Analytical Hysteresis Models

Updated 1 February 2026

Analytical hysteresis is a set of closed-form models that explicitly capture memory-dependent, nonlinear behavior in physical, electronic, and mechanical systems.
These models use operator-based, parametric, and differential equation frameworks to generate precise hysteresis loops, including major and minor loops as well as energy dissipation metrics.
They enable direct mapping of input-output relationships and support both online compensation and data-driven discovery, advancing control and optimization in complex dynamic systems.

Analytical hysteresis describes closed-form, mathematically explicit models that capture the memory-dependent, multi-valued, and nonlinear behavior characteristic of systems with hysteresis. Such models are distinguished by their ability to provide exact (or systematically approximable) expressions for hysteresis loops, minor loops, and rate-dependent or rate-independent phenomena, covering a range of physical, electronic, mechanical, thermal, and computational systems. Analytical hysteresis frameworks span from classical parametric, Preisach-type, and dynamic operator-based models, to explicit solution formulas for differential equations, symbolic regression–discovered laws, and stochastic energetics approaches.

1. Foundational Principles of Analytical Hysteresis

Analytical hysteresis models aim to express output–input relations as explicit mathematical mappings incorporating both instantaneous variables and the history of system inputs. These models typically fall into several classes:

Operator-based frameworks: The Preisach model and its extensions represent the system response as a functional superposition of elementary non-ideal relays (“hysterons”), parameterized over threshold spaces and often integrated against a density, allowing for exact construction of major and minor loops (Bermúdez et al., 2019, Tannous et al., 2013, &&&2&&&, Roussel et al., 2022).
Parametric models: Explicit models, such as the generalized play operator, Lapshin-type parametric curves, or piecewise-analytical forms, provide closed-form expressions for common loop shapes and their geometric characteristics (Lapshin, 2017, Lapshin, 2020, Kuehn et al., 2017).
Differential equation approaches: Analytical solutions of ODEs or DDEs governing the state dynamics of memristors, ferroelectric, magnetic, mechanical, or neural systems yield explicit or semi-explicit hysteresis laws and loop-area formulas (Georgiou et al., 2010, Ma et al., 2016, Zakharov et al., 2014, Chen et al., 2020, Wang et al., 25 Jan 2026).
Stochastic energetics: Heat-transport and related phenomena employ stochastic Langevin methods to obtain exact energy flux and loop-area expressions under dynamic drives (Chen et al., 22 Jan 2025).
Learning-based discovery: Recent data-driven approaches enable direct extraction of governing analytical hysteretic equations from high-dimensional or experimental data using internal variable learning and symbolic regression, bypassing the need for a fixed model structure (Yang et al., 2 Dec 2025).

Analytical hysteresis models provide not only the mapping from histories to outputs but also quantification of phenomena such as rate dependence, loop area (energy dissipation), pinched/crossed loops, phase properties, and the effect of stochastic or frequency-dependent drives.

2. Preisach and Generalized Relay Frameworks

The Preisach model and its generalizations are the canonical analytical framework for history-dependent, rate-independent hysteresis. The system output is constructed as a superposition of relay (rectangular) hysterons $\gamma_{\alpha,\beta}$ , each characterized by switching thresholds $\alpha$ and $\beta$ :

$y(t) = \iint_{\alpha\geq\beta} \rho(\alpha, \beta) \gamma_{\alpha,\beta}[u](t)\; d\alpha\,d\beta$

where $\rho(\alpha,\beta)$ is the Preisach density and $\gamma_{\alpha,\beta}$ changes state based on the input’s crossings relative to $\alpha$ and $\beta$ (Roussel et al., 2022, Schubert et al., 2017, Tannous et al., 2013).

Key attributes:

The dynamic Preisach model augments each relay with rate-dependent dynamics (finite velocity $k$ ) (Bermúdez et al., 2019). The relay state $r_{\alpha,\beta}(t)$ solves

$\dot{r}_{\alpha,\beta}(t) + \partial \chi_{[-1,1]}(r_{\alpha,\beta}(t)) \ni -k (H-\alpha)^- + k (H-\beta)^+,$

creating smooth, velocity-dependent loops that reduce to the classical case as $k\to\infty$ .

Generalized play operators arise as singular limits of planar fast–slow systems, where the slow variable is projected onto a time-varying strip and the play law emerges from the variational inequality or rate-independent inclusion (Kuehn et al., 2017).
The effective Preisach density $\tilde{\mu}(u,v)$ via input distribution determines emergent long-time correlations, 1/f noise, and scaling of autocorrelation and spectra (Schubert et al., 2017).

