Papers

The replicator equation is a central framework for studying frequency–dependent selection in ecology and evolutionary game theory. In well-mixed populations, the long-term outcome of a two-species system is determined by the signs of the pairwise invasion fitnesses, leading to dominance, coexistence, or bistability. However, many ecological systems are spatially structured, with environmental heterogeneity and dispersal shaping local interactions. How these spatial effects modify the classical replicator regimes remains incompletely understood. In this work, we study a spatially heterogeneous extension of the two-species replicator equation in which pairwise invasion fitnesses vary across space, and frequencies evolve under diffusion and advection. In this setting, the classical invasion fitnesses are replaced by spatial invasion rates given by the principal eigenvalues of the associated linearized operators. Using bifurcation theory, we show that these spatial invasion rates do not fully determine the qualitative dynamics of the system. In particular, spatial heterogeneity can induce a backward bifurcation, generating stable coexistence states even in parameter regimes where the well-mixed replicator predicts competitive exclusion. We derive an explicit local condition for this mechanism, characterizing when such coexistence states arise. These results show that spatial structure can fundamentally alter classical replicator dynamics and provide a concrete mechanism through which coexistence may emerge.

Geodesic flows on closed hyperbolic surfaces are a quintessential example of chaotic dynamics, i.e., systems whose long term behavior is very sensitive to initial conditions. The speed of such chaos is controlled by the spectral gap of the Laplace-Beltrami operator of the underlying hyperbolic surface. In this paper we give an overview of recent breakthroughs of Anantharaman and Monk showing that large genus closed hyperbolic surfaces have optimal spectral gap in a probabilistic sense. On the way we introduce and discuss the foundational works of many authors, from Selberg to Mirzakhani, that play a crucial role in the tour de force proof of Anantharaman and Monk.

The classical Gauss–Kuzmin distribution describes the asymptotic distribution of continued fraction digits. Geometrically, this may be interpreted as the equidistribution of faces of Klein sails in $\mathbb R^2$. In this paper, we establish a three-dimensional analogue of this phenomenon. We prove the equidistribution of local face structures in generic Klein sails in $\mathbb R^3$, thereby obtaining a higher-dimensional generalization of the Gauss–Kuzmin distribution. In addition, we resolve several open questions posed by Karpenkov~\cite{Ka17}. Our approach is based on homogeneous dynamics and is motivated from the work of Kontsevich and Suhov~\cite{KS99}. More precisely, we construct a cross-section for the diagonal flow on $\operatorname{SL}_3(\mathbb R)/\operatorname{SL}_3(\mathbb Z)$, such that visits to the cross-section encode the geometry of three-dimensional sails. The principal technical contribution is the proof that the associated cross-sectional measure is finite, which enables us to derive the limiting face statistics.

We prove that every $\mathcal{C}^{1,\alpha}$ self-map of a compact Riemannian manifold, in arbitrary dimension, admits a symbolic extension. This gives a positive answer to the conjecture of Downarowicz and Newhouse in arbitrary dimensions. Our method is based on analyzing the small singular values of the linearized orbit operator.

We adapt the theory of Ruelle-Pollicott resonances to annealed random dynamical systems generated by independent and identically distributed families of maps. Introducing annealed transfer and Koopman operators, we define resonances as elements of the point spectrum of the associated operators and establish a decorrelation formula relating these resonances to the asymptotic decay of annealed correlations. We then study several classes of systems for which the theory can be made explicit. First, we consider a family of piecewise expanding Markov maps of the interval, we construct Banach spaces adapted to the dynamics and obtain a complete description of the annealed resonance spectrum. Then we investigate an inverse spectral problem, proving realisability results for prescribed collections of complex numbers as resonances of suitably constructed annealed dynamical systems.

In this paper, we prove the existence of Lyapunov exponents for the geodesic flow on complete Riemannian manifolds without conjugate points whose sectional curvature is bounded below. Under suitable additional curvature assumptions, we establish Ruelle's inequality. Furthermore, assuming the same geometric conditions, we obtain the Pesin entropy formula for $C^1$-Hölder geodesic flows on finite-volume manifolds without conjugate points.

