero proves only order one. We show that this gap is intrinsic under the standard assumptions. We construct $3\times3$ matrix differential equations that are $C^2$ in time and smooth in the matrix variable, with rank-two initial data that satisfy uniform boundedness, Lipschitz, tangency-defect, and regularity bounds. Nevertheless, the exact-subflow Strang method has a nonzero $h^2$ term in the local error over one periodic forcing cycle consisting of four Strang steps. Repetition of that cycle rules out a global second-order bound whose constant and stepsize threshold are independent of the retained singular values. The mechanism is a rapid rotation of the factor directions associated with the small singular value: first-order consistency is preserved, but the changing projection spaces prevent the cancellation normally expected from a symmetric Strang composition. Hence the robust first-order result cannot, under these assumptions alone, be upgraded to robust second order for the classical projector-splitting method.

$\mathrm{PA}_{\mathrm{smu}}^{-}$ is a weak theory of arithmetic obtained by adding two principles concerning powers of two to basic axioms satisfied by the nonnegative part of a discretely ordered ring. We introduce a theory $\mathrm{PA}_{\mathrm{wit}}^{-}$ in which powers of two and the required witnesses are represented by primitive symbols. We show that, along a fixed sequence in the standard model of arithmetic, every one-variable term eventually agrees with a polynomial over the dyadic rationals. A finite avoidance lemma and the Compactness Theorem then yield a model of $\mathrm{PA}_{\mathrm{smu}}^{-}$ in which division by $3$ fails. In this model even the $n=3$ instance of the Standard Euclidean Division Principle pa16 fails. In fact, the model can be chosen to satisfy every universal $\mathcal{L}_0$-sentence true in the standard model. Consequently, neither pa16 nor the Euclidean Division Principle pa17 is derivable even after these universal truths are adjoined to $\mathrm{PA}_{\mathrm{smu}}^{-}$.

Given $s\in(0,1)$ and $n>2s$, we construct nontrivial solutions of the problem \begin{equation*} \begin{cases} (-\Delta)^s u=u^{2^*_s-1} &{\mbox{ in }}\mathbb{R}^n_+,\\ {\mathcal{N}}_s u=0&{\mbox{ in }}\mathbb{R}^n_-,\end{cases} \end{equation*}where ${\mathcal{N}}_s$ represents the nonlocal Neumann condition of exterior type.

We prove that every closed hyperbolic surface admits a sequence of finite covers with asymptotically optimal diameters.

In this paper we propose a robust nonconforming finite method for a gradient-elastic Kirchhoff plate (GEKP) model with a small parameter $0<\iota<1$. While the method employs an $H^3$ cubic finite element for the discrete space, it employs an interpolation mapping into an $H^2$ nonconforming finite element space in both the lower-order terms of the bilinear form and the right-hand side. Notably, the numerical method is robust with respect to the parameter $\iota$ and uniformly convergent to order $h$ without any stabilized terms. Finally, some numerical experiments are performed to verify the theoretical findings.

Fractional partial differential equations describe nonlocal dynamics, but discovering them from noisy data is difficult because fractional differentiation amplifies high-frequency measurement noise and the derivative orders are unknown. We propose Weak-Pareto, which combines an adjoint-consistent weak formulation of fractional terms with Pareto-based subset selection over discrete term types and continuous fractional orders. For linear right-hand-side terms, the adjoint transfers fractional operators from measured fields to smooth test functions, replacing noise-sensitive pointwise differentiation with smoothing integration; for nonlinear terms, the noise-suppression effect is partial yet useful. Coefficients are fitted by ridge regression within a branch-aware differential-evolution search over the orders. The support size is then selected at the validation-error-complexity elbow. We show that the variance of fixed linear right-hand-side weak features vanishes under grid refinement, whereas noise amplification in pointwise fractional features increases with derivative order. Across fractional advection-diffusion, reaction-diffusion, and Burgers benchmarks, Weak-Pareto recovers parsimonious structures from clean and noisy measurements. In controlled advection-diffusion and Burgers comparisons, it retains the correct support at every tested multiplicative-noise level, whereas the unregularised strong-form counterpart largely fails once noise is introduced; this advantage persists under additive Gaussian noise. Ablations show that the weak library drives noise robustness and that continuous-order Pareto search avoids the support-selection failure of a dense fixed dictionary. On the advection-diffusion benchmark, Weak-Pareto yields more consistent operator recovery and substantially lower measured runtime than a contemporary neural baseline.

