🌀 nonlinear sciences

Multiple hysteresis widths in inertial Kuramoto model

This paper demonstrates that the interplay of phase lag and triadic interactions in the inertial Kuramoto model generates multiple distinct hysteresis widths corresponding to different stable states, a phenomenon driven by saddle-node bifurcations that becomes more pronounced with increasing inertia and holds potential applications in power grids and memory systems.

Original authors: Jayesh C. Jain, Sarika Jalan

Published 2026-06-15

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CC BY 4.0

Original authors: Jayesh C. Jain, Sarika Jalan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a massive crowd of people, each trying to clap their hands to their own unique rhythm. In a standard scenario, if you ask them to listen to each other, they might eventually all clap in unison. This is the basic idea of the "Kuramoto model," a famous mathematical tool used to study how things sync up, from fireflies flashing together to neurons firing in a brain.

However, this paper looks at a more complicated version of that crowd: the Inertial Kuramoto Model. Think of "inertia" here like the weight of a heavy flywheel attached to each person's hands. Because of this weight, they can't change their rhythm instantly; they have momentum. They might overshoot, wobble, or keep going even after they try to stop.

The researchers added two new ingredients to this mix:

Phase Lag: Imagine a slight delay or a "lag" in how people react to each other, like a game of telephone where the message gets slightly distorted.
Triadic Interactions: Instead of just Person A listening to Person B, imagine a rule where Person A's rhythm is influenced by a specific trio of people (A, B, and C) acting together.

The Discovery: "Multiple Hysteresis Widths"

The main finding of the paper is about something called hysteresis. In everyday terms, hysteresis is like a thermostat. You might turn the heat up to 75°F to get the room warm, but you have to turn it down to 65°F before the heater actually turns off. The "switching point" depends on which direction you are coming from (heating up or cooling down).

In this study, the researchers found that when you have heavy inertia, phase lag, and group (triadic) interactions, the system doesn't just have one switching point. It has multiple different switching points depending on where the system started.

The Analogy of the Hilly Valley:
Imagine a ball rolling in a landscape with several valleys (stable states).

The Forward Path: If you start with the ball at the top of a hill and slowly push it down (increasing the connection strength), it rolls into a specific valley.
The Backward Path: If you start with the ball deep in a valley and slowly pull it back up (decreasing the connection strength), it gets stuck in a different valley than the one it would have chosen if you had started from the top.

The paper shows that because of the "lag" and the "group rules," there are different sizes of valleys for different starting points.

If you start from a chaotic, un-synced state, the "gap" between when the system syncs up and when it falls apart is wide.
If you start from a specific, partially-synced state, the "gap" is narrower.
If you start from a different partially-synced state, the gap is a different size again.

The authors call these different gap sizes "multiple hysteresis widths." It's like having a door that requires a different amount of force to open depending on which side of the room you are standing on.

Key Findings in Simple Terms

Inertia Makes it Messier: The heavier the "flywheel" (inertia), the more pronounced these different gaps become. The system becomes more stubborn and resistant to changing its state.
The "Forward" Branch is Chaotic: When the researchers tried to build up the connection strength from zero (the forward path), the system didn't settle into a calm, steady rhythm. Instead, it kept oscillating or wobbling. It was like trying to get a heavy swing to stop moving; it just kept swinging back and forth.
The "Backward" Branch is Stable: When they started with everyone already synced and slowly reduced the connection, the system held its steady rhythm for a while before suddenly snapping back to chaos. This snap-back happens at different points depending on the initial state.
Why It Happens: The math shows that these different "snap-back" points happen because the system hits different "tipping points" (called saddle-node bifurcations) at different connection strengths.

Why This Matters (According to the Paper)

The authors suggest that understanding these multiple "gaps" or switching points could be useful for:

Power Grids: Managing how electricity flows and stabilizes in a network.
Information Storage: Creating systems that can hold different states (like memory) depending on how they were set up.
Memory Selection: Helping real-world systems choose between different stable "memories" or modes of operation.

