Papers

Pro-nilsystems arise naturally in ergodic theory through the study of multiple ergodic averages. The Host-Kra structure theorem implies that every factor of an ergodic $s$-step pro-nilsystem is again an $s$-step pro-nilsystem. No proof of this fact independent of the structure theorem was previously known. In this thesis, we give such a proof, using only arguments intrinsic to nilsystems. The main new ingredient is a local rigidity result for self-joinings of nilsystems: every ergodic self-joining sufficiently close to the diagonal joining is necessarily the graph joining of an automorphism. We prove this geometrically by showing that nilmanifolds cannot contain arbitrarily small nontrivial subnilmanifolds. We also obtain a direct proof of the analogous factor-closure result for transitive topological pro-nilsystems, previously known as a consequence of the Host-Kra-Maass topological structure theorem. Finally, we give a detailed account of Tao's deduction of a weak structure theorem directly from the Green-Tao-Ziegler inverse theorem for the Gowers norms. Combined with the factor-closure of pro-nilsystems, this yields a new proof of the Host-Kra structure theorem. The principal new results of this thesis also appear in a joint paper with Asgar Jamneshan (arXiv:2604.02089). This thesis provides a more self-contained treatment, including the necessary background on dynamical systems and nilsystems, as well as full details of Tao's argument.

It is known that generalized hyperbolicity implies the shadowing property for invertible bounded linear operators on a Banach space. Whether the converse holds has been a central open problem in linear dynamics and has been conjectured to have a positive answer. We show that this conjecture fails on general Banach spaces by constructing a counterexample, whereas it holds on separable Hilbert spaces. The distinction is explained by the gap that may occur on Banach spaces between surjectivity and right invertibility, a gap that disappears on Hilbert spaces. The main ingredients of the proofs are a recent spectral characterization of shadowing in terms of the surjective spectrum and a new characterization of generalized hyperbolicity in terms of right resolvent functions near the unit circle.

We prove level-2 large deviation upper bounds for potentials on countable Markov shifts and for suspension semi-flows over countable Markov shifts. For strongly positive recurrent potentials, we establish equidistribution of weighted empirical measures toward the corresponding equilibrium state. We then apply these results to interval maps, obtaining, in particular, equidistribution and large-deviation estimates for measures supported on boundary points. For the Gauss map, this yields equidistribution results on rational numbers, including a theorem of David and Shapira.

The titular object of this paper is an analogue of the Mandelbrot set over the 3-dimensional real algebra whose multiplication is the bilinear extension of the rock-paper-scissors operation. In order to study the dynamics of the mappings $x\mapsto x^2+c$ in this setting, a notion of holomorphy for functions on general finite-dimensional real algebras is introduced. Under a mild assumption it is shown that for such algebras holomorphy is always equivalent to solving a finite system of linear first-order PDEs generalizing the Cauchy-Riemann equations. Code is provided for generating animations of these fractals and for exploring them in a video game format.

We introduce outer contact billiards as an odd dimensional counterpart to outer symplectic billiards. Until now, outer billiards have only been considered in even dimensional symplectic vector spaces. By projectivizing outer symplectic billiards, we obtain outer contact billiards, where the affine midpoint condition descends to its projective analog, namely harmonic conjugation. We prove that outer contact billiards generate contactomorphisms. For quadratic surfaces in $\mathbb{RP}^3$, we show that the correspondence is completely integrable: its domain is foliated by invariant quadrics and, on each leaf, the dynamics is determined by the iteration of an explicit linear transformation. We also establish two rigidity results for periodic trajectories: outer contact billiards admit no 3-periodic orbits and, among quadratic tables, only one admits 4-periodic orbits.

