| Citation: |
Wanjia Gong. A PERIODICITY DEGREE FOR TIME-T MAPS: DETECTION AND A POSTERIORI CERTIFICATION OF PERIODIC ORBITS[J]. Journal of Applied Analysis & Computation, 2026, 16(5): 2515-2539. doi: 10.11948/20260026
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A PERIODICITY DEGREE FOR TIME-T MAPS: DETECTION AND A POSTERIORI CERTIFICATION OF PERIODIC ORBITS
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Abstract
We develop a scalar interface between numerical detection and a posteriori certification of T-periodic solutions in time-periodic ODEs. Starting from the time-T map, we introduce the periodicity degree, a bounded indicator of one-period recurrence, and establish its basic analytical properties on compact sets. The periodicity degree serves as an optimization-friendly detector of near-fixed points of the time-T map. To convert such numerical evidence into a mathematically valid existence statement, we provide a contraction-based a posteriori criterion that upgrades a detected near-fixed point to the existence of an exact fixed point with an explicit error bound. Numerical illustrations for two forced benchmarks clarify the distinction between detection and certification, while a Lorenz double-scroll example shows how recurrence-guided lag selection and local Poincaré/Newton refinement can be combined to obtain and interpret a high-accuracy periodic-orbit candidate in a chaotic regime.
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Keywords:
- Periodic orbits /
- near recurrence /
- a posteriori validation /
- Poincaré map /
- Lorenz system
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References
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Wanjia Gong. A PERIODICITY DEGREE FOR TIME-T MAPS: DETECTION AND A POSTERIORI CERTIFICATION OF PERIODIC ORBITS[J]. Journal of Applied Analysis & Computation, 2026, 16(5): 2515-2539. doi: 10.11948/20260026
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Figure 1.
Benchmark Ⅰ. Filled markers indicate
$ \widehat x_{\max} $ $ \widehat x_{\min} $ -
Figure 2.
Benchmark Ⅰ. Local landscape of
$ J(x)=\dfrac12\|\widehat\Phi_T(x)-x\|^2 $ $ x^\sharp $ $ \widehat x_{\max}^{\mathrm{grid}} $ $ \widehat x_{\max} $ $ x^\sharp $ -
Figure 3.
Time evolution of
$ \rho(x(t, x_0)) $ $ [0, L] $ $ x_0\in\Omega $ $ \overline{\rho}_{[0, L]}(x_0) $ $ \underline{\rho}_{[0, L]}(x_0) $ -
Figure 4.
Benchmark Ⅱ on the cylinder
$ \mathbb{S}^1\times\mathbb{R} $ $ \widehat{x}_{\max} $ $ \widehat{x}_{\min} $ $ \rho_{\mathrm{cyl}} $ $ \Omega $ -
Figure 5.
Benchmark Ⅱ: Time evolution of
$ \rho_{\mathrm{cyl}}(x(t, x_0)) $ $ [0, L] $ $ \overline{\rho}_{[0, L]}(x_0) $ $ \underline{\rho}_{[0, L]}(x_0) $ -
Figure 6.
Two-pass lag selection for the Lorenz trajectory on
$ [T_{\min}, T_{\max}] $ $ T\mapsto \mathcal{S}(T) $ $ \mathcal{V} $ $ \mathcal{C} $ -
Figure 7.
Lag-driven event selection and Newton–Poincaré refinement on the section
$ \Sigma $ $ u_i=(y_i, z_i) $ $ u^{(0)}=u_{i^\star} $ $ u_{i^\star+p^\star} $ $ u^\star $ $ P^{p^\star}(u^\star) $ $ \|F_{p^\star}(u^{(k)})\|_2 $ $ \alpha_k $ $ \alpha_k $ $ u^{(k)} $ $ u^{(k+1)} $ -
Figure 8.
Phase-dependent continuous-time closure defect over one refined period. The curve shows
$ \kappa_{\mathrm{po}}(t)=\|\Phi_{t+T_{\mathrm{po}}^\star}(x^\star)-\Phi_t(x^\star)\|_2 $ $ t\in[0, T_{\mathrm{po}}^\star] $ $ x^\star $ $ u^\star $ -
Figure 9.
Lorenz attractor together with the refined periodic-orbit candidate
$ x^\star(t) $ $ t\in[0, T_{\mathrm{po}}^\star] $ $ x^\star(t+T_{\mathrm{po}}^\star) $ $ x^\star $ $ \Phi_{T_{\mathrm{po}}^\star}(x^\star) $ -
Figure 10.
Periodicity-degree signal evaluated at the refined period
$ T_{\mathrm{po}}^\star $ $ [50, 400] $