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Memory in Behavioral Models as Motion on a Slow Invariant Manifold

Nicholas B. Tufillaro

arXiv:2609.38771v1eess.SPmath.DS

TL;DR

Behavioral models describe memory under modulated drive, but existing approaches leave open how many states they need and what their identified dynamics represent. This paper derives the slow invariant manifold at a large-signal operating point and connects its reduced dynamics to behavioral models. In a GaN HEMT example, the trap’s 14 µs time constant comes from linearization, and the local trap-direction expansion is limited to a few thermal voltages.

  • Problem

    Existing behavioral approaches leave open how many memory states are needed, what an identified kernel or delay-coordinate map represents, and when a model built at one operating point can be trusted at another.

    Source: §I · p. 1

  • Method

    The paper applies the fixed-point parameterization theorem to the stroboscopic map at a stable periodic operating point and relates the slow manifold’s reduced dynamics to behavioral models.

    Source: §I · p. 1, §VII · p. 8

  • Results

    The GaN HEMT example has a 14 µs trap time constant from linearization, not the trap’s 6 ms emission time.

    Source: §I · p. 2

  • Takeaways & Limitations

    An exact reduced memory model needs as many states as there are slow Floquet exponents, while the memoryless model is the fixed-point family of the reduced dynamics.

    Source: §VII · p. 8

    The theorem is stated at fixed drive; modulated drive is covered only to first order in the ratio of envelope rate to electrical rates, and the ASM-HEMT trap-direction parameterization extends only a few thermal voltages.

    Source: §VII · p. 8

Abstract

from arXiv · show

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 (self-heating, trapping), 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. An existence and uniqueness theorem for that manifold follows from the parameterization method of Cabr\'e, 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 Floquet exponents, and the dynamic X-parameter kernel of Verspecht et al. identifies its reduced dynamics from step changes of the drive amplitude. Consequently, an exact reduced model has as many memory states as slow exponents, 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 a three-pole thermal network and a drain-lag trap: the trap contributes a $14\,\mu$s time constant set by the linearization and not by its $6$ ms emission time, the thermal submanifolds are nearly flat with linear reduced dynamics, and the expansion in the trap direction is valid only within a few thermal voltages ($nV_T\approx26$ mV), so trap memory needs a global representation of the manifold.

I. INTRODUCTION

The paper connects behavioral models of memory to the slow invariant manifold of a periodically forced circuit. It applies a fixed-point theorem at the large-signal operating point and relates the resulting reduced dynamics to existing memory models.

I. INTRODUCTION

  • Existing approaches leave open how many memory states are needed, what an identified kernel or delay-coordinate map represents, and when a model transfers between operating points.The approaches include a history-integrated dynamic X-parameter kernel, delay-sample state reconstruction, and a dynamic gain model with a choice of difference operators.Source: §I · p. 1
  • The slow-mode manifold provides the geometric basis for an exact memory model, with one state per slow Floquet exponent and the dynamic X-parameter kernel representing its flow between fixed points.The memoryless X-parameter surface is the fixed-point family, and the envelope-domain model captures the reduced dynamics.Source: §I · pp. 1–2
  • The manifold theorem applies the fixed-point result of Cabré, Fontich and de la Llave to the stroboscopic map at the orbit, without requiring small periodic forcing.The resulting manifold is a bundle over drive phase, with fiber dimension equal to the number of slow Floquet exponents.Source: §I · p. 1
  • For the GaN HEMT example, the theorem’s hypotheses hold for the full slow-mode set; the trap time constant is 14 µs from linearization, while its parameterization is locally valid only within a few thermal voltages.The expansion in the trap direction is limited by exponential diode gating, motivating a global representation of the manifold.Source: §I · p. 2

A. The circuit as a periodically forced analytic system

  • Time invariance makes the circuit’s drive phase enter through the instantaneous drive, giving the extended circuit equations phase-shift equivariance.The memoryless behavioral map is fixed by time invariance; memory appears when the drive is modulated.Source: §I · p. 1, A. The circuit as a periodically… · p. 2

B. The LSOP as a periodic orbit

A single-tone large-signal operating point is represented as a fixed point of the circuit’s stroboscopic map. Stability and analyticity conditions make that operating point persist locally as the drive changes.

