Virtual-Gyration Dynamics

• Virtual-gyration scenarios are defined as emergent rotation about an effective pivot rather than the geometric center, enabling off-axis dynamics.
• The approach is demonstrated in magnetic microparticles, vortex-state nanomagnets, and coupled skyrmions using tailored magnetic fields, spin-wave drives, and nonlinear mode coupling.
• This framework facilitates controlled signal propagation, mode-selective amplification, and retrograde locomotion through dynamic state transitions and dissipative coupling.

Searching arXiv for the cited papers and related uses of “virtual-gyration” across physical systems. A virtual-gyration scenario denotes a class of descriptions in which effective rotational or orbital dynamics are referenced to an emergent pivot, mode, or gyration-based descriptor rather than to a particle’s geometric center or to a directly driven low-frequency rotation. In the most concrete usage, anisotropic magnetic microparticles in a rotating magnetic trap behave as though they rotate about a virtual pivot located at a magnetization maximum inside the particle, and this local off-axis gyration can transition into large-area circular locomotion (Liu et al., 2021). Closely related usages appear in vortex-state nanomagnets, where polarity-selective vortex-core gyration, spin-wave-mediated gyration, and Floquet-sustained gyration are treated as continuous dynamical states [(Kumar et al., 2013); (Yan et al., 2014); (Philippe et al., 26 Jul 2025); (Devolder et al., 13 Nov 2025)]. The expression is also extended to coupled skyrmion gyration modes used as information carriers (Kim et al., 2016), to dissipatively coupled microscopic gyrators (Dutta et al., 2023), to gyration-based percolation descriptors built from the radius of gyration (Norizoe et al., 2020), and to holonomy-driven frame rotation in curved-space virtual reality (Weeks, 2020).

1. Definition and scope

In the context of magnetically actuated microparticles, a virtual-gyration scenario corresponds very closely to off-axis gyration and its transition into large-area circular locomotion. The particle behaves as if it is rotating and orbiting around a virtual center that is set by its internal magnetic morphology and the structure of the magnetic potential well, rather than by its geometric center (Liu et al., 2021). The essential ingredients are a rigid but internally heterogeneous object, a localized potential well whose characteristic size is comparable to the particle, and a rotating field that supplies torque while the gradient field supplies trapping.

In magnetic-vortex literature, the phrase is used differently but with the same structural idea: the operative state is a dynamical gyration mode rather than a static configuration. Kumar, Barman, and Barman treat polarity-dependent vortex-core gyration amplitude and phase as information carriers without switching polarity, so that logical “1” and “0” can be mapped to high versus low gyration amplitude or to amplitude/phase combinations relative to an external clock (Kumar et al., 2013). A further extension appears when a high-frequency spin-wave drive reshapes the energy landscape of a vortex and thereby excites or sustains a sub-GHz gyrotropic orbit indirectly, rather than by direct resonant forcing [(Yan et al., 2014); (Philippe et al., 26 Jul 2025); (Devolder et al., 13 Nov 2025)].

This breadth of usage suggests that virtual gyration is not a single mechanism but a unifying viewpoint. In every case, the operative “center” or “state” is emergent: a trapped magnetic core, a gyrotropic eigenmode, a Floquet-dressed comb, a coupled skyrmion displacement field, or a scalar descriptor based on radius of gyration.

2. Off-axis gyration in anisotropic magnetic microparticles

The clearest literal realization is the rotating magnetic trap studied for anisotropic core-shell microparticles made of one or several magnetizable cores embedded in a larger polymeric shell (Liu et al., 2021). Two facts govern the kinematics: the geometric center of the composite particle does not necessarily coincide with the centers of its magnetic cores, and the magnetic potential well has a lateral size similar to the particle dimension. Under a rotating external field, the particle can rotate about its geometric center or gyrate about one of the magnetic cores, which is offset from the geometric center. The effective rotation center then becomes a virtual gyration center located at the core that is strongly trapped and magnetized.

The dynamic magnetic trap is generated by a pair of rotating permanent magnets. The field rotates at 0–1500 rpm, corresponding to approximately $0$–$25\ \mathrm{Hz}$, with field strength $5$–$40\ \mathrm{mT}$ at the trap center. The gradient components $\nabla B_x$ and $\nabla B_z$ create, respectively, a dipolar-shaped lateral well and a peaked vertical component that provides lift or levitation. Because the particle size is comparable to the width of this well, different parts of the same particle experience different local gradients, which allows a symmetric trapping mode and an off-axis trapping mode. In the latter, one core sits at the well center and acts as the virtual pivot (Liu et al., 2021).