We prove that the conformal dimension of every locally finite Borel measure is either zero or infinite. The main ingredient is a quasisymmetric dimension-reduction theorem: every full-support probability measure of finite Hausdorff dimension on a separable metric space admits quasisymmetrically equivalent metrics in which its Hausdorff dimension is arbitrarily small. In particular, every locally finite Borel measure on a doubling metric space has conformal dimension zero. We also prove that for every $n\geq 1$ and $p>0$ there are a metric space $X$, a quasisymmetric homeomorphism $f:[0,1]^n\to X$, and a Borel set $E\subset[0,1]^n$ such that $\dim_H f(E)\leq p$ and $\dim_H([0,1]^n\setminus E)\leq n-1+p$. The second bound is sharp up to $p$: if $\dim_H f(E)<1$, then $\dim_H([0,1]^n\setminus E)\geq n-1$.

We prove a linear operator of a finite-dimensional Banach space has the topological shadowing property, as introduced in \cite{lny}, if and only if its spectrum lies either within the open unit complex disk or outside the closure of that disk. Moreover, the spectrum of every uniformly expansive linear operator with the topological shadowing property lies either within the open unit complex disk or outside the closure of that disk. Lastly, we prove that a normal operator with the topological shadowing property on a Hilbert space does not have any nonzero nonwandering points.

For $N\geq 1$, $p\geq 1$, and $\delta>0$, consider the nonlocal threshold functional \[ I_{\delta,p}(u)=\iint_{\{|u(x)-u(y)|>\delta\}} \frac{\delta^p}{|x-y|^{N+p}}\,\dd x\,\dd y. \] Nguyen and Squassina \cite{NguyenSquassina} asked whether $I_{\delta,p}$ decreases under Schwarz rearrangement. We give a negative answer in every dimension. For each $N\geq1$ and $\delta>0$, one and the same explicit counterexample works for every $p\geq1$: it is nonnegative, bounded, compactly supported, and takes only five values. Its four nontrivial superlevel sets are nested balls whose centers alternate between two points. Once the active threshold interactions are isolated, the energy difference reduces to comparing the interaction of a unit ball with a concentric annulus and with an eccentric shell. The sign is strict because the potential generated by the unit ball decreases with the radius. We also compute the energy gap in closed form in dimension one. This settles Open Problem~2.2 of Nguyen and Squassina.

A vertex of a graph is called a top vertex if it has maximum degree in the graph. A vertex of a graph is called a Gallai vertex if it belongs to every longest path of the graph. Golan and Shan proved that every top vertex of any connected induced-$2P_2$ free graph is a Gallai vertex. Long, Milans, and Munaro subsequently showed that if every connected induced-$H$ free graph has a Gallai vertex, then $H$ must be a linear forest of order at most nine. They also proved that, for every linear forest $H$ of order at most four, every top vertex of any connected induced-$H$ free graph is a Gallai vertex. In this paper, we determine the graphs $H$ for which every top vertex of any connected induced-$H$ free graph is a Gallai vertex. Our result shows that this holds precisely when $H$ is a linear forest of order at most four or $H=P_3+2P_1$.

In 1972, M. Sato proposed an approach to proving modularity of forms like Thetanullwerte by characterizing them via certain differential operators of infinite order (DOI) in the modular variable(s) alone. A DOI is an infinite series in derivatives decreasing so fast that it acts on holomorphic functions by a sheaf morphism. This approach was developed by several authors including Kashiwara, Kawai, Takei and Yoshida. We give an interpretation of this approach using supersymmetry which provides a natural source of DOIs: the naive exponential of any odd supersymmetry generator is a DOI. The case of the Riemann theta function of genus n is governed by the supergroup OSp(1|2n) (and its metaplectic cover) acting on a natural super-thickening of the Siegel plane. For n=2 this is the 3-dimensional N=1 superconformal group and the structure at hand is precisely the free massless scalar supermultiplet (combining the Laplace and Dirac equations). For n>2 we get a super-extension of the generalized conformal structure existing on the Lagrangian Grassmannian as on any Hermitian symmetric space. An additional interesting feature here is that the odd supersymmetry generators acting “on-shell” (i.e., in the space of solutions of the equations of motion) satisfy even-style Heisenberg commutation relations. These equations of motion upgrade to a complex of differential operators corresponding to a natural BGG-type resolution of the super-Weil representation of osp(1|2n).

We show that for every conformal map $\phi:\mathbb{D}\to\Omega\subsetneq\mathbb{C}$ from the unit disk onto a simply connected proper domain $\Omega$, the length of $\phi^{-1}(\Omega\cap L)$ is at most $\pi^2$ for every line $L$.