B. The LSOP as a periodic orbit

  • A single-tone large-signal operating point is a periodic solution, and its state at the reference phase is a fixed point of the stroboscopic map.The map advances the circuit state by one drive period.Source: B. The LSOP as a periodic orbit · p. 2
  • The circuit equations are analytic on the operating region under the stated assumptions; the ASM-HEMT’s piecewise clamps are inactive at the operating points considered.The compact model and added thermal and trap networks are described as analytic there.Source: B. The LSOP as a periodic orbit · p. 2
  • The theorem assumes asymptotic stability: every Floquet multiplier has magnitude below one, equivalently every Floquet exponent has negative real part.The multipliers and exponents come from the linearization of the stroboscopic map at the operating point.Source: B. The LSOP as a periodic orbit · p. 2
  • When the operating point exists and is stable, it persists analytically with the drive on a neighborhood of that drive value.This follows from the implicit function theorem because 1 is not an eigenvalue of the monodromy matrix.Source: B. The LSOP as a periodic orbit · p. 2

C. Slow modes and the spectral quotient

The slow spectral subspace isolates thermal and trap dynamics from much faster electrical modes. The spectral quotient sets the order relevant to the theorem’s non-resonance condition.

C. Slow modes and the spectral quotient

  • Slow modes are thermal and trapping dynamics, with rates of 10^2–10^6 s^-1, while electrical modes are 10^10 s^-1 and faster.The slow subspace E is spanned by the q slowest modes.Source: C. Slow modes and the spectral quotient · p. 2
  • The absolute spectral quotient is the integer part of the ratio of the fastest circuit decay rate to the slowest decay rate in the slow subspace.It sets the upper order considered in the slow-mode non-resonance condition.Source: C. Slow modes and the spectral quotient · p. 2
  • The non-resonance assumption excludes products of up to the spectral-quotient number of slow multipliers matching a multiplier outside the slow subspace.The stated exponent condition is sufficient because differing real parts imply differing multiplier moduli.Source: C. Slow modes and the spectral quotient · p. 2

III. THE SLOW MANIFOLD OF AN LSOP

Applying the parameterization method to the stroboscopic map yields a unique smooth slow invariant manifold at a stable large-signal operating point. Its phase-indexed fibers carry the slow Floquet directions, and its dynamics has a polynomial representation.

III. THE SLOW MANIFOLD OF AN LSOP

  • The theorem gives a locally invariant manifold tangent to the slow spectral subspace and unique among locally invariant manifolds of class C^Σ(E)+1; in particular, it is the unique analytic manifold.The result follows by applying Cabré, Fontich and de la Llave to the stroboscopic map.Source: §III · pp. 2–3
  • The phase-indexed manifold is analytic and has dimension q+1, with each fiber tangent to the slow Floquet subspace transported along the periodic orbit.It contains the large-signal operating point and is locally invariant under the extended flow.Source: §III · p. 2
  • The stroboscopic dynamics restricted to the manifold is analytically conjugate to a polynomial map of degree at most Σ(E).The map can be linear when the specified internal non-resonance condition holds; the theorem requires analyticity and slow-mode non-resonance.Source: §III · pp. 2–3
  • No small-forcing parameter is needed: the theorem applies directly to the stroboscopic map at the periodic orbit being simulated or measured.This is the one-frequency form of the spectral submanifold result of Haller and Ponsioen.Source: §III · p. 3
  • The manifold attracts trajectories after the faster transverse electrical transient decays, leaving their subsequent evolution on the slow manifold.The rates transverse to the slow subspace are strictly faster than rates within it.Source: §III · p. 3
  • For manifolds inferred from measured data, uniqueness among C^Σ(E)+1 manifolds is vacuous, so a uniqueness claim must instead rely on attraction.For realistic devices, Σ(E) can be of order 10^7.Source: §III · p. 3

IV. CONSEQUENCES FOR BEHAVIORAL MODELS

The slow Floquet modes determine the exact reduced model’s memory-state count, while the memoryless X-parameter model is the fixed-point family on the slow manifold.

