Trigonometric-Polynomial Disturbances

• Trigonometric-polynomial disturbances are finite sums of sine and cosine functions, often with polynomial coefficients, used to model periodic influences in dynamical systems.
• They influence both deterministic and stochastic models, aiding in stability analysis, spectral root equidistribution, and robust control through internal model techniques.
• Algorithmic methods like Sturm sequences verify nonnegativity and enable efficient evaluation, forming the basis for advanced signal processing and moment closure in complex systems.

A trigonometric-polynomial disturbance is any signal or system input expressible as a finite sum of sines and cosines, or equivalently as a finite Fourier sum, possibly with polynomial coefficients or time-varying arguments. These structures permeate dynamical systems theory, nonlinear control, perturbation theory, spectral analysis, and stochastic processes, being central in the study of both deterministic and stochastic dynamics, disturbance rejection methodologies, spectral root properties, and algebraic algorithms for positivity or moment analysis.

1. Mathematical Definitions and Algebraic Structure

A trigonometric polynomial is a finite sum of functions of the form $$\sum_{k=0}^N a_k \cos(k t) + \sum_{k=1}^N b_k \sin(k t)$$, or, equivalently, a finite sum $\sum_{k=-N}^N c_k e^{i k t}$ with $c_{-k}=\overline{c_k}$ for real-valuedness. In perturbation theory and celestial mechanics, such expressions are typically embedded in "Poisson series," indexed by multi-indices in both angle and action variables:

$$f(\theta, I) = \sum_{k\in \mathbb{Z}^n,\,\alpha\in\mathbb{N}^m} a_{k,\alpha} I^\alpha e^{i\langle k, \theta\rangle}$$

with $\theta\in\mathbb{R}^n/2\pi\mathbb{Z}^n$, $I\in\mathbb{R}^m$ (Giorgilli et al., 2013).

Such trigonometric-polynomial signals (hereafter TP disturbances, Editor's term) are closed under addition, multiplication, and differentiation, and their algebraic manipulation underlies normalization, Lie series, and Poisson bracket computations essential in perturbative and control analyses. The nonzero coefficients are captured via sparse or dense integer-indexed arrays or tree data structures, supporting efficient evaluation and symbolic algebra. In practical settings, an explicit degree bound is maintained, and various orderings are used for index compression and to manage computational complexity (Giorgilli et al., 2013).

2. Analysis and Impact in Dynamical Systems

Trigonometric-polynomial disturbances strongly influence the behavior of both ordinary and stochastic differential equations.

• Deterministic ODEs: For an autonomous system

$$x'(t) = P(x(t)),\qquad P:R^n \to R^n \text{ a trigonometric polynomial}$$

any trajectory $x(\cdot)$ admits a rotation vector $\rho\in\mathbb{R}^n$ such that $x(t) - \rho t$ is uniformly bounded in $t$, and this deviation is a weakly almost periodic function of slope $\rho$. This generalizes the rotation number in the circle map to higher dimensions and nonlinear cases (Oukil, 2019). No small-divisor condition is needed; the finitely-supported Fourier spectrum renders the flow $C^\infty$ and globally Lipschitz.

• Perturbation and Normal-Form Theory: In celestial mechanics, TP series naturally arise as the algebra for constructing normal forms, averaging transformations, or Lie exponentials. The presence of TP disturbances implies the persistence of bounded, oscillatory corrections to drift solutions, with stability times typically exponential in the order of truncation (Giorgilli et al., 2013).
• Moment Dynamics in SDEs: For stochastic systems driven by TP nonlinearities, the dynamics of statistical moments become unclosed (the time derivative of a moment includes higher moments and trigonometric moments). Moment-closure techniques such as derivative matching provide systematic ways to obtain approximate closed systems, treating exponential-monomial expectations by augmenting the state with corresponding moments of $e^{i k x}$ (Ghusinga et al., 2017).

3. Internal Model and Disturbance Rejection

Trigonometric-polynomial disturbances play a foundational role in the internal model principle for robust control and adaptation:

• Modeling via Exosystems: Any TP disturbance can be generated as the output of a block-diagonal linear ODE ("exosystem"). For $\sin(\omega t+ \phi_k)$ components, this system is captured via a companion-matrix realization whose state is constructed from the disturbance and its derivatives:

$$d_i(t) = a_{i0} + \sum_{j=1}^{\rho_i} a_{ij} \sin(\omega_{ij} t + \phi_{ij})$$

and satisfies an $r_i$-order ODE, embeddable as $\dot{\upsilon}_i = \Phi_i\upsilon_i$, $d_i = \Gamma_i\upsilon_i$ (He et al., 14 Jan 2026).

• Canonical Internal Model Embedding: Bypassing the standard regulator equation, the canonical internal model directly clones the dynamics of the exosystem into an augmented observer. Using controllable pairs $(M, N)$, the adaptive observer ensures that the estimated disturbance asymptotically converges to the true disturbance, under global conditions without the need for persistent excitation. Under PE, exponential convergence is achieved (He et al., 14 Jan 2026).
• Feedforward Compensation: The disturbance estimate is then used for feedforward cancellation:

$u = \varpi(\cdots) - \hat{d}$

to achieve trajectory tracking and disturbance rejection, with rigorous guarantees provided by an ISS-Lyapunov criterion.