A finite-dimensional two-field system for divergence-free edge cochains is introduced. Its MHD-type designation refers only to a quadratic exchange pattern and exact total-energy cancellation; it is not a physical MHD discretization. A general cancellation class is separated from a corrected explicit realization: the anticommutator $D(a)J+JD(a)$ is skew-symmetric for diagonal $D(a)$ and skew-symmetric $J$, and its projected bilinear map has the required trilinear antisymmetry. The central result is a Hodge coercivity criterion: the full divergence-free space admits the Poincaré-type estimate needed for dissipativity if and only if its harmonic $1$-cochain space is trivial. Under this condition, global existence, an exact energy identity, an absorbing ball, and a compact global attractor follow. When harmonic modes are present, a harmonic-decoupled interaction class yields invariant harmonic affine fibres and fibre-wise attractors. Deterministic disk, annular, and two-hole examples illustrate the spectral criterion, energy law, and distinction between general harmonic exchange and harmonic-fibre invariance.

We prove the existence of periodic solutions for Hamiltonian differential inclusions under growth conditions involving a G-function.

We show that the existence of a growth gap for infinite-index subgroups of a given finitely genrated group can depend on the finite generating set. More precisely, for any irreducible lattice $\Lambda$ in a higher rank semisimple Lie group $G$ with Kazhdan's property (T), the group $\Lambda \times \Lambda$ admits one finite symmetric generating set with a growth gap and another without a growth gap. We also prove that the growth gap can be made arbitrarily small. In contrast, for a non-elementary hyperbolic group the existence of a growth gap is independent of the finite generating set.

Noise-activated switching between coexisting stable states is a fundamental mechanism underlying stochastic dynamics in systems ranging from chemical reactions to neural networks. While this phenomenon is well understood for stationary attractors, it remains largely unexplored for limit cycles, whose periodic motion cannot be described by a static potential landscape. Here we experimentally demonstrate activated switching between two coexisting limit-cycle attractors in a driven nonlinear system of coupled resonators. Specifically, we introduce controlled fluctuations to directly observe the rare stochastic transitions between two limit cycles and measure their dependence on noise intensity and driving strength. The measured switching rates are well described by a large-deviation theory, which replaces the conventional activation barrier by the action along the most probable transition path. Our results extend the concept of activated dynamics from stationary to limit-cycle attractors and establish a framework for modeling stochastic transitions between limit cycles in driven-dissipative systems.

Let $X$ be a vector field on an open set of $\mathbb{R}^n$ with $X(p)=0$ and Jacobian $J=DX(p)$ of rank $n-1$ having $0$ as an eigenvalue of algebraic multiplicity two. Let $q_0$ span $\ker J$ and let $e_k(A)$ denote the sum of the principal $k\times k$ minors of $A$. Under the usual hyperbolicity assumption on the transverse block, we prove that the quadratic coefficients $a,b$ of the Bogdanov–Takens normal form on the centre manifold are $a=-\frac{1}{2}\frac{D_{q_0}e_n}{e_{n-2}}$ and $b=\frac{D_{q_0}e_{n-1}}{e_{n-2}}-\frac{e_{n-3}D_{q_0}e_n}{e_{n-2}^2}$. The underlying spectral identity requires only invertibility of the transverse block and remains valid without hyperbolicity. Both coefficients arise as the lowest-order terms of a generating identity for the first-order spectral jet of $DX$ along $\ker J$; all higher coefficients depend on the transverse block. We prove that this dichotomy is sharp. The planar formulas $a=-\frac{1}{2}D_{q_0}\det$ and $b=D_{q_0}\operatorname{tr}$ are recovered when $n=2$. We also obtain a coordinate-free nondegeneracy test and a geometric interpretation in terms of the transversality of the kernel line to two invariant hypersurfaces. A self-contained Wolfram Language notebook accompanies the paper and verifies the results symbolically.