A. The number of memory states

  • An exact reduced model needs one state per slow mode; retaining only a subset is unique only when the corresponding non-resonance condition holds.Slow-to-fast rate ratios make that condition generically true, but the reported 1:15 ratio among slow modes makes a slowest-mode-only model fail it.Source: A. The number of memory states · p. 3

B. The memoryless model is the fixed-point family

  • The memoryless X-parameter model is the fixed-point family on the slow manifold, whose fibers are transported around the drive-phase circle.The static spectral map is obtained by composing the periodic steady-state family with the output map.Source: B. The memoryless model is the… · p. 3

C. The reduced dynamics is the dynamic X-parameter kernel

The reduced dynamics gives the dynamic X-parameter kernel’s step response and, when driven by a slowly varying envelope, the envelope-domain model.

C. The reduced dynamics is the dynamic X-parameter kernel

  • A step in envelope amplitude traces a reduced-dynamics trajectory from the previous fixed point to the new one, and the kernel is minus the output’s time derivative along it.The memoryless model is the output at the fixed point.Source: C. The reduced dynamics is the dynamic… · pp. 3–4
  • Kernel dependence on the previous amplitude reflects reduced-dynamics nonlinearity; only in the small-step limit does it become a sum of exponentials with slow Floquet rates.In compact models, physical slow states can serve as fiber coordinates to leading order in the rate ratio, with period-averaged equations describing their reduced dynamics.Source: C. The reduced dynamics is the dynamic… · p. 4

D. Modulated drive

  • For a slowly varying envelope, the reduced dynamics follows the frozen family of slow manifolds under a uniform spectral-gap condition.The resulting attracting invariant manifold is O(ε)-close in C1 to that frozen family.Source: D. Modulated drive · p. 4
  • The envelope-domain model is the frozen reduced dynamics driven by the envelope, accurate to first order in the envelope-to-electrical rate ratio.The paper gives 10^-3 for a 10 MHz envelope and assumes periodicity for compactness.Source: D. Modulated drive · p. 4

V. THE BUNDLE IN TIME-DOMAIN COORDINATES

Forced-system delay embedding applies to the slow-manifold bundle, giving bounds on the number of waveform samples needed for fiber or bundle coordinates.

V. THE BUNDLE IN TIME-DOMAIN COORDINATES

  • Stark’s forced-system theorem, rather than the autonomous Takens theorem, applies because the device is driven and its slow manifold is invariant.A node voltage supplies an observable depending on the circuit state alone.Source: §V · p. 4
  • Generically, 2q+1 period-spaced delays embed the fiber, while 2q+3 incommensurately spaced delays embed the whole bundle.Here q is the number of slow modes.Source: §V · p. 4
  • The time-domain memory-model dimension is set by the slow-exponent count, which false-nearest-neighbor methods estimate as q.This complements the model-order rule from the companion paper [1].Source: §V · p. 4
  • The extra two delays for a full time-domain model account for carrying drive phase, unlike an envelope-domain model on a fixed-phase fiber.The circle equivariance transports equivalent fibers across drive phase, so a model identified at one phase applies at every phase.Source: §V · p. 5

VI. THE HYPOTHESES CHECKED ON A GAN HEMT WITH MEMORY

The GaN HEMT example combines a three-pole thermal network and a drain-lag trap, then computes an envelope-domain surrogate from period-averaged slow-state equations.