The dynamical modes are organized by field strength and frequency. At low field strength and low frequency, the particle shows synchronous rotation about its geometric center, with phase lag $< \pi/2$. When the frequency exceeds a critical value for a given field strength, the phase lag exceeds $\pi/2$ and the particle enters asynchronous oscillation. At higher field strength and reduced magnet–particle distance, one core is firmly trapped at the potential well center and the particle undergoes synchronous off-axis gyration. At higher rotation frequencies, above approximately $7\ \mathrm{Hz}$ or in the $400$–$25\ \mathrm{Hz}$0 range, centrifugal forces due to off-axis rotation can exceed local trapping plus friction, and the system enters large-area circular locomotion with radii up to approximately $25\ \mathrm{Hz}$1 (Liu et al., 2021).

The local torque balance is written as

$25\ \mathrm{Hz}$2

and, for rotational dynamics in a viscous environment,

$25\ \mathrm{Hz}$3

For the transition to large-area motion, the relevant balance is between centrifugal force, magnetic gradient trapping, and substrate friction. The normal force is

$25\ \mathrm{Hz}$4

and the friction force is

$25\ \mathrm{Hz}$5

The paper states that large-area circular motion becomes possible when centrifugal force exceeds the sum of trapping and friction, and that the orbit radius can be tuned by field frequency, particle length, and off-axis geometry (Liu et al., 2021).

A distinctive feature of the large-area mode is retrograde locomotion. The field may rotate clockwise while the net orbit of the particle is anticlockwise. The mechanism is a brachiation-like switching of the local pivot between different cores: one core supports the particle while the other is slightly lifted by the vertical gradient, and after about half a cycle the roles swap. The combined effect of clockwise local rotations and pivot swapping yields a net anticlockwise orbital drift. For orbits with radii $25\ \mathrm{Hz}$6–$25\ \mathrm{Hz}$7, the corresponding centripetal forces are reported as approximately $25\ \mathrm{Hz}$8–$25\ \mathrm{Hz}$9, with periods $5$0–$5$1 (Liu et al., 2021).

The dependence on magnetic morphology is central. Dual-core particles with unequal cores, such as $5$2 and $5$3, preferentially trap the larger core because it carries a larger induced magnetic moment. Three-core particles, being longer, provide a larger effective step length and maintain efficient circular motion with speed scaling with the field up to approximately $5$4. This establishes a design principle already explicit in the paper: to achieve controllable virtual-gyration centers and large-area motion, the particle should present a dominant, localized magnetization maximum that can be pinned selectively by the potential well (Liu et al., 2021).

3. Vortex-core implementations: polarity selectivity, spin-wave mediation, and Floquet magnons

In vortex-state nanomagnets, virtual gyration is formulated as a dynamical-state representation. For a $5$5 thick Permalloy disk of diameter $5$6, Kumar, Barman, and Barman numerically obtained a gyrotropic frequency $5$7, and showed that the direction of gyration is controlled by core polarity rather than chirality (Kumar et al., 2013). A rotating in-plane field at or near $5$8 excites a large-amplitude response when the field rotation sense matches the intrinsic gyrotropic sense set by the polarity, and a much smaller response when it does not. In that framework, information is encoded in gyration amplitude and phase without switching polarity. The same study extended the idea to dipole-coupled vortices and short vortex chains, where opposite polarities can produce asymmetric energy transfer and effective amplification. In a two-vortex system at $5$9, for example, the $40\ \mathrm{mT}$0 configuration gave $40\ \mathrm{mT}$1 in the excited vortex and $40\ \mathrm{mT}$2 in the passive vortex, and at $40\ \mathrm{mT}$3 the same polarity relation gave $40\ \mathrm{mT}$4 versus $40\ \mathrm{mT}$5 (Kumar et al., 2013).

A second route to virtual gyration is indirect excitation by spin waves. In a $40\ \mathrm{mT}$6 diameter, $40\ \mathrm{mT}$7 thick Permalloy disk with an in-plane bias field $40\ \mathrm{mT}$8, an out-of-plane oscillating field

$40\ \mathrm{mT}$9

excites spin-wave modes that reshape the energy landscape seen by the shifted vortex core (Yan et al., 2014). At $\nabla B_x$0, the radial spin-wave modes occur at approximately $\nabla B_x$1, $\nabla B_x$2, and $\nabla B_x$3. When $\nabla B_x$4 shifts the vortex off center, cylindrical symmetry is broken, the spin-wave spectrum splits, and resonant excitation of these modes can induce obvious vortex gyration even though the gyrotropic frequency is far below the drive frequency. The Thiele equation is written as

$\nabla B_x$5

and the paper shows that the dominant effect of spin waves is a modification of $\nabla B_x$6 and $\nabla B_x$7, not a large change in the gyrovector. Under simultaneous gyrotropic motion and spin-wave drive, the effective damping can be enhanced or reduced, and can even become negative at selected $\nabla B_x$8 combinations (Yan et al., 2014).

