Nonlinear Oscillators Research Articles

Classical chaos in quantum computers

The development of quantum computing hardware is facing the challenge that current-day quantum processors, comprising 50–100 qubits, already operate outside the range of quantum simulation on classical computers. In this paper we demonstrate that the simulation of limits can be a potent diagnostic tool for the resilience of quantum information hardware against chaotic instabilities potentially mitigating this problem. As a testbed for our approach we consider the transmon qubit processor, a computing platform in which the coupling of large numbers of nonlinear quantum oscillators may trigger destabilizing chaotic resonances. We find that classical and quantum simulations lead to similar stability metrics (classical Lyapunov exponents vs quantum wave function participation ratios) in systems with O(10) transmons. However, the big advantage of classical simulation is that it can be pushed to large systems comprising up to thousands of qubits. We exhibit the utility of this classical toolbox by simulating all current IBM transmon chips, including the 433-qubit processor of the Osprey generation, as well as devices with 1121 qubits (Condor generation). For realistic system parameters, we find a systematic increase of Lyapunov exponents with system size, suggesting that larger layouts require added efforts in information protection. Published by the American Physical Society 2024

Frequency response based identification of nonlinear oscillators

Experimental modal analysis is commonly used to identify models for the vibratory behavior of structures. This is done by conducting a set of experiments to obtain the structure’s governing equations and information in the form of eigenfrequencies, mode shapes, and damping. However, linearity of the test structure is assumed within this identification procedure. Hence, as it stands, experimental modal analysis is not readily applicable to build models when nonlinearities are present through, for example, friction, (electro-) magnetic fields, or large deformations. To identify governing equations for such systems, a robust and systematic identification procedure is proposed in this article. The identification routine is formulated in the frequency domain, and a noise reduction scheme and a simplification routine are employed to obtain sparse and robust models. The identification procedure is implemented in an automated script (FrID), which is applied to forced response measurements stemming from structures with magnets, clamps, and bolted joints as well as systems with multiple active modes and internal resonances. The identified governing equations accurately fit the experimentally obtained frequency response measurements and can also be utilized to extrapolate the response for different forcing amplitudes. Moreover, nonlinear modes of the underlying conservative system can be computed from the identified governing equations.

Relaxation and Noise-Driven Oscillations in a Model of Mitotic Spindle Dynamics

During cell division, the mitotic spindle moves dynamically through the cell to position the chromosomes and determine the ultimate spatial position of the two daughter cells. These movements have been attributed to the action of cortical force generators which pull on the astral microtubules to position the spindle, as well as pushing events by these same microtubules against the cell cortex and plasma membrane. Attachment and detachment of cortical force generators working antagonistically against centring forces of microtubules have been modelled previously (Grill et al. in Phys Rev Lett 94:108104, 2005) via stochastic simulations and mean-field Fokker–Planck equations (describing random motion of force generators) to predict oscillations of a spindle pole in one spatial dimension. Using systematic asymptotic methods, we reduce the Fokker–Planck system to a set of ordinary differential equations (ODEs), consistent with a set proposed by Grill et al., which can provide accurate predictions of the conditions for the Fokker–Planck system to exhibit oscillations. In the limit of small restoring forces, we derive an algebraic prediction of the amplitude of spindle-pole oscillations and demonstrate the relaxation structure of nonlinear oscillations. We also show how noise-induced oscillations can arise in stochastic simulations for conditions in which the mean-field Fokker–Planck system predicts stability, but for which the period can be estimated directly by the ODE model and the amplitude by a related stochastic differential equation that incorporates random binding kinetics.