A. The device and the two-timescale computation

  • The ASM-HEMT example uses a three-pole thermal network and drain-lag trap, with temperatures and trap voltage as its four slow states.The thermal time constants are 6 µs, 100 µs, and 1.5 ms; the trap has 1 ns capture and 6 ms emission times.Source: A. The device and the two-timescale… · p. 5
  • The operating point is computed by pinning slow states during short transient runs, then iterating their period-averaged equations to a fixed point.At 1.5 V gate drive, the reported junction-temperature rise is 30.4 K and output is 30.0 dBm; power balance closes to 0.05%.Source: A. The device and the two-timescale… · p. 5
  • The envelope-domain surrogate tabulates slow maps over drive, temperature, and trap voltage at 440 periodic steady states, then interpolates them with a tensor-product cubic spline.Its slow equations integrate in milliseconds, and it serves as the computed parameterization of the slow manifold.Source: A. The device and the two-timescale… · p. 5

B. The spectrum

The four-mode slow spectrum supports a four-dimensional memory fiber, and the full-set hypotheses hold; a one-mode thermal reduction, however, encounters an order-15 resonance.

B. The spectrum

  • The thermal exponents closely match network poles, and a full-circuit transient validates the fastest slow mode to 0.02%.The transient and averaged-equation time constants are 6.015 µs and 6.016 µs, respectively.Source: B. The spectrum · p. 5
  • The trap exponent is about 6.9 × 10^4 s−1, set by RF-operating-point linearization rather than the trap’s 6 ms emission time.The estimate from the linearized diode relation is within 10% of the Jacobian result; the emission time does not appear in the spectrum.Source: B. The spectrum · pp. 5–6

C. The hypotheses

  • The full four-mode slow set satisfies non-resonance, yielding a four-dimensional fiber and four-state reduced dynamics; nine delays embed the fiber and eleven the bundle.The eleven-delay result requires an incommensurate sampling interval.Source: C. The hypotheses · p. 6
  • An exact order-15 resonance between thermal modes is permitted, but it adds a degree-15 resonant monomial to the reduced dynamics, otherwise linear.The resonance follows from the thermal fit’s round time constants.Source: C. The hypotheses · p. 6
  • Keeping only the slowest thermal mode fails the theorem’s non-resonance condition at order 15, so that one-state model is not covered.The exact ratio comes from the round time constants of the thermal fit.Source: C. The hypotheses · p. 6

D. The linearized kernel and its range of validity

The linearized kernel matches small drive steps but not large ones, where trap voltage slews; the resulting trap-memory expansion is local and needs a global manifold representation.

D. The linearized kernel and its range of validity

  • The response rises from 0.612 below 1 kHz to 0.690 above 1 MHz, about 1 dB of memory, essentially all due to the trap, with a corner near 10 kHz.The kernel’s static gain of 0.612 matches the independently computed fixed-point-family slope to four digits.Source: D. The linearized kernel and its range… · p. 6
  • A −0.07% drive step follows the linearized kernel to 2.5%, whereas a −3% step differs by 68% and ±6 dB steps by 85–96%.The trap voltage slews instead of relaxing for the larger steps, while the thermal modes remain linear.Source: D. The linearized kernel and its range… · p. 6
  • Because the trap-direction expansion is informative only within a few millivolts of drive, the manifold needs a global representation rather than the local normal form.The paper describes tables, splines, or a fitted network as global representations; a level-dependent linear-memory model works for thermal memory but fails for trap memory beyond a few millivolts.Source: D. The linearized kernel and its range… · p. 7

E. Explicit polynomial reduced models on the thermal submanifolds

The paper constructs polynomial reduced models on thermal submanifolds and finds that these are nearly flat, while trap-direction nonlinearity limits local polynomial representation. Across examined operating points, thermal modes remain nearly fixed and the trap mode accounts for most memory.

