Memory in Behavioral Models as Motion on a Slow Invariant Manifold

Abstract

A single-tone large-signal operating point of a nonlinear two-port is a periodic orbit of a periodically forced circuit. When the device has memory, the Floquet exponents of that orbit separate into fast (electrical) and slow (thermal and trapping) modes, and long-term memory is motion on the invariant manifold attached to the slow modes. Existence and uniqueness of that manifold follow from the parameterization method of Cabré, Fontich and de la Llave, applied to the stroboscopic map at the orbit; the manifold is the spectral submanifold of Haller and Ponsioen, without a small-forcing parameter. The manifold is a bundle over the circle of drive phase, its fiber dimension the number of slow exponents, and the dynamic X-parameter kernel of Verspecht et al. identifies its reduced dynamics from the transient after a step in drive amplitude. An exact reduced model has one memory state per slow exponent, the memoryless X-parameter surface is the fixed-point family of the reduced dynamics, and the envelope-domain model is the reduced dynamics driven by the envelope. The hypotheses are verified and the manifold constructed for a GaN HEMT compact model with thermal and trap memory: the trap's time constant at the operating point, $14\mu$s, is set by the linearization, not by its $6$ ms emission time, and the trap's nonlinearity confines a polynomial representation of the manifold to a few millivolts of drive, so trap memory must be represented over the operating range, by tables or a fitted network, not by an expansion about the operating point.

10 pages, 2 figures, 3 tables. v2: prior work attributed (the hidden-variable formulation of Root et al.; the dynamic gain model of Verspecht et al.); the trap's emission time and its linearized rate distinguished explicitly; wording revised throughout; results unchanged. Companion papers: arXiv:2609.35828 and arXiv:2610.01031. Toolkit v1.1.0: doi:10.5281/zenodo.23107935