Extensive simulation on flexible-joint robotics demonstrates sub-second convergence of observer, parameter, and disturbance estimate, with trajectory tracking error decaying to zero—a concrete verification of the analytic guarantees (He et al., 14 Jan 2026).

4. Root Distribution, Equidistribution, and "Disturbances" in the Spectral Domain

The real zeros ("disturbances") of a TP are closely linked to the angular distribution of the roots of associated algebraic polynomials on the unit circle. The main result (Steinerberger) establishes that the number of sign changes of the real part $q(\theta) = \Re\sum_{k=0}^n a_k e^{ik\theta}$ controls the possible clustering of roots of $p(z) = \sum_{k=0}^n a_k z^k$ on $|z|=1$ at small scales (Steinerberger, 2019):

• Erdős–Turán Theorem: If the "height" $h(p)$ is moderate, zeros on $|z|=1$ are equidistributed in angle at scale $n^{-1/2}$.
• Main Theorem: If $q(\theta)$ has $X=O(n^{\delta})$ real zeros with $\delta<1/2$, then, except for an $O(n^{-\delta})$ fraction of arcs, the root density of $p(z)$ is within a factor-5 of uniform on scales $n^{-(1-\delta)}$—much finer than $n^{-1/2}$ (Steinerberger, 2019).
• Implication: When $q(\theta)$ is nonnegative (as for $\cos(n\theta)+1$), $X$ is constant and root equidistribution is exact at the smallest scale $n^{-1}$.

This spectral perspective links the time-domain "disturbances" of TP signals to the uniformity properties of their spectral roots, interfacing analytic, geometric, and number-theoretic properties.

5. Algorithmic Verification and Nonnegativity Testing

The nonnegativity of trigonometric-polynomial signals over an interval is algorithmically certifiable via the Sturm–based method (Kwong, 2015):

• Reduction to Algebraic Polynomial: By tangent half-angle substitution, each $f(t)=\sum a_k \cos(k t) + \sum b_k \sin(k t)$ on $t\in[\alpha, \beta]$ is mapped to a rational function with a polynomial numerator $P(T)$ in $T=\tan(t/2)$.
• Sturm Sequence Construction: The classical Sturm sequence $(P_0,P_1,\ldots, P_m)$ is computed via polynomial Euclidean division, providing a sequence whose sign changes at the endpoints encode the number of real roots in the interval.
• Root Counting and Positivity: The total variation $V(\alpha) - V(\beta)$ is the number of real roots; $P(\alpha),P(\beta)$ positive and zero variation certify interval-wise nonnegativity.
• Complexity: Naive implementation is $O(n^2)$ in the degree $n$; subresultant and modular methods mitigate coefficient swell for large degrees.

This method enables rigorous certificates of nonnegativity for general trigonometric polynomials involving both sine and cosine components.

6. Stochastic Systems and Moment Closure under TP Disturbances

For stochastic systems with TP drift or input nonlinearities, computation of statistical moments is hindered by unclosed moment dynamics:

• General SDE Structure:

$$dx(t) = f_p(x(t))\,dt + f_t(x(t))\,dt + G(x(t), t)\,dW(t)$$

with $f_p$ polynomial and $f_t$ trigonometric-polynomial components (Ghusinga et al., 2017).

• Unclosed Moments: The time derivative $d\langle x^{[m]}\rangle/dt$ generally involves moments of higher order (e.g., $\langle x^{m+2}\rangle$) or mixed exponential-monomial moments (e.g., $\langle e^{j x_1} x_2^2 \rangle$).
• Derivative-Matching Closure: Mixed moments are systematically approximated by products of lower-order moments, determined by matching time-derivatives at initial conditions. Trigonometric terms are rewritten in terms of $e^{i k x}$, extending the closure procedure to these mixed moments.
• Validation: Empirical accuracy is demonstrated for canonical oscillatory and pendulum systems, closely tracking Monte Carlo simulations for mean trajectories and statistics (Ghusinga et al., 2017).

This approach gives a scalable recipe for moment computation in the presence of TP disturbances in continuous-state, nonlinear stochastic dynamics.

7. Historical and Applied Contexts

Trigonometric-polynomial disturbances are omnipresent in celestial mechanics, robust and adaptive control, signal processing, and stability theory:

• Celestial Mechanics: TP (Poisson) series encode perturbations in nearly-integrable Hamiltonian systems, with algebraic frameworks and Diophantine assumptions governing the convergence properties and timescales of normal-form expansions (Giorgilli et al., 2013).
• Nonlinear and Adaptive Control: Internal model construction for disturbance rejection synthesized for broad classes of nonautonomous nonlinear systems is intrinsically wedded to the TP exosystem framework (He et al., 14 Jan 2026).
• Spectral Analysis and Random Polynomials: Distribution and clustering of zeros of polynomials on the unit circle are explicated via the disturbance (zero-crossing) count of associated TPs (Steinerberger, 2019).
• Computation: Efficient algorithms for positivity, root-counting, and algebraic manipulation of TPs are foundational in symbolic and numeric computation, with practical applications in optimization and verification (Kwong, 2015).

The enduring relevance of trigonometric-polynomial disturbances thus spans theory, computation, and engineering, underpinning both foundational results and implementable algorithms across mathematics and systems science.