Curriculum of Recurrences: Dynamics & Sequences

• Curriculum of Recurrences is a dual-themed topic unifying chaotic returns in Hamiltonian systems with generalized Narayana-type linear sequences.
• Computational techniques such as the Ulam, Arnoldi, and Survival Monte Carlo methods reveal algebraic decay and mode localization in area-preserving maps.
• The Narayana-family recurrences extend classical sequences like powers of two and Fibonacci numbers, underpinning integer decompositions and combinatorial identities.

The "Curriculum of Recurrences" encompasses two distinct but thematically related research domains addressed in the literature: (1) the statistical structure and computation of Poincaré recurrences in dynamical systems, specifically area-preserving maps such as the Chirikov standard map, and (2) the algebraic and combinatorial properties of a general family of linear recurrences that extend the classical sequences from powers of two to Fibonacci and Narayana numbers. Both perspectives analyze recurrence as a foundational motif—either as a temporal return in phase space or as an arithmetic process generating integer sequences—which enables unified treatments of phenomena across mathematics and dynamic systems.

1. Poincaré Recurrences in Area-Preserving Maps

Poincaré recurrences refer to the return of a trajectory in a bounded Hamiltonian system to a neighborhood of its initial condition. In two-dimensional symplectic maps, specifically the Chirikov standard map—$(x,p)\in[0,2\pi)^2$, iterated by

$$p_{n+1} = p_n + K \sin(x_n),\quad x_{n+1} = x_n + p_{n+1}\pmod{2\pi},$$

—the recurrence phenomenon is structurally governed by the interplay between chaotic seas, KAM tori, cantori, and resonance islands. At a critical parameter value ($K_g\approx 0.9716354$), the destruction of the last "golden" invariant torus results in a cantorus, promoting complex, long-time trapping ("sticking") that induces algebraic (power-law) tails in the recurrence time distribution. Quantitatively, the recurrence-time distribution $P(t)$ satisfies: $$P(t) = \frac{N(\tau>t)}{N_\text{tot}} \xrightarrow[t\to\infty]{} C t^{-\beta},$$ with numerically observed exponents $\beta$ typically in the range 1.5–1.8 (Frahm et al., 2013).

2. Numerical Approaches: Ulam Method and Survival Monte Carlo

The computation and analysis of recurrences leverage two principal numerical methodologies:

Generalized Ulam Method:

• The relevant phase space is partitioned into $M\times M$ cells. A single chaotic trajectory of up to length $10^{12}$ is used to construct a stochastic (Ulam) matrix $S$ of dimension $N \times N$ (where $N$ is the number of occupied cells), preserving invariant measures without artificial diffusion.
• Imposing absorbing boundaries—e.g., discarding cells below a momentum cutoff $p < p_\text{cut}$—results in a "projected" Ulam matrix $S_\text{abs}$. The largest eigenvalue $\lambda_0 < 1$ dictates the exponential escape regime.

Arnoldi Method:

• Direct diagonalization is feasible for $N \lesssim 10^5$; for larger $N \sim 10^6$, the Arnoldi method retrieves the leading eigenvalues and eigenvectors.

Survival Monte Carlo Method (SMCM):

• $N_i \sim 10^6$ trajectories are initialized in a minimal cell near an unstable fixed point and evolved in parallel. Surviving trajectories are re-injected with small perturbations ($\varepsilon \sim 10^{-14}$) as population drops below a threshold ($N_f \sim 10^3$), facilitating estimation of $P(t)$ over up to ten decades in time.

These methods allow for accurate measurement of recurrence statistics up to distinct regimes: the Ulam method is applicable up to crossover times $t_\text{exp} \sim 10^4 – 10^5$, while the SMCM extends results six to seven decades further in $t$, reliably revealing algebraic decay (Frahm et al., 2013).

3. Main Quantitative Results and Dynamical Mechanisms

Applying the above techniques yields explicit exponents for power-law decay in recurrence probabilities. For the standard map at the critical golden torus ($K_g$), the SMCM yields $\beta \approx 1.587 \pm 0.009$ over the interval $10^6 \le t \le 10^{10}$; the separatrix map at its critical value ($A_c \approx 3.1819316$) produces $\beta \approx 1.706 \pm 0.004$. These values are summarized as follows:

Map	Critical parameter	$\beta$
Chirikov standard	$K = K_g \approx 0.9716354$	$1.587 \pm 0.009$
Separatrix	$A = A_c \approx 3.1819316$	$1.706 \pm 0.004$

Long-time algebraic tails originate from trajectories sticking in the vicinity of the critical golden cantorus and secondary resonance islands (e.g., ratios 1/2, 2/7, 1/3), with density plots demonstrating cluster formation along these structures as time increases.

