Universal Dimensionless Invariants in Period-Doubling Dynamics
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| Format: | Recurso digital |
| Langue: | anglais |
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2026
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| _version_ | 1866901661810688000 |
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| author | Sepiashvili, David |
| author_facet | Sepiashvili, David |
| contents | <p>This work presents a compact yet numerically robust investigation of the period-doubling cascade in one-dimensional nonlinear maps.<br>The primary objective is to confirm the universality of scaling behavior without a priori fixing critical parameter values and without relying on a single numerical procedure.</p> <p>In <strong>Phase I</strong>, the logistic map</p> <p><span></span><br></p> <p>is analyzed, where superstable parameters <span></span>are defined by the condition that the critical point belongs exactly to a stable cycle of period <span></span>.<br>In <strong>Phase II</strong>, an analogous procedure is applied to the quadratic map</p> <p><span></span><br></p> <p>providing an independent test of universality for a family with different normalization and phase-space geometry.</p> <p>For both families, sequences of superstable parameters, associated observables, Feigenbaum ratios <span></span>, and accumulation points are computed.<br>In addition, an independent <em>harmonic sweep</em> analysis based on parameter scanning and the detection of harmonic activation thresholds at high cascade levels is employed as a consistency check.</p> <p>All three procedures consistently converge to the universal Feigenbaum constant</p> <p><span></span><br></p> <p>while the accumulation points <span></span>and <span></span>remain family-dependent.<br>This triple numerical agreement excludes algorithmic artifacts and demonstrates that the universality of the period-doubling cascade represents a genuine structural property of nonlinear dynamical systems.</p> <p> </p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18157684 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Universal Dimensionless Invariants in Period-Doubling Dynamics Sepiashvili, David period-doubling cascade universality Feigenbaum constant dimensionless invariants superstable orbits logistic map quadratic map nonlinear dynamics route to chaos numerical analysis <p>This work presents a compact yet numerically robust investigation of the period-doubling cascade in one-dimensional nonlinear maps.<br>The primary objective is to confirm the universality of scaling behavior without a priori fixing critical parameter values and without relying on a single numerical procedure.</p> <p>In <strong>Phase I</strong>, the logistic map</p> <p><span></span><br></p> <p>is analyzed, where superstable parameters <span></span>are defined by the condition that the critical point belongs exactly to a stable cycle of period <span></span>.<br>In <strong>Phase II</strong>, an analogous procedure is applied to the quadratic map</p> <p><span></span><br></p> <p>providing an independent test of universality for a family with different normalization and phase-space geometry.</p> <p>For both families, sequences of superstable parameters, associated observables, Feigenbaum ratios <span></span>, and accumulation points are computed.<br>In addition, an independent <em>harmonic sweep</em> analysis based on parameter scanning and the detection of harmonic activation thresholds at high cascade levels is employed as a consistency check.</p> <p>All three procedures consistently converge to the universal Feigenbaum constant</p> <p><span></span><br></p> <p>while the accumulation points <span></span>and <span></span>remain family-dependent.<br>This triple numerical agreement excludes algorithmic artifacts and demonstrates that the universality of the period-doubling cascade represents a genuine structural property of nonlinear dynamical systems.</p> <p> </p> |
| title | Universal Dimensionless Invariants in Period-Doubling Dynamics |
| topic | period-doubling cascade universality Feigenbaum constant dimensionless invariants superstable orbits logistic map quadratic map nonlinear dynamics route to chaos numerical analysis |
| url | https://doi.org/10.5281/zenodo.18157684 |