In this paper, we investigate the Lyapunov exponents and hyperbolicity of random billiard systems under small stochastic perturbations. Our study begins with a review of the necessary theoretical background on Lyapunov exponents for stationary sequences of matrices and Markov processes. We then consider several classical billiard systems and demonstrate that random perturbations can lead to significant changes in their dynamical properties. Specifically, we show that non-circular elliptic and lemon billiards exhibit both ergodicity and hyperbolicity under random perturbations, whereas circular billiards, while ergodic, maintain zero Lyapunov exponents. We also establish conditions under which the largest Lyapunov exponent becomes positive, indicating the emergence of hyperbolic behavior in these systems. These results provide new insights into the stability and chaotic properties of dynamical systems subjected to random noise, with implications for the broader study of smooth dynamical systems.

In 1986, Ivić and Tenenbaum introduced arithmetic functions with squarefull kernel, which are also called $s$-functions. Later, Erdős and Ivić gave an asymptotic estimate on the shifted convolution sums of $s$-functions. Recently, Bergelson and Richter studied the orbits along the prime Omega function in a uniquely ergodic topological dynamical system and established a new dynamical generalization of the prime number theorem (PNT). These orbits can be viewed as functions of invariant average under multiplications. In this paper, we show that both $s$-functions and their shifted convolutions are asymptotically uncorrelated to the orbits along the prime Omega function in a uniquely ergodic system. As a consequence, we obtain a refinement of the PNT via the local distribution of $s$-functions. Furthermore, several variants of these results are established as well.

We present a computer assisted proof of the existence of the oscillatory motion to collision and to infinity in the Earth-Moon planar circular restricted three body problem. In particular, we show that all types of motion (hyperbolic, parabolic, oscillatory to infinity, oscillatory to collision and terminating in collision) can be combined. We also prove the existence of arbitrarily large ejection/collision orbits. The results are achieved with the use of the Levi-Civita regularization, the McGehee regularization, topological tools and rigorous numerical computations.

We show that the weak Markus–Yamabe conjecture fails in every dimension $n\geq14$. We first prove a chain realization theorem: every polynomial Keller map $F=\I+H$ of $\R^n$ with component degrees $d_1,\ldots,d_n$ yields an explicit polynomial vector field on $\R^N$, with $N=\sum_i\max(d_i,2)-n$, whose Jacobian matrix has spectrum $\{-1\}$ at every point and whose singularities are in bijection with any prescribed fiber of $F$. Applied to the recent counterexample to the Jacobian conjecture, this gives a Hurwitz vector field of degree seven on $\R^{14}$ with three rational singularities. A rank-reduced dehomogenization of the associated cubic stabilization gives, independently, a Hurwitz field of degree three on $\R^{18}$. The dimensions $3\leq n\leq13$ remain open.

We consider the persistence for invariant graphs of twist maps that exhibit the strongest possible dynamics, namely those real-analytically conjugate to rigid rotations, under Gevrey-$\gamma$ ($\gamma\in [0,1]$) perturbations. By enhancing the regularity of the perturbation itself, we show that invariant graphs with the strongest dynamics can persist even when the size of the perturbation and the constraints on the frequency go beyond the requirements of classical KAM theory and the theory of normally hyperbolic invariant manifolds. The proofs of these results are based on a parameterized direct KAM method.

We develop a port-Hamiltonian framework for coupled rigid/flexible multibody systems. The rigid dynamics may be nonlinear and subject to configuration and velocity constraints, while the flexible components are described by linear port-Hamiltonian partial differential equations on one-dimensional spatial domains. The subsystems interact through boundary or distributed ports. On the flexible side, the differential operator and its domain remain fixed, whereas state dependence enters only through finite-dimensional coupling components of the Dirac structure. We show that, under a natural surjectivity condition, the port-Hamiltonian interconnection with a modulated Dirac structure of a finite-dimensional rigid subsystem again yields a modulated Dirac structure. The Hamiltonians of the subsystems add, while the internal coupling powers cancel. The framework is illustrated by a planar moving Euler–Bernoulli beam and a slider–crank mechanism with a flexible connecting member.