E. Explicit polynomial reduced models on the thermal submanifolds

  • The degree-four slow-field fit used 49 periodic steady states, kept trap softplus gating exact, and solved the invariance equation order by order through order seven.The fit covered ±12 K and ±0.12 V about the operating point; reduced dynamics remained linear unless an inner resonance appeared.Source: E. Explicit polynomial reduced models… · p. 7
  • The two-dimensional thermal manifold is nearly flat, has linear reduced dynamics through order seven, and slaves trap voltage to temperature at −4 mV/K.The order-seven invariance residual is 3 × 10−5 of |f| at a 10 K excursion.Source: E. Explicit polynomial reduced models… · p. 7, Figure 2 · p. 8
  • At four 5–10 K excursions, ngspice steady states confirm tangency to the true vector field, with tangential components agreeing within 0.2%.Transverse components correspond to displacements of 10−3–10−2 eigen-units, the simulation noise floor.Source: E. Explicit polynomial reduced models… · p. 7
  • A trajectory started 2 eigen-units off the manifold is 5 × 10−4 away at 100 µs and 5 × 10−8 at 300 µs, then moves along it.The fast modes decay at their own rates.Source: E. Explicit polynomial reduced models… · p. 7, Figure 2 · p. 8
  • The one-dimensional manifold violates the theorem’s non-resonance condition at order 15, but its resonant term contributes only 10−5 K at the 12 K box edge.The resonant term becomes comparable to the linear term only near a 30 K excursion, beyond the local expansion’s validity.Source: E. Explicit polynomial reduced models… · p. 7
  • The trap-direction Taylor field matches the exact trap equation to 10−3 at ±50 mV, to 3% at ±80 mV, and fails at ±120 mV (37%).Its expansion radius is two to three nVT; polynomial reduction is adequate for thermal memory but inadequate for trap memory, which requires a global representation.Source: E. Explicit polynomial reduced models… · p. 7

F. Dependence on the operating point

  • Across examined operating points, thermal exponents move by less than 0.4%, while the trap time constant shifts from 25 µs to 13 µs with drive and is 16 µs at deeper bias.The trap kernel weight is 30–200 times each thermal weight; the required model order is four, and the operating-point-dependent model must track trap exponent and weight.Source: F. Dependence on the operating point · pp. 7–8

VII. DISCUSSION

The discussion frames long-term memory as motion on a unique attracting finite-dimensional invariant manifold, with dimension set by slow Floquet exponents. It identifies limits on the theorem’s scope and outlines measurement and computational work needed to apply the reduction.

VII. DISCUSSION

  • The manifold’s fiber dimension equals the number of slow Floquet exponents, which sets the exact envelope model’s state count and the intrinsic dimension reconstructed by time-domain models.The memoryless model is its fixed-point family, and the dynamic X-parameter kernel is its reduced flow between fixed points.Source: §VII · p. 8
  • The theorem does not bound the manifold’s invariant neighborhood; for the ASM-HEMT, parameterization extends only a few thermal voltages, nVT ≈26 mV, in the trap-voltage direction.Modulated drive is covered only to first order in the ratio of envelope rate to electrical rates; the theorem also assumes analyticity on the operating region.Source: §VII · p. 8
  • The theorem assumes an analytic model on the operating region, a condition met by compact models by construction but not by every fitted behavioral model.For the memoryless case q = 0, the manifold is the periodic orbit alone; memory adds a q-dimensional fiber at each drive phase.Source: §VII · p. 8
  • A physical-device measurement should test whether measured rates are slow Floquet exponents and whether the trap rate shifts with drive and bias rather than staying at its emission time constant.The cited 0.3–30 µs envelope periods are shorter than the two slowest time constants in Table I.Source: §VII · p. 8
  • The proposed reduction integrates q ordinary differential equations with tabulated right-hand sides instead of solving a harmonic-balance problem at every envelope sample.Automation still depends on tabulation-grid growth with envelope inputs and assignment of intermediate-rate electrical-memory modes.Source: §VII · p. 9
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