We introduce a finite stochastic spiking-neuron network with Elephant-type memory, in which past firing activity modifies future excitability through a reinforcement-dependent threshold. For a bounded hard-threshold firing rate, we prove non-explosion of the finite system and obtain conditional exponential contraction in (1)-Wasserstein distance on a truncated potential space. We then formulate the corresponding replica mean-field dynamics and establish global existence, uniqueness in law, and non-explosion of the nonlinear process, together with a characterization of its invariant measures. Numerical experiments show that Elephant memory produces a (p)-dependent decline in firing activity, alters extinction behaviour, and yields finite-network dynamics closely matched by the replica mean-field approximation.

We study estimation of the drift matrix in a continuously observed high-dimensional Ornstein-Uhlenbeck process when the drift is exactly or approximately low rank. In this setting, exact low rank induces non-stable directions and hence a non-ergodic regime, resulting in a poorly conditioned empirical covariance matrix. To address this difficulty, we introduce a Weighted Nuclear Elastic Net Estimator that combines ridge regularization with a nuclear-norm penalty expressed in the empirical likelihood geometry. Under a general diagonalizable spectral framework, we establish oracle inequalities relative to arbitrary low-rank comparison matrices. For near low-rank drifts, the approximation error is naturally measured through the singular-value decay of the drift after weighting by the regularized empirical covariance. The stochastic term is controlled by self-normalized martingale arguments under appropriate choice of the tuning parameter. For a symmetric positive-semidefinite exact low-rank model, we verify the empirical-curvature condition required to translate the weighted bound into a Frobenius-norm bound. With an appropriate choice of tuning parameters, the resulting estimator satisfies, up to a logarithmic factor, the standard rank-$r$ matrix-estimation scaling $r d/T$: specifically, its squared Frobenius error is of order $r d\log(T)/T$ with high probability, under an explicit dimension-horizon condition.

The linearized Cahn-Hilliard-Cook equation is discretized in the spatial variables by a standard finite element method. Strong convergence estimates are proved under suitable assumptions on the covariance operator of the Wiener process, which is driving the equation. The backward Euler time stepping is also studied. The analysis is set in a framework based on analytic semigroups. The main effort is spent on proving detailed error bounds for the corresponding deterministic Cahn-Hilliard equation. The results should be interpreted as results on approximation of the stochastic convolution, which is a part of the mild solution of the nonlinear Cahn-Hilliard-Cook equation.

A brick is a 3-connected graph $G$ such that $G-u-v$ has a perfect matching for any two distinct vertices $u,v\in V(G)$. An edge $e$ in a matching covered graph $G$ is removable if $G-e$ is matching covered. We say that a removable edge $e$ in a brick $G$ is $b$-invariant if $b(G-e)=b(G)=1$, where $b(H)$ denotes the number of bricks in the tight cut decomposition of a matching covered graph $H$. An edge of a graph is solitary if it lies in precisely one perfect matching. Lucchesi and Murty proposed the problem of characterizing bricks, distinct from $K_4$, $\overline{C_6}$ and the Petersen graph, in which every $b$-invariant edge is solitary. Note that every $b$-invariant edge is removable. In this paper, we strengthen the condition by requiring that every removable edge is solitary. We show that every nonsolid brick satisfying this strengthened condition can be obtained by repeatedly splicing odd wheels (up to multiple edges). Moreover, properties of such bricks imply that "repeatedly splicing odd wheels" cannot be replaced by "repeatedly splicing copies of $K_4$".

Distribution steering seeks feedback laws that drive the state law of a dynamical system between prescribed initial and terminal distributions. Optimal transport provides a natural geometric approach, but its implementation generally requires a transport map or coupling in the full state space. Sliced optimal transport avoids this full-dimensional construction through one-dimensional projections. Yet, the resulting projected maps specify only directional displacements and do not by themselves prescribe a realizable feedback law. To this end, we develop a finite-horizon control framework based on sliced optimal transport. At each sampling instant, a projected optimal transport map defines a directional terminal condition, whose minimum-energy realization yields a randomized single-direction controller. Averaging over projection directions gives a deterministic sliced feedback. For the single-integrator dynamics, the averaged feedback makes the sliced Wasserstein distance to the target non-increasing. For Gaussian endpoint laws, it is affine, preserves Gaussianity, and steers the mean and covariance to their prescribed terminal values. We further identify a law-dependent gain that yields linear decay of the sliced Wasserstein distance together with an explicit characterization of the control energy. We also prove that the randomized controller converges to the averaged sliced flow as the sampling period vanishes. Finally, we extend the construction to linear dynamical systems. Reachability-normalized coordinates allow instantaneous realization of the sliced velocity for uniformly fully actuated systems, while local controllability Gramians provide exact finite-step realization for general controllable systems. Numerical examples illustrate the resulting distributional flows.