The projective spectrum of a tuple $A=(A_0,A_1,\dots,A_n)$ with elements in a unital Banach algebra $\mathcal{A}$ is the collection of $[z_0:z_1:\cdots:z_n]\in \mathbb{P}^n$ such that $z_0A_0+z_1A_1+\dots+z_nA_n$ is not invertible in $\mathcal{A}$. For the tuple $A_{\rho}=(\rho({\bf e}),\rho({\bf a}),\rho({\bf b}))$, where ${\bf e},{\bf a},{\bf b}$ are the three states of the automaton generating the Basilica group $\mathcal{B}$ and $\rho$ is the Koopman representation, a dynamical map $F$ preserving the projective spectrum of $A_{\rho}$ is established. Further, a substantial common subset of the Julia set of $F$ and the projective spectrum of $A_{\rho}$, as well as a large common subset of the Fatou set of $F$ and the projective resolvent set of $A_{\rho}$ is identified. The result provides partial support for the conjecture of equality of the Julia set of $F$ and the projective spectrum of $A_{\rho}$.

We consider evolution of piecewise linear curves with boundary by linear hyperbolic curvature flows. Given appropriate boundary behaviours we find corresponding self-similar solutions. More generally, given any specific boundary trajectories we write down general solutions using finite Fourier expansions. Finally we consider the problem of evolving one piecewise linear curve to another using nonhomogeneous linear hyperbolic flow.

The classical theory of nonlinear observer error linearization—the nonlinear observer canonical form—has attracted sustained attention since its inception in the 1980s. Under the existing theoretical framework, the verification and construction for a nonlinear system to achieve observer error linearization admits a systematic, efficient implementation, severely limiting the applicability of the theory to high-dimensional systems. To address this issue, inspired by the definition of high-order fully measured systems, this paper proposes the Pascal-Hessian condition for single-output systems. This condition equivalently converts the necessary and sufficient condition for observer error linearization into a structural test on the Hessian matrix of the nonlinear term in high-order fully measured systems: the coefficients in the upper-left corner of Hessian matrix form a Pascal's triangle, while the lower-right corner vanishes identically. Simultaneously, we provide explicit integral formulas for all output-dependent univariate functions in the canonical form, eliminating the need to solve partial differential equations. Compared with the existing theory, our method reduces the computational complexity of condition verification from $O(n^4)$ to $O(n^2)$, and replaces the intricate process of solving partial differential equations with explicit indefinite integral for canonical form construction. We further extend the result to multi-output systems with equal observability indices. Beyond its computational advantages, this work reveals a fundamental structural connection between the nonlinear observer canonical form and Pascal's triangle—a link that has remained unnoticed since the inception of the theory in the 1980s. Numerical examples validate the effectiveness of the proposed method.

We show that small planar perturbations of families of unimodal maps have maximal (equal to the intersection of the iterates of their trapping region) attractors at parameters near homoclinic tangency of one of the fixed points. The results hold for general unimodal maps exhibiting certain hyperbolic properties. Hence, admissible families considered include Hénon-like families, Lozi-like families, or perturbations of border collision normal forms. Therefore, our results generalize results of Viana in the case of Hénon-like families and Glendinning and Simpson in the case of border collision normal forms.

Let $X$ be a Cantor space and let $Q\subset M_{fc}(X)$ be a compact Choquet simplex of atomless full-support probability measures. We introduce the associated Affine Evaluation Map (AEM), which assigns to each clopen set $A\subset X$ the affine function \[ \widehat A(\mu)=\mu(A),\qquad \mu\in Q, \] and study the geometric subset condition obtained by comparing these evaluation functions pointwise on $Q$. For a good geometric AEM we construct the ordered group \[ G_Q=C(X,\mathbb Z)/N_Q, \qquad N_Q=\left\{f\in C(X,\mathbb Z): \int f\,d\mu=0\ \text{for every }\mu\in Q\right\}, \] and show that it is a simple dimension group whose normalized state space is canonically $Q$. We prove that its order interval $[0,u]$ is precisely the clopen scale and that \[ J_Q=N_Q, \] where $J_Q$ is generated by the elementary relations $\mathbf 1_A-\mathbf 1_B$ with $\widehat A=\widehat B$. We also show that the full stabilizer $\mathcal H_Q$ has invariant-measure simplex exactly $Q$. Using the clopen-scale property and the Herman–Putnam–Skau realization theorem, we obtain a dimension-group proof that every good geometric AEM is realized by a minimal Cantor homeomorphism $T$ with $Q=M_T(X)$. For a Cantor minimal system we identify $G_Q$ with the classical dimension group modulo infinitesimals. We further distinguish goodness, fairness, ergodicity, and minimality of the measure stabilizer by explicit examples. Finally, we apply the AEM framework to minimal Cantor actions of countable amenable groups. If $Q=M_G(X)$, then $Q$ canonically defines a proper geometric AEM, and we clarify which parts of the preceding theory depend only on $Q$ and which are specifically $\mathbb Z$-dynamical. In particular, $Q$ is good if and only if there exists a minimal homeomorphism $T$ of $X$ such that \[ M_T(X)=M_G(X). \]