4. Spectral Properties and Localization in the Ulam Matrix

The projected Ulam matrix encodes the dynamics of recurrences in its spectrum and eigenstates:

• Diffusive modes ($\lambda_j$ real and near unity) are delocalized in the chaotic sea but vanish near absorbing regions.
• Resonant modes ($\lambda_j$ complex or negative real) are sharply localized around unstable periodic orbits correlated with secondary resonances.

For each eigenvector $\psi_j(x,p)$, exponential localization in the $p$-direction manifests as

$$A_j(p) = \frac{1}{\Delta x} \sum_{x\in[x_0-\Delta x/2, x_0+\Delta x/2]} |\psi_j(x,p)|$$

with rapid decay away from the principal chaotic region. Decomposition of an initial delta-state over eigenmodes leads to heavily fluctuating coefficients $p_j$ (spanning up to ten orders of magnitude), a direct reflection of mode localization.

Survival probability can be reconstructed via

$P(t) = \sum_j p_j (\lambda_j)^t,$

with the largest eigenvalue dominating at asymptotic times (Frahm et al., 2013).

5. The General Narayana Family and Curriculum of Linear Recurrences

A parallel thread within the "Curriculum of Recurrences" concerns the algebraic structure and application of the Narayana-type family of linear recurrences, defined by the characteristic polynomial

$g_q(x) = x^q - x^{q-1} - 1, \qquad q \geq 1,$

with initial conditions

$$G^{(q)}_0 = G^{(q)}_1 = \cdots = G^{(q)}_{q-2} = 0; \quad G^{(q)}_{q-1} = \cdots = G^{(q)}_{2q-2} = 1,$$

and recurrence relation

$$G^{(q)}_{n+q} = G^{(q)}_{n+q-1} + G^{(q)}_n, \qquad n \geq 0.$$

This structure subsumes:

• $q = 1$: Powers of 2, $G^{(1)}_n = 2^n$.
• $q = 2$: Fibonacci numbers.
• $q = 3$: Narayana numbers.

The ordinary generating function is

$$G^{(q)}(x) = \sum_{n \geq 0} G^{(q)}_n x^n = \frac{x^{q-1}}{1 - x - x^q},$$

admitting a Binet-type formula with roots $\alpha_j$ of the characteristic polynomial: $G^{(q)}_n = \sum_{j=1}^q c_j \alpha_j^n.$

6. Applications: Integer Decompositions and Combinatorial Identities

The Narayana-type recurrences underpin a unified theory for several classical topics:

• Zeckendorf-type decompositions: Any positive integer $n$ can be written as a sum of non-adjacent terms from $\{a_i=G^{(q)}_{2q-2 + i}\}$.
• Compositions and Binomial-Sum Identities: The structure admits generalizations of combinatorial interpretations, including connections to the Pascal triangle and digital-sum theorems.
• Nim-Game Variants and Beatty Sequences: The family provides arithmetic and combinatorial models relevant to these game-theoretic and sequence-enumeration contexts (Ballot, 2017).

7. Synthesis and Outlook

The Curriculum of Recurrences integrates the rigorous paper of time-return events in Hamiltonian systems and the development of a general arithmetic and combinatorial architecture encapsulated in the Narayana family. In the dynamical context, algebraic decay exponents emerge from complex phase-space structures—especially golden cantori and resonance islands—while computational advances such as the generalized Ulam method and survival Monte Carlo method enable precision analysis across broad regimes. In the algebraic-combinatorial context, a single parametric family unifies classical sequences, decompositions, and identities, offering organizing principles for integer representations and discrete dynamics. Ongoing research is directed at deeper synthesis, including potential theoretical bridges—such as renormalization models linking localization in dynamical systems with properties of recurrence-generated number systems—within the broader landscape of recurrence phenomena (Frahm et al., 2013, Ballot, 2017).