The homogeneous Dirichlet boundary value problem captures the core difficulty of solving Dirichlet problems with non-interpolatory methods. We propose BTF-PINN (Boundary-Training-Free Physics-Informed Neural Network), an interior-only strategy for solving homogeneous Dirichlet boundary value problems that requires no boundary training, boundary penalties, or boundary-conforming parametrizations. The key idea is to embed the essential boundary condition into a newly designed boundary-free loss function. We prove the equivalence between the proposed interior-only variational formulation and the original boundary value problem, establish a sharp threshold condition for the residual weight, and develop a convergence analysis based on the coercivity of the functional. Numerical experiments on high-dimensional problems, irregular geometries, and anisotropic elliptic equations demonstrate the effectiveness of BTF-PINN. Comparisons with standard boundary-penalty PINNs further show that BTF-PINN achieves superior boundary trace accuracy.

For a connected graph $G = (V, E)$, a bijective edge labeling $f:E \to\{1,\ldots ,|E|\}$ is a local antimagic labeling of $G$ if it induces a vertex labeling $f^+$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(xu)$, with $u$ ranging over all the vertices adjacent to $x$. The minimum number of distinct induced vertex labels over all local antimagic labelings of $G$ is the local antimagic chromatic number of $G$, denoted $\chi_{la}(G)$. In this paper, we make use of algebraic analysis to obtain necessary and sufficient conditions for every bipartite graph with all vertices of degree 2 except exactly three vertices of degree at least $3$ (say, $u,v,w$, such that every $(u,v)$-, $(u,w)$- and $(v,w)$-path is of even length and not all are of length 2) to have local antimagic chromatic number 2. Moreover, we showed that the consecutive edge labels of every induced path of each case is unique.

Large language models have begun refuting long-standing conjectures and solving long-open problems. The introspection this has prompted about the future of mathematical discovery is well under way, and the anxiety accompanying it legitimate – but both, we claim, are attached to the wrong loss. What machines now produce is the countable part of mathematics – theorems, proofs, refutations – which was always the work's residue, not its product. The distinction is old, and not economic: a result can be taken in its finished essence, or in the operations that engendered it. The product is human understanding: not a stock of results but a collective, hard-won way of deciphering the world and acting upon it. The two are arcs of a single loop: understanding tells us where to look; looking produces the residue; and taking it up again, one journey at a time, rebuilds shared understanding. Machines are strong on the countable arc, absent from the one that feeds it. The peril is to leave the loop open. AI did not create the confusion between residue and product; it has called a bluff long on the books, driving the cost of the residue towards zero and making the scarce thing visible at last. A new instrument makes a new way of working before it makes a new result. The pressing questions are therefore institutional: who can check an announced result, whoever announces it; what work and training become for the next generation of researchers; and whether the one thing that cannot be mass-produced – the journey that nourishes a shared understanding – continues to be funded. Mathematics, we argue, is uniquely placed among the sciences on the first – a proof answers to no one's permission – and uniquely exposed on the other two: teaching cannot go on as before, and no collective position yet exists; and the journey has never had a price our institutions knew how to pay. The decision is ours.

In this paper, we prove the local null controllability of the Boussinesq system in dimensions two and three with a reduced number of localized controls. More precisely, the controls act on the temperature equation and on only $N-2$ components of the velocity equation. Our proof is based on a spectral approach in the spirit of the Lebeau–Robbiano method. The main difficulty comes from the coupled structure of the system. The velocity and temperature equations are governed by the Stokes operator and the Dirichlet Laplacian, respectively. Their natural spectral decompositions are different, and there is no common spectral localization naturally adapted to the coupled system. Therefore, the usual componentwise spectral argument cannot be applied directly. We show that the cascade structure of the linearized system makes it possible to overcome this difficulty. The main ingredient is a mixed observability estimate in which only the Stokes component is spectrally localized, while the heat component is treated without any frequency restriction. Combining this estimate with the dissipation of the high Stokes frequencies allows us to carry out a Lebeau–Robbiano iteration for the linearized system. The resulting linear controllability estimate is transferred to the nonlinear Boussinesq system through a time-iteration argument. As an important consequence, the controls have a small-time cost bounded by $C\exp(C/T)$, which recovers the expected parabolic order while preserving the reduced number of controls.