A fundamental problem in the algebraic study of biochemical reaction networks is to characterize degeneracy. Two-dimensional zero-one networks, where each reactant appears with stoichiometric coefficient zero or one, constitute the smallest biologically relevant class capable of exhibiting degeneracy. In this paper, we provide a complete classification: a two-dimensional zero-one network without trivial species is degenerate if and only if it is a consistent subnetwork of a species refinement of one of two prototypical networks, namely the complete paired-exchange network or the catalytic-pair conversion network. Equivalently, in matrix-theoretic terms, degeneracy is determined entirely by the row patterns of the stoichiometric and reactant matrices, and can be verified by purely structural inspection without computation. Remarkably, every such degenerate network has a steady-state system consisting entirely of binomials, so its steady-state variety is toric. These results also bear on absolute concentration robustness: since any network exhibiting this property necessarily contains a degenerate subnetwork, the minimal degenerate networks characterized here serve as fundamental building blocks for constructing such networks.

We give a finite symbolic characterization of the strong topological Rokhlin property in terms of globally realizable finite pattern systems. We use this characterization to prove finite-index ascent: if $H\leq G$ has finite index, $G$ is finitely generated, and $H$ has STRP, then $G$ has STRP. Consequently every finitely generated virtually free group has STRP, whereas among countable locally virtually free groups STRP holds exactly for the finitely generated ones. In the compact coding space $\operatorname{NSub}(F_\omega)$ of countable groups, the STRP locus belongs to ${\boldsymbol \Pi}^0_4$, is ${\boldsymbol \Sigma}^0_3$-hard; the locus of finitely generated virtually free groups is ${\boldsymbol \Sigma}^0_3$-complete. For quotients $F_\omega/N$ with $N$ recursively enumerable, every projectively isolated subshift has decidable finite tuple-language and contains an $N$-recursive configuration. This yields effective SFT obstructions to STRP, including subgroup and direct-product obstructions, and implies that $\mathrm{SL}_n(\mathbb Q)$ does not have \STRP for any $n\geq2$.

A theorem of Dougherty–Jackson–Kechris states that any equivalence relation generated by a single Borel function is hypersmooth. A well-known open problem is whether this can be generalized to equivalence relations generated by countable families of pairwise commuting Borel functions. We give an affirmative answer in the case where the functions are bounded-to-one. This generalizes the theorem of Gao–Jackson on Borel actions of countable abelian groups.

We compute the codimension-one stable norm for a natural class of cohomogeneity-one metrics on tori. In every dimension $n\ge3$, the formula yields smooth nonflat metrics for which each primitive codimension-one homology class is represented by a foliation of calibrated tori, giving a negative answer to a question of Bangert. On $\mathbb T^3$, we construct an infinite-dimensional family of nonflat metrics whose codimension-one stable norm agrees exactly with that of the unit cubic flat torus and whose total volume is fixed. An explicit two-parameter subfamily contains pairwise non-isometric metrics. These examples also show that the Euclidean-stable-norm-and-volume data are not locally injective near the cubic flat metric. Conversely, among smooth metrics on $\mathbb T^3$ admitting a free isometric circle action and having the cubic Euclidean codimension-one stable norm, we prove that volume is at most one, with equality only for the cubic flat metric up to an isometry isotopic to the identity.