Let $G$ be a connected real semisimple linear algebraic group with finite center and no compact factors, let $X$ denote its associated Riemannian symmetric space, and let $h_{\mathrm{vol}}(X)$ be the volume growth entropy of $X$. We show that there exists an $\epsilon = \epsilon(G) > 0$ so that the following holds: if $\Gamma < G$ is a discrete subgroup with critical exponent greater than $h_{\mathrm{vol}}(X) - \epsilon$, then $\Gamma$ is Zariski dense in $G$. If $G$ is further assumed to be isomorphic to one of the isometry groups of the real, complex, or quaternionic hyperbolic spaces, we determine the smallest possible value of the critical exponent to guarantee Zariski density of the associated subgroup (in other words, the largest possible value of $\epsilon = \epsilon(G)$). In the setting of discrete subgroups of real semisimple Lie groups with no compact factors, this generalizes Borel's Density Theorem for lattices. The key ingredient is input from the theory of unitary representations, particularly the recent work of Benoist–Liang (which was inspired by earlier work of Benoist–Kobayashi) on general temperedness criteria for the quasi-regular representation of $L^2(G/H)$ for any closed subgroup $H$ of $G$.

The classical normally hyperbolic invariant manifold theorem asserts that a \(C^1\) normally hyperbolic invariant manifold persists under sufficiently small \(C^1\) perturbations. For a family of standard-like dissipative twist maps with Jacobian determinant \(\lambda\in(0,1)\), we prove that \(C^1\) perturbations of Lipschitz seminorm at most \((1-\sqrt{\lambda})^2\) preserve a unique \(C^1\) invariant graph, including at equality. Under the strict bound, this graph is normally hyperbolic. Conversely, for every prescribed \(a^*>(1-\sqrt{\lambda})^2\) and every rational frequency, we construct a \(C^\infty\) perturbation of \(C^1\)-norm exactly \(a^*\) whose unique invariant graph is Lipschitz but not differentiable everywhere. We also give an analytic perturbation at the critical norm whose invariant graph is \(C^1\) but not normally hyperbolic. Thus the closed bound for \(C^1\) regularity and the strict bound for normal hyperbolicity are both sharp.

The main objective of this work is to investigate the traveling wave solution and dynamic characteristics of the (2 + 1)-dimensional seventh-order Caudrey-Dodd-Gibbon-KP (sCDG-KP) equation. Applying the ($\frac{G^{\prime}}{G^{\prime}+G+A}$) method, we examine the exact solution of the (2 + 1)-dimensional seventh-order Caudrey-Dodd-Gibbon-KP (sCDG-KP) equation by altering it into a reduced ODE via a suitable wave transformation. Graphical representations, such as 2D, 3D, and a heat map of the ascertained solution, are present to facilitate comprehension of the empirical relevance of the obtained solutions. As a result, we acquired a bright and anti-kink soliton solution. Next, we alter the ODE into a 2D system of equations to analyze the dynamical behavior of the reduced system via bifurcation analysis, phase portrait, and attractor analysis. During this process, we portray the graphical visualization of the bifurcation phase portrait, 2D phase portrait, 3D phase portrait, time series, chaotic attractor, sensitive analysis, fractal dimension, recurrence plot, and power spectrum of the dynamical system.

We study a two-dimensional semiclassical Schrödinger operator whose potential admits one reflection symmetry. Under a rational independence assumption on the harmonic frequencies and a nonvanishing condition \(a_{30}\ne0\), we show that the first two layers of the quantum Birkhoff normal form, together with the sign of \(a_{30}\) and the transverse data \(\{a_{1,2k}\}_{k\ge1}\), determine the Taylor series of the potential at the bottom of the well. The proof is constructive: the \(\hbar^2\)-layer gives a triangular recursion for odd-degree terms, while the classical layer recovers even-degree terms from their resonant projections.

We propose a mathematical model that represents the influence of an external actor (influencer) on a network of actors (influenced). The model is an adaptation of control systems within the family of formulations inspired by the Kuramoto Model of synchronisation. Capturing influence as a capacity to affect the character or behaviour of another, we study an external node's ability to influence the synchronisation of the Kuramoto system while its global behaviour is pulled to the external node's frequency and away from its natural mean frequency. In our work, stochastically generated dynamical weights assigned to the links between the network and the external influencer whereby links from one system to the other are assigned via one-sided heavy tail noise, generated by the Gamma distribution. We perform numerical experiments to examine transition points in the ability of the external node to alter the behaviour of the influenced network; either to disrupt synchronisation or to drive it to collective frequencies determined by the external node. We examine the dependence of transition points on the external node's frequency, where too ambitious a driving frequency fails to influence the system while retaining a synchronised state, and reducing achieves a state of synchronisation at a frequency shifted from the mean natural frequency. We also look at the analytic approximation for the system close to synchronisation and fragmentation to understand its behaviour around this limit.