We consider a countably infinite collection of linear, scalar control systems, where each system is represented by the pair $(a_n,b_n)$ with $a_n>0$ for $n\in \mathbb{N}$, and all systems are forced by a common scalar control input. We refer to this collection of systems as a discrete linear ensemble system. The ensemble system is feedback stabilizable if there exists a common feedback control input that asymptotically stabilizes every system simultaneously. Unlike finite-dimensional linear systems, an infinite-dimensional linear system is not guaranteed to be stable if its poles lie in the open left half-plane. Stability is guaranteed, however, if (i) its poles are contained in the closed left half-plane and (ii) its infinitesimal generator is diagonalizable. In this paper, we provide necessary and sufficient conditions for the existence of a static, linear feedback control law that ensures the closed-loop ensemble system satisfies (i) and (ii). In particular, we show that the exponential decay of the sequences $(|b_n|)_{n\in \mathbb{N}}$ and $(a_n/|b_n|)_{n\in \mathbb{N}}$ is necessary and, under an assumption on the desired poles, sufficient.

Different long-time patterns can prevail in different regions of a reaction–diffusion system's parameter space. We study the transition curves between such regimes for a one-dimensional Brusselator in the two-parameter plane $(\sigma,b)$, focusing on wave/stripe-like, spiral/source-defect-like, and target-like states. We develop a feature-based continuation framework built on time-dependent PDE simulations. Scalar observables extracted from late-time solution data distinguish the regimes and define threshold level sets where the feature crossings are regular. A secant predictor and a local one-dimensional sweep corrector are used to trace these level sets. For the spiral transition, we introduce a branch-adapted spacetime symmetry-defect feature that separates asymmetric source-like patterns from more symmetric wave patterns. For the target transition, we use the minimum of a core spatial-variance score and a tail temporal-variance score to detect the characteristic structure of half-target states. The method recovers robust side portions of both transition curves. In lower parameter regions, where mixed and irregular patterns make a single scalar feature less specific, we instead report transition estimates obtained from vertical parameter sweeps and direct inspection of spacetime plots. These results show how simulation-based continuation and direct pattern classification can be combined to map regime boundaries while preserving the different levels of numerical evidence.

In this paper, we derive quenched scaling limits for linear functionals and the empirical spin field of Ising models on inhomogeneous random graphs generated by a graphon (encompassing both dense and sparse graphs), in the high-temperature regime. We first prove a joint central limit theorem (CLT) for finite collections of linear statistics of the spin configurations, where the limiting covariance is characterized by the resolvent of the associated graphon integral operator. Building on this result, we establish functional CLTs for the average magnetization and for the spin field indexed by suitable classes of regular test functions. We further prove convergence of the full empirical spin field, viewed as a random generalized function in negative Sobolev spaces. These scaling limits provide applications to both Bayesian neural networks and causal inference. Specifically, for the former, we derive infinite-width Gaussian-process limits for two-layer Bayesian neural networks with Ising-dependent output-layer signs, while for the latter, we establish the asymptotic normality of Hájek estimators for average treatment effects under network interference.

For every Lie–Rinehart algebra, we construct an $F$-manifold algebra on the direct sum of its base algebra and module. Contrary to the assertion in \cite[Proposition 13.3.26]{LodayVallette}, the resulting structure is generally not Poisson. We determine when powers of the positive-degree ideal in the associated symmetric Poisson algebra are Poisson ideals, and relate the Leibnizator to the Lie–Rinehart differential. For a finite projective module of constant rank, the trace of the Leibnizator recovers the anchor and yields a rigidity result for injective anchors. We conclude with algebraic and geometric examples.

We analyze the dynamics of two harmonic oscillators with nonlinear, non-Hermitian coupling between them. Specifically, we consider a synthetic antiferromagnet composed of two ferromagnetic layers with antiferromagnetic interactions between the two layers, leading to the emergence of two oscillation modes of the combined system. We analyze the response of this system to external driving fields. We determine the allowed steady states under various driving conditions. We then analyze the dynamics of small deviations away from the different steady states, which reveals the normal modes that can be probed using spectroscopic techniques in experiment. As expected, the nonlinear system exhibits multistability, in which multiple steady states can exist for the same driving conditions. We find that the dynamical response can be drastically different depending on which one of the two modes is driven and on the strength of the driving. Specifically, we can obtain level-repulsion or level-attraction patterns in the spectrum. In addition to the rich variety of spectra, we find that some steady states exhibit normal modes that contain multiple frequencies. Our results explain recent experiments on synthetic antiferromagnets and provide guidance for designing experiments that can explore previously unseen phenomena in these systems.