The variational principle is a key tool in the study of invariant measures for chaotic dynamical systems. In recent times, dynamicists have developed techniques in random and nonstationary systems to better model real-world phenomena. Here, we prove a relative variational principle for a class of expanding iterated function systems. In particular, we use a nonstationary Ruelle–Perron–Frobenius theorem to show that the marginal entropy given an ergodic invariant measure on the symbolic base equals the average topological entropy along fibers in the induced skew product.

We study quasi-irreducibility, expansion, and their cotangent duals for random $C^1$ (local) diffeomorphisms on compact invariant sets of a Riemannian manifold. Quasi-irreducibility is defined through stationary lifts to the projective tangent bundle: every lift must realize the Furstenberg–Kifer formula for the top Lyapunov exponent. We prove that, for ergodic stationary measures, this is equivalent to the absence of equators, namely non-random invariant subbundles on which the top exponent drops. Under simplicity of the first Lyapunov exponent for every stationary measure, quasi-irreducibility is also equivalent to vertical mostly contraction, contraction on average, a vertical spectral gap, and uniqueness of stationary projective lifts. We then apply these criteria to continuity of the top Lyapunov exponent and to expansion on average. In particular, expansion is characterized by positivity of the top Lyapunov exponent on all non-random invariant subbundles; under quasi-irreducibility, it is equivalent to positivity of the top Lyapunov exponent for every stationary measure. Dual statements hold for coquasi-irreducibility, coequators, coexpansion, and continuity of the bottom Lyapunov exponent. We also describe the interplay between expansion and coexpansion and extend the formalism to Grassmannian bundles, obtaining higher-dimensional versions controlling intermediate sums of Lyapunov exponents.

In this article, we consider an energy-critical quintic wave equation on a bounded domain $\Omega\subset\mathbb{R}^3$ with nonlinear and nonlocal Kelvin–Voigt damping of the form $-\|\nabla u_t\|_{L^2(\Omega)}^\alpha\Delta u_t$, where $\alpha\in\mathbb{R}_+=[0,\infty)$. Under suitable hypotheses on the quintic source term, we establish the well-posedness of the problem and investigate its long-time dynamics in the natural energy space $\mathcal H=H_0^1(\Omega)\times L^2(\Omega)$. For every $\alpha\in\mathbb{R}_+$, we show that the associated dynamical system $(\mathcal H,S^\alpha(t))$ is gradient and dissipative, and we prove a stabilization estimate that yields asymptotic smoothness and, consequently, the existence of a compact global attractor $\mathcal A_\alpha$. The same estimate provides an upper bound for the Kolmogorov $\varepsilon$-entropy of $\mathcal A_\alpha$ and, in the limiting case $\alpha=0$, reduces to a quasi-stability inequality, which implies that $\mathcal A_0$ has finite fractal dimension. Furthermore, we prove that the family $\{\mathcal A_\alpha\}_{\alpha\in\mathbb{R}_+}$ is uniformly bounded in the higher-regularity space $\mathcal H_1=(H^2(\Omega)\cap H_0^1(\Omega))\times H_0^1(\Omega)$. Finally, we establish the upper semicontinuity of $\{\mathcal A_\alpha\}_{\alpha\in\mathbb{R}_+}$ at $\alpha=0$, showing that the attractors associated with the nonlinear and nonlocal Kelvin–Voigt damping converge to the global attractor of the limiting problem with classical linear Kelvin–Voigt damping.

