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Nicholas B. Tufillaro

Publications and source records attributed to Nicholas B. Tufillaro.

14 recordsLinked to original sources
arXiv2026

A comparison of three models of a whirling string as it becomes chaotic

A stretched wire driven near its fundamental frequency can leave the plane of the driving force and whirl. At larger amplitudes the whirl becomes modulated, reverses its direction, and becomes chaotic. Three models of this motion are in common use: the geometrically exact string equations; a two-mode model -- the Kirchhoff-Carrier equation with one transverse mode in each plane; and the averaged equations for the slow amplitudes of those two modes. We integrate all three under similar conditions and compare them along the route to chaos, which starts with a Hopf bifurcation of the steady whirl and passes through period doublings to chaotic attractors organized by a Shil'nikov homoclinic orbit. The small parameter of the averaging, the ratio $\varepsilon = c/ω_1$ of the modulation rate to the fundamental frequency, runs from 0.005 to 0.02. Qualitatively the three models agree in the sequence of bifurcations, the coexisting attractors, the return maps and the Lyapunov exponents. Quantitatively, averaging places each local bifurcation $0.7\varepsilon$ to $1.2\varepsilon$ higher in the scaled detuning. The geometrically exact and two-mode models agree with each other to within 0.002 at the local bifurcations for $\varepsilon \le 0.01$. The Shil'nikov ratio, which decides whether chaos accompanies the homoclinic orbit, agrees among the three models to within 1.5% for $\varepsilon \le 0.01$, yet the homoclinic bifurcation itself moves by about $\varepsilon$ at the lower damping and by 6 to 9 times $\varepsilon$ at the higher; the difference is set by how quickly the return of the homoclinic loop changes with detuning.

nlin.CD↗
arXiv2026

Time Invariance, Circle Symmetry, and the Completeness of the Cardiff Behavioral Model

Frequency-domain behavioral models of nonlinear microwave devices (the Cardiff model, X-parameters, higher-order sinusoidal describing functions, and the baseband models of power-amplifier predistortion) share one functional form, and each modeling framework justifies the form by its own argument, time invariance among them. Here we give the mathematical argument by which time invariance alone determines the model form, and state the result as a theorem for two-port devices, with its extension to any number of ports. Under a shift of the time origin each harmonic phasor rotates by a multiple of the fundamental's angle, so the spectral map of a time-invariant device is equivariant under a weighted circle action, and classical invariant theory gives its general form. The result is a completeness theorem for the form already in use: every time-invariant two-port response in single-tone periodic steady state is a phase factor attached to the fundamental multiplying a function of the wave magnitudes and their relative phase. The Cardiff exponents become counters: $m$ is the order of the load-side nonlinearity, $n$ is set by the drive-side harmonic, and $r$ is the number of conjugate pairs. The bound $r_{\max}=\lfloor K/2\rfloor$ for a load-side nonlinearity of degree $K$ makes the familiar restriction $r\le1$ a hypothesis about the device, exact for a cubic nonlinearity; for a loaded device the bound becomes a measurable decay in $r$. The results are illustrated with a published measurement and two simulations.

eess.SP↗
arXiv2026

Symmetry and the Form of Nonlinear Behavioral Models: A Tutorial for Microwave Engineers

Behavioral models of nonlinear microwave devices -- the Cardiff model, X-parameters, the higher-order describing functions of the mechanical-systems literature -- all share a functional form. This tutorial derives that form from time invariance alone. It develops the consequences of that derivation using only the harmonic-balance description of a single-tone periodic steady state, in which each port waveform is a finite set of harmonic phasors, and uses them to give the Cardiff model's three exponents physical interpretations. In the monomial rewriting the magnitude exponent $m$ is the order of the device's load-side nonlinearity; the phase exponent $n$ is set by the drive-side harmonic; and the conjugate index $r$ obeys $r_{\max}=\lfloor K/2\rfloor$, where $K$ is the degree of the load-side nonlinearity. The familiar restriction $r\le1$ is therefore a statement about the device, exact whenever $K\le3$. The final sections show that a tailored A-pull measurement displays this decomposition directly: each spectral cluster's half-width is the order of the nonlinearity that produced it. The tutorial is pedagogical and is meant as a gentler introduction to the results treated in the companion paper "Time invariance, circle symmetry, and the completeness of the Cardiff behavioral model", which states and proves the theorems in full.

eess.SP↗
arXiv2026

Memory in Behavioral Models as Motion on a Slow Invariant Manifold

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, $14 μ$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.

eess.SP↗
arXiv1994

Braid analysis of chaos

The (forced) periodic orbit spectrum is calculated for a chaotic time series (BZ reaction).

chao-dyn↗
arXiv1994

Braid analysis of a bouncing ball

Prunning front is calculated for a bouncing ball system and the results are compared to a braid analysis. To Appear in Physical Review E.

chao-dyn↗
arXiv1993

Teardrop and heart orbits of a swinging Atwood's Machine

An exact solution is presented for a swinging Atwood's machine. This teardrop-heart orbit is constructed using Hamilton-Jacobi theory. The example nicely illustrates the utility of the Hamilton-Jacobi method for finding solutions to nonlinear mechanical systems when more elementary techniques fail.

chao-dyn↗
arXiv1993

Discrete Dynamical Models Showing Pattern Formation in Subaqueous Bedforms

A new class of ``toy models'' for subaqueous bedform formation are proposed and examined. These models all show a similar mechanism of wavelength selection via bedform unification, and they may have applications to bedform stratigraphy. The models are also useful for exploring general issues of pattern formation and complexity in stochastically driven far from equilibrium systems.

patt-sol↗
arXiv1993

Topological organization of (low-dimensional) chaos

Recent progress toward classifying low-dimensional chaos measured from time series data is described. This classification theory assigns a template to the time series once the time series is embedded in three dimensions. The template describes the primary folding and stretching mechanisms of phase space responsible for the chaotic motion. Topological invariants of the unstable periodic orbits in the closure of the strange set are calculated from the (reconstructed) template. These topological invariants must be consistent with any model put forth to describe the time series data, and are useful in invalidating (or gaining confidence in) any model intended to describe the dynamical system generating the time series.

chao-dyn↗