Analytical results afford:

Closed-form loop construction for arbitrary densities or meshings (with smooth or differentiable surrogates enabling gradient-based identification (Roussel et al., 2022)).
Quantitative analysis of minor/major loops, memory loss, energy dissipation, and even full stochastic-driven output statistics.
Systematic extensions to angle-dependent, multi-modal, or FMR behavior via explicit integration over parameter spaces (Tannous et al., 2013).

3. Differential Equation Systems and Explicit Solutions

Many systems exhibiting hysteresis permit exact or semi-exact analytical solutions for their state variables and output–input characteristics:

Memristors and resistive-switching devices: The current-voltage relation under generic input is governed by a Bernoulli ODE, yielding explicit solutions for $i(v)$ under sinusoidal, triangular, or piecewise inputs, and closed-form expressions for normalized hysteresis as a function of a dimensionless design parameter $\beta$ (Georgiou et al., 2010).

$\bar{H}_{\text{sin}} = \frac{2}{\pi} \left( \int_0^\pi \frac{\sin^2 x}{\sqrt{1-\beta (1-\cos x)}} dx - 2 \int_0^{\pi/2} \frac{\sin^2 x}{\sqrt{1-\beta (1-\cos x)}} dx \right)$

Dynamic bifurcation systems: Hysteresis between attractors (equilibrium and limit cycles) is captured analytically through normal form reduction (via multi-scale expansions), providing explicit amplitude equations to fifth order that reveal hysteresis loop width and fold bifurcations in parameter space for delay ODE/DDE systems (Chen et al., 2020).
Mechanical and traffic models: Phase-space analysis yields closed-form parametric expressions (Lissajous ellipses) for hysteresis between variables, with precise relations among amplitude, phase lag, and loop area, directly connected to time-delay, synchronization, and energy dissipation metrics (Wang et al., 25 Jan 2026).

Analytical solutions enable:

Direct derivation of performance metrics, e.g., work per cycle, dissipation, and phase boundaries.
Control and design guidance: explicit links between device/material parameters and hysteresis response.

4. Explicit Parametric, Self-Similar, and Relaxation Models

A broad class of physically-motivated systems admits analytical hysteresis representation through explicit parametric or self-similar forms:

Parametric and phase-shift models: Explicit software– and hardware–ready parametric models can generate leaf, crescent, classical, double, triple, minor, and mixed loops with less than 1% fitting error, and supply closed expressions for coercivity, remanence, energy, phase, and linearization coefficients (Lapshin, 2017, Lapshin, 2020).
Self-similar free-energy approaches: For ferroelectric and related systems, models based on a homogeneous logarithmic–cosh free-energy function yield analytic polarization–field relations with tunable parameters. These compact forms naturally capture a family of loop morphologies and their control by external fields, temperature, and stress (Ma et al., 2016).
Relaxation-type switching models: The analytical solution for the evolution of polarization (or magnetization) in response to arbitrary time-dependent fields, including responses to sinusoidal drives, is available if the domain switching rate is prescribed as a function of field and its time derivative (Zakharov et al., 2014).

Such models provide

Unified, exact mappings for experimental loop fitting or online compensation (e.g., scanning probe piezo actuators).
Direct expressions for design-by-parameter, e.g., controlling loop area, phase lag, and saturation properties.
Reduced computational cost over purely numerical or high-order polynomial Landau–type expansions.

5. Energy Dissipation, Loop Areas, and Power-Law Scaling

Analytical treatment is essential for quantifying the energetics of hysteresis:

Area and power laws: Closed-form or systematically approximated expressions are available for major and minor loop areas in magnetic, ferroelectric, frictional, heat-transfer, and memristive systems. Key results include:
- Loop area as a function of field amplitude and frequency, e.g., $A(\omega) \sim 1/(1+(\omega \tau)^2)$ for linear dynamic hysteresis (Chen et al., 22 Jan 2025).
- Power-law ( $A \sim H_0^n$ ) behavior with exponent $n$ varying between 0.6 and 2 depending on dynamic regime and loop type (Carrey et al., 2010).
- Exact analytical expressions for major/minor loop areas in friction models (Dahl, LuGre) and sharp characterizations of dissipation (Ikhouane et al., 2018).