Recent work then reframed this interplay in explicitly Floquet terms. Time-resolved measurements in vortex-state magnetic tunnel junction nanopillars showed that forced excitations in the frequency range of the first-order azimuthal spin waves scatter into the vortex gyration mode and a Floquet spin wave, producing a comb at

$\nabla B_x$9

with spacing equal to the gyration frequency (Devolder et al., 13 Nov 2025). In a representative device, $\nabla B_z$0 was approximately $\nabla B_z$1, while the thermally observed azimuthal modes occurred near $\nabla B_z$2 and $\nabla B_z$3. The paper concluded that the first-to-occur scattering mechanism is a three-wave splitting of a regular azimuthal eigenmode into a coherent pair formed by a gyration magnon and a Floquet spin wave, and reported a common incubation delay for the gyration and the dominant Floquet sideband that diverges at the scattering threshold and can be as short as $\nabla B_z$4 at the maximum investigated power (Devolder et al., 13 Nov 2025).

A complementary theoretical and computational treatment in a $\nabla B_z$5 diameter, $\nabla B_z$6 thick CoFeB-like disk wrote the core dynamics in Thiele form,

$\nabla B_z$7

with $\nabla B_z$8, and analyzed how driven GHz azimuthal modes generate a nonlinear Floquet bath that sustains core gyration (Philippe et al., 26 Jul 2025). The resulting radial equation,

$\nabla B_z$9

shows that steady radii satisfy $< \pi/2$0. Under suitable field conditions, the work found multiple steady-state gyration radii, distinct Floquet frequency comb spectra, hysteretic effects, and, in one regime, up to three stable radii at the same drive. In this precise sense, the gyration is a driven, dissipative state engineered by Floquet magnons rather than a direct low-frequency response (Philippe et al., 26 Jul 2025).

4. Coupled skyrmion gyration as virtual transport

A skyrmion-based version of the virtual-gyration scenario was established in one-dimensional periodic arrays of Néel-type skyrmions in narrow-width nanostrips (Kim et al., 2016). The micromagnetic model used $< \pi/2$1, $< \pi/2$2, $< \pi/2$3, $< \pi/2$4, and thickness $< \pi/2$5. A single confined skyrmion showed a clockwise gyrotropic mode at approximately $< \pi/2$6 and a breathing mode at approximately $< \pi/2$7. For coupled skyrmions, the guiding-center coordinates $< \pi/2$8 were extracted from the topological charge density

$< \pi/2$9

In a five-skyrmion chain, local excitation of the first skyrmion generated five discrete coupled gyration modes at $\pi/2$0, $\pi/2$1, $\pi/2$2, $\pi/2$3, and $\pi/2$4. In a $\pi/2$5-skyrmion strip of length $\pi/2$6, width $\pi/2$7, and thickness $\pi/2$8, the coupled gyration wave packet propagated with group velocity approximately $\pi/2$9. The dispersion was concave up, with minimum at $7\ \mathrm{Hz}$0 and maximum at the Brillouin-zone boundary $7\ \mathrm{Hz}$1, and the dominant interaction was exchange rather than dipolar coupling. The paper states explicitly that the temporal variation of the exchange energy density is two times larger than that of the magnetostatic energy during the skyrmion cores’ motion, indicating exchange-dominated coupling (Kim et al., 2016).

This system is virtual in the specific sense that information is carried by oscillatory displacement of skyrmion cores rather than by physical translation of skyrmions along a racetrack. A local excitation of one skyrmion launches a gyration wave packet that propagates along the chain while each skyrmion remains near its equilibrium position. The approach thereby avoids the skyrmion Hall effect and corner annihilation issues associated with translational schemes. The same study showed propagation through an L-shaped nanostrip without skyrmion annihilation at the bend, and demonstrated that a perpendicular field $7\ \mathrm{Hz}$2 tunes the dispersion, the zone-boundary frequency, and the propagation speed (Kim et al., 2016).