Trust assessment is a fundamental component of cooperative and connected vehicle systems. However, existing approaches operate primarily at the level of individual vehicles, making it difficult to reason about trust evolution across road segments. In this paper, we propose a spatio-temporal trust-field framework that aggregates microscopic vehicle-level trust into a continuous representation over space and time. The trust field is formally defined on road segments. We conducted simulation-based experiments using synthetic trajectories generated under controlled conditions, enabling analysis of trust-field behavior in simple road scenarios. Beyond theoretical modeling, we study an implication of the trust-field concept: reconstructing the full trust field from sparse roadside-unit (RSU) measurements. We compare (i) a coordinate-based deep learning baseline that learns a generic trust field from sparse samples and (ii) a field-informed deep learning method that treats trust as a latent quantity carried by vehicles and enforces measurement consistency through the aggregation mechanism. The field-informed approach more accurately recovers trajectory-aligned low-trust patterns and yields improved reconstruction error.

We introduce Ordered Diffusion Kernels (ODKs), a novel class of local kernels that can approximate the infinitesimal generator of an arbitrary Itô Stochastic Differential Equation (SDE). ODKs are designed to be applied to data sampled from dynamical systems where little dynamical information is available a priori. The Laplacian of classical diffusion kernels approximates the Laplace-Beltrami operator on the underlying manifold; adjusting the normalisation introduces an advection term that depends on the sampling density; recently, TMDmap generalised this normalisation to target an arbitrary measure, but at the cost of coupling advection to diffusion. More general local kernels can learn arbitrary second-order elliptic operators but are formulated in terms of known velocity fields — making the first step of any analysis a potentially ill-posed inference problem. To formulate ODK, we first relax the problem of potential estimation to the more tractable task of inferring an ordering of the data, which we represent through an ordering function. We prove ODK's Laplacian converges to the infinitesimal generator of a gradient-flow SDE with state-dependent isotropic diffusion, without coupling advection and diffusion. We provide various extensions of ODK to: arbitrary drifts via local ordering functions; anisotropic diffusions via a Strang splitting scheme; multiple ordering functions; and self-tuning bandwidths. In addition, we introduce two loss functions which exploit the structure of ODKs to solve a non-parametric inference problem. We validate this framework on synthetic data from deterministic and stochastic systems, demonstrating accurate recovery of operators, velocity fields, extrinsic curvature, and spatially dependent drift and diffusion coefficients.

We study the critical exponent of the Falconer functional for projective Anosov representations with Lipschitz limit sets and its relationship to the Hausdorff dimension of the limit set. Extending a result of Pozzetti, Sambarino, and Wienhard, we show, under density and irreducibility assumptions on the representation, that this exponent equals the Hausdorff dimension of the limit set. We also show that these assumptions are necessary for the strategy used there and in the present paper. To this end, we study the action of $\mathrm{SO}(p,p)$ on the space of maximal isotropic subspaces of $\mathbb R^{p,p}$ and provide examples of $\mathbf H^{p,q}$-convex-cocompact representations of lattices with virtual cohomological dimension $p$. Besides yielding counterexamples to seemingly innocent claims about irreducible representations in $\mathrm{SO}(p,q+1)$, these examples are of independent interest: they admit equivariant spacelike embeddings of Riemannian symmetric spaces into $\mathbf H^{p,q}$ on which uniform lattices act cocompactly.

Chaotic itinerancy (CI), brought to attention, among others, by K. Ikeda, I. Tsuda and K. Kaneko in the early 1990s, is a phenomenon in which trajectories in a dynamical system experience periods of ordered motion near quasi-attractors interspersed with chaotic transitions between them. Possible maps in which CI was found include coupled map lattices (CML) and globally coupled one-dimensional chaotic maps (GCM). We study such maps using numerical methods and graph algorithms. Specifically, we partition the state space into a finite grid of compact subsets, and we represent the map using a multivalued mapping of grid elements. This mapping can be perceived as a directed graph, with grid elements as vertices and individual mappings between them as weighted edges. This setup provides a coarse view of global dynamics and opens the opportunity for using Markov chains and efficient graph algorithms to study dynamical features. In particular, invariant sets can be found by computing strongly connected components in the graph. Analysis of the transition matrix of the graph makes it possible to find its stationary distribution and to compute local entropy as a measure of expansion or instability in the system. Using these tools, we propose an algorithm for assessing whether a certain map possesses the CI property and show its application to dynamical systems: a globally coupled system of logistic maps and a variant of a CML system for which we conduct computations for a large range of parameters.