In short, the paper reveals that complex systems with momentum, delays, and group interactions don't just have one way to switch on or off. They have a whole menu of different switching behaviors, and which one you get depends entirely on where you started.

Technical Summary: Multiple Hysteresis Widths in Inertial Kuramoto Model

Problem Statement
The inertial Kuramoto model (KMI) is a standard framework for describing synchronization in systems with inertia, such as power grids and neuronal networks. While multistability (the coexistence of multiple attractors) and hysteresis are known features of KMI, the specific dynamics arising from the simultaneous presence of phase lag and higher-order (triadic) interactions remain unexplored. Standard reduction techniques like the Ott-Antonsen approach are inapplicable to KMI because the density function depends explicitly on velocity. Consequently, there is a gap in understanding how the interplay of inertia, phase lag, and triadic coupling influences the stability of synchronized states and the nature of synchronization transitions.

Methodology
The authors investigate a system of $N$ globally coupled oscillators governed by a second-order Kuramoto-Sakaguchi equation. The model incorporates:

Inertia ( $m$ ): A second-order time derivative term.
Pairwise and Triadic Interactions: Coupling strengths $K_1$ (pairwise) and $K_2$ (triadic).
Phase Lag ( $\alpha$ ): A constant phase shift in the coupling terms.

To analyze the system, the authors employ a self-consistent mean-field approach rather than dimensional reduction. They define order parameters $r_1$ (global coherence) and $r_2$ (2-cluster state) and transform the variables into a rotating frame. This allows the derivation of an analytical solution for the steady-state behavior. The dynamics are further simplified by introducing auxiliary variables to reduce the equation of motion to a form resembling a damped driven pendulum.

The study utilizes two distinct numerical procedures to map the parameter space:

Forward Branch: Starting from an incoherent state (random phases) and adiabatically increasing coupling strength $K_1$ .
Backward Branch: Starting from a coherent state and adiabatically decreasing $K_1$ .

Key Results

Multiple Hysteresis Widths: The central finding is that the interplay between phase lag and triadic interactions leads to distinct hysteresis widths corresponding to different stable states. Unlike the standard KMI with only pairwise interactions (which typically shows a single hysteresis loop), the inclusion of triadic terms and phase lag causes the backward transition points to diverge depending on the initial multistable state.
Dependence on Initial Conditions: The backward transition points ( $\tilde{K}_{1c}$ ) are not unique; they vary based on which multistable branch the system occupies. The study demonstrates that hysteresis widths associated with multistable states are generally smaller than those associated with the globally coherent state.
Forward vs. Backward Dynamics:
- Forward Branch: The system does not admit a steady-state solution; instead, it exhibits time-dependent, oscillatory behavior (limit cycles) before transitioning.
- Backward Branch: The system admits steady-state solutions (fixed points). The transition from the synchronized state to the incoherent state occurs via a saddle-node bifurcation.
Analytical Characterization: Theoretical calculations confirm that the multiple hysteresis widths arise because saddle-node bifurcations occur at different coupling strengths for different initial conditions. The analysis identifies three distinct regions in the parameter space: fixed-point regions, limit-cycle regions, and bistable regions where both coexist.

Significance and Claims
The paper claims that the study of multiple hysteresis widths provides a more nuanced understanding of synchronization transitions in complex systems. Specifically, the authors suggest that these findings may be useful for:

Modeling Power Grid Systems: Where inertia and complex interactions are critical.
Information Storage and Memory Selection: The existence of multiple stable states with distinct transition thresholds offers a mechanism for selecting specific memory states or storing information in real-world systems.

The authors emphasize that their work fills a theoretical gap by characterizing the combined effect of higher-order interactions and phase lag in inertial systems, a combination previously unexplored in the literature.

The Discovery: "Multiple Hysteresis Widths"

Key Findings in Simple Terms

Why This Matters (According to the Paper)

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