Tables of analytical formulas for the area for various input forms and device types are frequently provided in the primary literature (Georgiou et al., 2010, Zakharov et al., 2014, Lapshin, 2017, Ikhouane et al., 2018, Carrey et al., 2010).

6. Algorithmic and Data-Driven Analytical Discovery

Recent advances extend analytical hysteresis modeling to data-driven discovery:

Symbolic regression frameworks: Direct equation discovery from measured data, using embedded internal variable learning and symbolic regression, has yielded explicit forms matching or exceeding classical analytical models in fidelity and interpretability. This removes the dependence on predefined model libraries and provides natural dynamic prediction of complex, multi-modal, or fractional-exponent laws (Yang et al., 2 Dec 2025).
Differentiable operator learning: Adaptive, differentiable Preisach networks enable gradient-based inference of high-dimensional, nonparametric densities, leveraging backpropagation and regularization to fit arbitrary loop shapes and integrate online with accelerator or beam control systems (Roussel et al., 2022).

These innovations facilitate:

Automated modeling and compensation of hysteresis in high-stakes applications (e.g., particle accelerators).
Rapid adaptation to previously unmodeled behaviors and mechanisms.
Expanded domains for analytical hysteresis beyond classical or low-dimensional systems.

7. Representative Applications Across Domains

Analytical hysteresis finds application across magnetism, piezoelectrics, memristors, thermal/heat-transfer, neurodynamics, structural mechanics, and even traffic dynamics:

Preisach/dynamic Preisach: Analysis and numerical simulation of electromagnetic field equations and parabolic PDEs with dynamic operator coupling (Bermúdez et al., 2019).
SQUID multiplexers: Closed-form power- and hysteresis-parameter dependence for resonator frequency response and experimental device matching (Wegner et al., 2021).
Magnetic hyperthermia and nanomagnets: Loop area and SAR optimization via explicit major/minor loop formulas (Carrey et al., 2010).
Probe microscopy and control: Real-time analytic inversion and hardware embedding for compensation (Lapshin, 2020, Lapshin, 2017).
High-precision accelerator systems: Online differentiable Preisach modeling for beam quality optimization under actuator hysteresis (Roussel et al., 2022).
Nonlinear system identification: Full symbolic discovery of hysteresis laws for mechanical, seismic, and physical systems (Yang et al., 2 Dec 2025).

These diverse applications showcase the universality and technical utility of analytical hysteresic modeling across scales and disciplines.

References:

Mathematical analysis and numerical solution of models with dynamic Preisach hysteresis (Bermúdez et al., 2019)
Analytical model of the readout power and SQUID hysteresis parameter dependence of the resonator characteristics of microwave SQUID multiplexers (Wegner et al., 2021)
Angular Preisach analysis of Hysteresis loops and FMR lineshapes of ferromagnetic nanowire arrays (Tannous et al., 2013)
Quantitative Measure of Hysteresis for Memristors Through Explicit Dynamics (Georgiou et al., 2010)
Mathematical modeling of hysteresis loops in ferroelectric materials (Ma et al., 2016)
A unified framework for equation discovery and dynamic prediction of hysteretic systems (Yang et al., 2 Dec 2025)
Minor loops of the Dahl and LuGre models (Ikhouane et al., 2018)
Hysteresis bifurcation and application to delayed Fitzhugh-Nagumo neural systems (Chen et al., 2020)
An improved parametric model for hysteresis loop approximation (Lapshin, 2017)
Modeling of hysteresis phenomena in crystalline ferroelectrics: hysteresis loops shape control by means of electric field parameters (Zakharov et al., 2014)
Preisach models of hysteresis driven by Markovian input processes (Schubert et al., 2017)
Analytical model for the approximation of hysteresis loop and its application to the scanning tunneling microscope (Lapshin, 2020)
Simple models for dynamic hysteresis loops calculation: Application to hyperthermia optimization (Carrey et al., 2010)
Heat Transport Hysteresis Generated through Frequency Switching of a Time-Dependent Temperature Gradient (Chen et al., 22 Jan 2025)
Differentiable Preisach Modeling for Characterization and Optimization of Accelerator Systems with Hysteresis (Roussel et al., 2022)
Generalized Play Hysteresis Operators in Limits of Fast-Slow Systems (Kuehn et al., 2017)
Synchronization in Traffic Dynamics: Mechanisms of Hysteresis (Wang et al., 25 Jan 2026)