We investigate the emergence of traveling front solutions in a spatial SIRS epidemic model with diffusion acting on the infected population. The model exhibits a natural slow-fast structure due to the presence of a small parameter governing the loss of immunity, which induces a separation of scales in the dynamics. Using a traveling wave reduction, the PDE system is transformed into a singularly perturbed system of ODEs, which we analyze within the framework of Geometric Singular Perturbation Theory. In the singular limits, we study the fast excursions governed by the layer problem, and the slow evolution close to the critical manifold. In particular, we identify an entry-exit mechanism tracking the transitions between slow and fast regimes, and derive a quantitative characterization of the entry-exit dynamics. Numerical simulations of the full system confirm the validity of the proposed geometric picture. The traveling front is shown to consist of a concatenation of local, fast, and slow segments, in agreement with the theoretical analysis.

Echo State Networks (ESNs) offer an efficient framework for temporal prediction, but their randomly initialized reservoirs are often over-parameterized and dynamically redundant. Existing pruning methods largely rely on static connectivity or activation statistics, which may overlook neurons that shape input-driven state transitions. We propose Dynamical Mode Pruning (DMP), a reservoir pruning method that ranks neurons by their contribution to dominant transition modes obtained from a trajectory-averaged Jacobian Gramian. DMP removes low-impact units and retrains only the readout. Experiments on chaotic and real-world time-series benchmarks show that DMP improves or preserves forecasting accuracy while reducing redundant reservoir components. Our results suggest that dynamical influence is a useful criterion for reservoir refinement beyond static structural importance alone.

In this paper, we investigate the relationships between the $\beta$-$n$-sensitivity and other forms of sensitivity in discrete dynamical systems. And we present some different sufficient conditions (equivalent condition) to be (strongly) $\beta$-$n$-sensitive. These results improve and extend some existing ones. Moreover, we introduce and study the semi-strong $\beta$-$n$-sensitivity and (strong) multi-$\beta$-$n$-sensitivity with respect to $\alpha$. Besides, we proved that strongly multi-transitivity implies strongly multi-sensitivity, which improves some existing ones. Finally, we obtain that (semi-)strongly $\beta$-$n$-sensitivity is preserved under iterations but $\beta$-$n$-sensitivity fails.

Near a planar fast–slow fold, a local maximal canard is selected by the parameter at which the attracting and repelling slow manifolds meet. We compare this threshold for a physical flow and a Runge–Kutta map, using actual invariant manifolds on a common fold section. An order-two Runge–Kutta method has two independent order-three rooted-tree defects. Both enter the pointwise one-step residual, but Gaussian fold transport acts on their leading contribution by $(\alpha,\beta)\mapsto-3\beta\Xi(J)/8$. Here $\Xi(J)$ is an explicit functional of the fold jet. Thus the singular passage filters the numerical defect space: it annihilates the bushy-tree direction and can retain only the chain-tree direction. For compact analytic classes of affinely normalizable folds and every fixed compact, uniformly finite-stage family of real Runge–Kutta methods of order at least two, the actual flow and map splittings obtained from independent continuations have unique roots whose displacement satisfies a uniform absolute estimate throughout the full small-step rectangle. Whenever the step-independent, exponentially small selection ambiguity is $o(h^2\varepsilon^2)$, the joint-fold law is $\lambda_{\mathrm{RK}}-\lambda_{\mathrm{flow}}=K_\theta(J)h^2\varepsilon^2+o(h^2\varepsilon^2)$. Fold-matched continuations additionally give ordinary second-order convergence as $h\to0$ with $\varepsilon$ fixed. The leading joint-fold bias therefore vanishes under $b^T A c=1/6$, without classical third order. This is cancellation in one nonlinear observable, not an increase in trajectory order or, in general, in fixed-$\varepsilon$ threshold order. Affine covariance transfers the coefficient to physical fold germs, and a van der Pol invariant-graph computation illustrates the sign change, cancellation, and fixed-$\varepsilon$ convergence.

