Unnikrishna Pillai | Search arXiv

arXiv2026

Analog Realization and Measurement of a Forced Aliev-Panfilov Excitable Oscillator

A single resistor moves a discrete analog realization of the forced Aliev-Panfilov excitable oscillator from a chaotic operating point of the model to a regular one. In this model a rate parameter multiplies the complete right-hand side of the excitation equation, forcing term included. The input resistor of the excitation integrator sets that parameter, so no state, delay element, or controller is added. Sampled waveforms were recorded at two resistor values, with two independent runs each. The measured excitation excursion agrees with numerical integration to within three percent at the baseline setting and five percent at the rate-controlled setting. Between the two settings the excitation excursion contracts by a factor of thirteen. At the rate-controlled setting the recovery excursion exceeds prediction by a factor of eight. This excess traces to a displaced operating point on the state-dependent recovery rate. Thermal drift in the multiplier chain and leakage at the high-impedance recovery-rate divider could each produce it, and the present data do not separate them. A tested bipolar input stage strongly loads that divider, a constraint absent from the ideal equations. The divider, not the sub-hertz signal bandwidth, sets the amplifier input-stage requirement.

eess.SY↗

arXiv2026

Nonlinear springs with orbits in radicals: a solvability criterion

An odd polynomial oscillator has a periodic orbit given by a quadrature. When the zeros of the energy polynomial are all simple or all of even multiplicity, that quadrature is elementary exactly when the polynomial is a perfect square, and the time is a sum of inverse tangents. Recovering the displacement inverts that relation, algebraic when their coefficients are commensurate, making a closed form a Galois question. This paper identifies the group as the monodromy group of a rational map, computes its branch data from the parameters, and shows the branch value at the phase-angle singularity never vanishes, so every decomposition pairs the amplitudes reciprocally. In the commensurate family with distinct real factors, the orbit is radical exactly when the primitive inversion degree is at most four, or when reciprocal pairs leave a reduced spring of that degree. Radical orbits occur at the fifth, ninth, thirteenth and seventeenth force orders, the last only for reciprocal members, an expression recorded for each. The multiplicity hypothesis cannot be dropped. Factoring the energy polynomial into square and squarefree parts turns the quadrature into a differential of the third kind on a hyperelliptic curve, where elementarity is a torsion condition on the divisor of its poles with a vanishing holomorphic remainder. The first nonsquare elementary member appears at the ninth force order, and the power pullback inverts in closed form at every such order, giving a radical orbit for a confining spring of arbitrarily high degree with a sufficient condition for positive coefficients.

math.DS↗