In this work, we discover an intriguing Liénard system which is integrable near infinity, integrable near zero and having Ilyashenko-Kotova lips in between. Moreover, we give a necessary and sufficient condition of polynomial Liénard system with center at infinity. In a different direction we improve a lower bound on Hilbert number for Liénard systems. The improvement is due to development of Brudnyi method of calculation of cyclicity of centers near infinity and near zero.

Synchronizing nonlinear dynamical networks typically requires solving matrix inequalities or detailed system models, which fail for large networks. This paper offers a simple fix : a purely graph-theoretic framework using only a single Lipschitz-like bound on the dynamics. Coupling strengths are computed directly from the digraph, bypassing inequality solvers entirely. The method succeeds where existing approaches encounter infeasibility due to connectivity patterns. It examines only $n-1$ directed paths per strongly connected component versus $\frac{n(n-1)}{2}$ undirected paths before, achieving $O(n^3)$ complexity. Results show network connectivity can be exploited to synchronize a large class of nonlinear dynamical networks.

This work proposes a semi-analytical theory for improving orbit design of the GARATÉA-L Brazilian lunar mission. The dynamical model incorporates the lunar gravitational potential up to degree 12 and order 3 harmonics and a third-body perturbation model accounting for the eccentricity and inclination of the Earth's orbit with respect to the Moon. Through Hori's method, short- and medium-period terms are eliminated from the Hamiltonian to derive a first-order theory in mean orbital elements. It is demonstrated that nominal frozen orbits designed under simplified circular and equatorial assumptions for the motion of the third body are not maintained in this more realistic dynamical environment, leading to significant eccentricity growth and potential collisions of the probe with the lunar surface. The inclusion of the third body's inclination and eccentricity breaks the system's axial symmetry, introducing the longitude of the ascending node into the double-averaged variational equations and rendering stationary frozen orbits practically unattainable. Consequently, the problem is addressed through a parametric study around the simplified model's frozen conditions to identify “low-drift" states where eccentricity variations are highly decreased. A new nominal orbit is proposed for GARATÉA-L. Numerical validation against high-fidelity propagation and JPL ephemeris confirms that this new configuration maintains the “low-drift" behaviour throughout the mission, ensuring orbital safety and avoiding the collision predicted by simpler models.

Across complex systems science, networks, spatial structures, populations, spectra, and dynamical flows are often represented by object-level measures such as connectivity, frequency, geometric isolation, spectral strength, or deformation. Determining whether these values contain distinct scales commonly requires a chosen cutoff, a prescribed number of groups, or an assumed prevalence. We introduce Scale Partitioning by Incremental Nested Entropy (SPINE), a deterministic framework that identifies scale structure without these inputs. The central theory reduces an arbitrary positive prefix of ordered measure values to two entropy-effective descriptors: an effective number of contributing components and a characteristic measure scale. This reduction yields an exact finite-size criterion for when the next value produces an entropy-supported scale transition. As the effective size grows, the critical ratio converges to \(e\), which emerges from the balance between increasing diversity and increasing dominance rather than from threshold calibration. The same critical relation also determines the exact aligned perturbation margin of a detected boundary. Successive local transitions produce a data-supported number of scale strata, while a measure with no resolvable separation returns a single stratum. Numerical experiments verify the critical and stability relations to numerical precision across heterogeneous prefixes and recover two, three, and four generated scales without being supplied their number once separation is sufficient. In a double-gyre flow, SPINE identifies a high-expansion structure occupying \(12.85\%\) of the domain without prescribing a retained percentile. The framework applies to geometric, categorical, network, spectral, and dynamical measures.