Spectral submanifold (SSM) reduction has emerged as a mathematically principled route to reliable nonlinear reduced-order models, capturing dynamics beyond the reach of linear techniques such as Dynamic Mode Decomposition (DMD). The computation of SSMs, however, remains computationally expensive, particularly for high-dimensional systems. In this work, we introduce equivariant spectral submanifold (eSSM) reduction, a novel extension of the SSM framework that explicitly incorporates symmetries of the full-order model into the reduction process. We establish the mathematical foundations of this approach by showing that SSMs are naturally equivariant submanifolds and that the associated charts and reduced dynamics inherit the appropriate induced group actions. Building on this framework, we develop a novel equivariant SSM reduction algorithm that exploits these symmetries to achieve substantially faster computations while also improving model robustness. We demonstrate the advantages of this approach on several benchmark problems including a test from the Common Task Framework for Science.

Control theory and dynamical systems are closely intertwined fields. Pontryagin's maximum principal offers a strong connection by providing a constructive way to synthesize feedback control laws via lifting the control system to a (Hamiltonian) dynamical system. Feedback laws can then be encoded as invariant manifolds of this induced dynamical system - under the condition that these manifolds project diffeomorphically back to the base control system. While feedback stabilization is impossible for many control systems, the above procedure can still be carried out. In this setting, the invariant manifold no longer projects diffeomorphically which results in the emergence of caustics. This paper offers an overview of the above connection by translating target sets from controls to isotropic submanifolds in symplectic geometry and associates feedback controllability to singularities of the induced invariant manifolds. Multiple low-dimensional examples are included to elucidate the theory.

In 2010, Levine found that the activity functions of parallel chip-firing games on the complete graph $K_n$ converge to a devil's staircase pattern: adding chips causes activity to progress through open intervals in which it is locally constant. In 2022, Bu, Choi, and Xu improved on an earlier bound by Kominers and Kominers to show that there exists a strict lower bound below which all games have activity $0$, and a strict upper bound above which all games have activity $1$. They thereby generalized the bottom and topmost rungs of the devil's staircase to all graphs. In 2024, Ji, Li, and Wang conjectured that a similarly general bound exists for games with activity $\tfrac12$. We use GPT-5.6-Sol to prove this conjecture, unifying existing results for trees, cycles, complete graphs, and complete bipartite graphs. This generalizes the middle rung of the devil's staircase.

Homeostasis is a biological phenomenon in which a function of the state of the system remains approximately constant as an input parameter varies. Recent mathematical work has developed a framework for studying homeostasis using methods from singularity theory. In this framework, the requirement that the function of the state remain approximately constant is replaced by the condition that the derivative of the function with respect to the input parameter vanishes at an isolated point. This zero-derivative condition is called infinitesimal homeostasis. Much of the existing theory concerns steady states. In this paper, we develop an analogous theory for oscillatory systems in which the homeostatic quantity is the period of a stable periodic solution. We refer to this behavior as period homeostasis and to the corresponding zero-derivative condition as infinitesimal period homeostasis. Experimental studies in cyanobacteria and cultured human cells suggest that circadian rhythms can disappear through Hopf bifurcation as temperature decreases to a critical value, with the oscillation amplitude tending to zero. Motivated by this evidence, we focus on stable periodic solutions arising from nondegenerate Hopf bifurcation. We show that bifurcation theory and singularity theory can be used to find period homeostasis in parameterized ODE models. Our results connect the onset of oscillations with homeostasis of the period and yield a geometric description of parameter space that locally organizes the qualitative behavior of solutions of the ODE.

Phase reduction provides a principled relationship between high-dimensional oscillator dynamics and low-dimensional phase models such as the Kuramoto model. However, connecting analytical results derived from the Kuramoto framework to experimental data remains an open challenge, as many physical platforms violate the weak-coupling assumptions underlying standard phase reduction. Here, we present an experimental platform based on electronic quadrature oscillators whose phase dynamics are, under certain assumptions, equivalent to the Kuramoto model. Crucially, this equivalence is achieved for both weak and strong coupling through a non-diffusive coupling circuit, and the design approach supports arbitrary (weighted and directed) network topologies while preserving scalability and oscillation regularity. Across different coupling schemes, we demonstrate that the platform reliably reproduces phase transitions predicted by the Kuramoto model for both global and cluster synchronization. This cost-effective platform enables systematic experimental validation of analytical and numerical results for synchronization in complex networks.