5. Dissipative and statistical formulations

Virtual-gyration ideas also appear in stochastic systems where steady rotation is generated not by conservative coupling but by dissipative coupling. For an overdamped Brownian chiral ellipsoid in an isotropic harmonic trap,

$7\ \mathrm{Hz}$3

the effective translational dynamics in the small-coupling limit is

$7\ \mathrm{Hz}$4

with $7\ \mathrm{Hz}$5 encoding anisotropic friction and a finite tilt due to chirality (Dutta et al., 2023). The trap is isotropic; the coupling is carried by the friction or mobility tensor. The resulting tangential current in polar coordinates is nonzero only when $7\ \mathrm{Hz}$6 and $7\ \mathrm{Hz}$7, and the steady-state angular velocity is

$7\ \mathrm{Hz}$8

The same paper then considered an inertial granular chiral ellipsoid with no trapping force, anisotropic athermal fluctuations $7\ \mathrm{Hz}$9, and dissipative coupling in velocity space,

$400$0

and obtained an analogous steady-state gyration in $400$1-space. The dominant contribution toward the gyrating frequency was attributed to the Coriolis force acting on the granular ellipsoid (Dutta et al., 2023).

A more abstract use of virtual gyration appears in ideal molecular percolation, where the radius of gyration functions as an effective control parameter rather than a literal rotational motion (Norizoe et al., 2020). In three-dimensional single-component ideal gas systems of penetrable homogeneous rigid molecules, the paper defined

$400$2

and showed that the percolation threshold obeys

$400$3

with fitted values $400$4 and $400$5 after effective-size correction. The systems with the same parameter set of the molecular volume and radius of gyration, but in different molecular shapes, were reported to show the identical value of the percolation threshold. In this literature, “virtual-gyration” therefore names an effective gyration-based descriptor that replaces detailed geometry by the scalar ratio $400$6 (Norizoe et al., 2020).

6. Holonomy, body coherence, and curved-space virtual reality

A geometrically distinct but conceptually related scenario arises in curved-space virtual reality, where motion around loops produces holonomy: a net rotation of the local frame under parallel translation (Weeks, 2020). The paper models spherical, Euclidean, and hyperbolic spaces as

$400$7

$400$8

and

$400$9

For a small square loop in $25\ \mathrm{Hz}$00, the product of four $25\ \mathrm{Hz}$01 rotation matrices gives, to lowest nontrivial order, an orientation change of order $25\ \mathrm{Hz}$02 about the $25\ \mathrm{Hz}$03-axis, while the positional error is third order in $25\ \mathrm{Hz}$04. The paper interprets this as a holonomy effect that, in real curved space, would be felt as a mysterious torque in the neck that the player must resist.

The central technical problem is body coherence: what the player sees with her eyes must agree with what she feels with her hands. A naive mapping of head motion into curved space would make the virtual head undergo holonomy while the virtual hands remained inconsistent with the real hands. The solution is an anchor-based tracking algorithm. In the billiards implementation, one chooses an anchor point on the table edge, expresses head and hand poses relative to that anchor, updates a corresponding virtual anchor by the same local motion interpreted in the curved manifold, and then places the virtual head and hands relative to the virtual anchor. This absorbs the global holonomy into the evolving anchor frame rather than into visible head–hand misalignment (Weeks, 2020).

The same paper also introduced a native-inhabitant view, implemented by an azimuthal equidistant map from the manifold onto the Euclidean tangent space at the bridge of the nose, in order to correct the tourist misperception of distances in $25\ \mathrm{Hz}$05 and $25\ \mathrm{Hz}$06. This is not a magnetic or micromechanical gyration, but it preserves the same structural feature as other virtual-gyration scenarios: a global rotational effect is real in the geometry, yet local coherence is maintained by a carefully chosen effective representation (Weeks, 2020).

Across these literatures, the common content of a virtual-gyration scenario is therefore precise. A localized or effective center of motion is defined by internal morphology, nonlinear mode coupling, collective texture dynamics, dissipative cross-coupling, a gyration-based descriptor, or the parallel transport of a frame. The immediate consequence is controlled nontrivial motion—off-axis rotation, retrograde orbiting, mode-selective amplification, Floquet comb formation, exchange-mediated signal propagation, steady-state probability currents, scale-free percolation collapse, or holonomy with preserved body coherence—whose natural reference point is not the obvious geometric center but an emergent one [(Liu et al., 2021); (Kumar et al., 2013); (Yan et al., 2014); (Kim et al., 2016); (Dutta et al., 2023); (Norizoe et al., 2020); (Weeks, 2020); (Devolder et al., 13 Nov 2025); (Philippe et al., 26 Jul 2025)].