We study the composition of temporal ergodic averaging with spatial averaging over shrinking metric balls, and determine sharp Orlicz endpoints for the corresponding unrestricted joint limit. For consecutive Birkhoff averages normalized by $N\Lambda_q(N)$ ($q\ge 0$), the sharp Orlicz endpoint is $L\log_{q+1}L$. In particular, the ordinary case $q=0$ yields an $L\log L$ local joint convergence theorem under the Lebesgue differentiation property alone, while $L^1$ fails even on the Euclidean interval, answering two questions of Young. The same $L\log_{q+1}L$ endpoint holds for prime averages for every $q\ge 1$. For arbitrary time sequences, $L\log_qL$ always suffices at the same normalization $N\Lambda_q(N)$ ($q\ge 1$), and this endpoint is sharp in the Orlicz sense for fixed-base exponential sequences $k^n$ ($k\ge 2$) and for sequences with polynomial ratio separation, including $n!$. The positive results rest on a local stability principle: under the ball Lebesgue differentiation property, pointwise temporal convergence lifts to the local joint limit whenever the associated temporal maximal function admits an $L^1$ majorant. The additional logarithm in the regular-time case comes from lifting restricted logarithmic maximal estimates from sets to general functions. The arbitrary-sequence theorem uses a dyadic decomposition instead. The lower bounds are local $\infty$-sweeping out constructions, obtained by weighted local sweeping out for polynomial-growth regular times, by residue constructions for polynomially ratio-separated sequences, and by digit constructions for fixed-base exponentials.

A classical result by Neubert, Caswell and Murray states that reactivity of a spatially homogeneous equilibrium is a necessary condition for diffusion-driven (Turing) instability. In this work, we investigate whether the same conclusion remains valid in the presence of chemotaxis. We consider a general reaction–diffusion system coupled with a chemotactic flux and establish necessary conditions for asymptotic instability. We show that the classical requirement of reactivity can be relaxed when the chemotactic contribution is sufficiently strong. In particular, while reactivity remains necessary for Turing instability, it is not a necessary condition for chemotaxis-driven instability. From a computational viewpoint, we extend the matrix-oriented formulation developed for reaction–diffusion systems to the more general class of reaction–diffusion–chemotaxis models. The chemotactic transport term is discretized in a form compatible with the matrix-oriented approximation of the diffusion operator, yielding an efficient numerical framework for the simulation of chemotaxis-driven pattern formation. A geometric interpretation of the instability region is presented, highlighting the distinct roles played by diffusion and chemotaxis. The theoretical and numerical developments are illustrated through two representative examples: a chemotaxis-extended Schnakenberg model, showing how chemotaxis modifies classical Turing patterns, and a predator–prey model, demonstrating that chemotaxis alone can induce pattern formation in the absence of both reactivity and diffusion-driven instability. These results reveal a fundamental difference between diffusion-driven and chemotaxis-driven mechanisms of spatial self-organization and provide new theoretical and computational insights into the role of non-symmetric transport processes in biological pattern formation.

In bioreactor experiments, budding yeast can manifest stable metabolic oscillations. In some instances these oscillations involve the cell cycle and are associated with a dynamical phenomenon called temporal clustering. Since yeast divide asymmetrically, mother cells may be able to divide sooner than the smaller daughter cells. We show that asymmetric division plus positive feedback can result in stable temporal clustering. In this scenario, the cells self-organize into $p$ clusters of mother cells and $q$ clusters of daughter cells, $p \le q$. In simple numerical simulations of a population model with asymmetric division and positive feedback, we show that $p:q$ periodic arrangements form spontaneously from random populations of cells. Our main result is that these structures can be stable.

We study Braess-type paradoxes in coupled oscillator networks where the addition of an edge to the network leads to an increase in the critical coupling required for the existence of a stable phase-locked synchronized solution. By introducing a continuous edge-weight parameter and constructing an augmented system that allows application of the Implicit Function Theorem, we derive an explicit first-order formula for the sensitivity of the critical coupling, $K'_c(0)$, to small changes in the network structure. This formula reveals that an edge is locally Braess if the ordering of the oscillators being coupled is opposite the ordering of their components in the critical eigenvector at the saddle-node bifurcation, providing a simple criterion for Braess paradox to occur. We then employ this local approximation as a predictor for the occurrence of Braess paradox when the edge is fully incorporated in the network. This prediction is tested numerically in several classes of random networks, demonstrating both high predictive fidelity and the nonlinear limitations of the local approximation. We also examine several explicit examples, highlighting possible motifs by which Braess-type paradoxes occur. Together, these results establish a predictive framework for understanding how incremental changes in network topology can produce counterintuitive synchronization outcomes.