Second Harmonic Generation from exciton-polaritons: Strong coupling between monolayer WS[formula omitted] and multilayer WSe[formula omitted] metasurfaces Quasibound states in the continuum

The second harmonic generation (SHG) from exciton-polaritons, produced by the strong coupling between quantum emitters and cavity photons, can provide a unique insight for controlling nonlinear responses on the nanoscale. Achieving SHG generally depends on phase matching, limited by nonlinear materials and nanostructure fabrication. Here, we demonstrate a novel way to obtain tunable SHG from van der Waals metasurfaces composed of nanostructured multilayer WSe2 strong coupling with monolayer WS2. By breaking symmetry via varying the metasurface unit tilted angle and period, quasibound state in the continuum (q-BIC) mode resonances energy is tuned across the monolayer WS2 exciton resonance energy, enabling anticrossing behavior of strong coupling in both reflection spectrum and SHG spectrum, which means that SHG was successfully observed by strong coupling. Furthermore, the SHG spectrum can be tuned by changing the armchair direction of the monolayer WS2 and pump polarization direction. Moreover, the mechanism of exciton-polaritonic SHG is investigated by a coupled nonlinear oscillator model, which reveals the SHG origin from exciton-polaritons, providing a new rule for fine-tuning the SHG. Our work provides a new strategy and additional freedom in the nonlinear light-matter interaction and coherent control of the SHG in integrated photonic devices.

Weakly nonlinear shape oscillations of viscoelastic drops

Axisymmetric shape oscillations of a viscoelastic drop in a vacuum are studied by a weakly nonlinear analysis. The two-lobed initial drop deformation mode is studied. The Oldroyd-B model is used for characterizing the drop liquid rheological behaviour. The equations of motion and the solutions up to second order are developed, where elastic effects appear. Solutions of the characteristic equation are validated against the decay rate and frequency of damped shape oscillations of polymer solution drops in an acoustic levitator. The theory shows enhancing or dampening of the nonlinear behaviour and enhanced mode coupling, as compared with the Newtonian case. The study reveals an excess time in the prolate shape and a frequency change, together with a quasi-periodicity of the oscillations. The Fourier power spectra of traces of the drop north pole position in time show mode coupling as a nonlinear effect. At moderate stress-relaxation Deborah number, the resultant drop motion for large Ohnesorge number, suggesting aperiodic drop behaviour, may nonetheless be oscillatory, where the drop elasticity is owing to the liquid bulk elasticity instead of surface tension. A method is developed to predict the oscillatory or aperiodic behaviour of a drop of a given size from a rheologically characterized viscoelastic Oldroyd-B liquid.

Realization of logic gates in bi-directionally coupled nonlinear oscillators.

Implementation of logic gates has been investigated in nonlinear dynamical systems from various perspectives over the years. Specifically, logic gates have been implemented in both single nonlinear systems and coupled nonlinear oscillators. The majority of the works in the literature have been done on the evolution of single oscillators into OR/AND or NOR/NAND logic gates. In the present study, we demonstrate the design of logic gates in bi-directionally coupled double-well Duffing oscillators by applying two logic inputs to the drive system alone along with a fixed bias. The nonlinear system, comprising both bi-directional components, exhibits varied logic behaviors within an optimal range of coupling strength. Both attractive and repulsive couplings yield similar and complementary logic behaviors in the first and second oscillators. These couplings play a major role in exhibiting fundamental and universal logic gates in simple nonlinear systems. Under a positive bias, both the first and second oscillators demonstrate OR logic gate for the attractive coupling, while exhibiting OR and NOR logic gates, respectively, for the repulsive coupling. Conversely, under a negative bias, both the first and second oscillators display AND logic gate for the attractive coupling, and AND and NAND logical outputs for the repulsive coupling. Furthermore, we confirm the robustness of the bi-directional oscillators against moderate noise in maintaining the desired logical outputs.

Response analysis of the Duffing oscillator in dielectric elastomer transducers

To elucidate the nonlinear oscillations inherent to dielectric elastomer transducers (DETs), this study delves into the dynamic behaviour exhibited by a one-dimensional asymmetric Duffing oscillator, which may exhibit either hardening or softening stiffness. The investigation utilizes hyperelastic and visco-hyperelastic constitutive models to capture the nonlinear stiffness properties of dielectric elastomers within compliant transducers. The canonical Duffing equation is employed to articulate the oscillator’s motion, complemented by both asymptotic and quantitative analyses of unforced vibrations and harmonic excitation. Our results reveal that the distinctive stiffness characteristics observed in DETs arise from the complex interplay among hyperelasticity, rheology and applied electric fields. Contrary to conventional oscillators, Maxwell stress substantially modifies the nonlinear dynamics of DETs, an aspect that has been quantitatively dissected in this study. Furthermore, the influence of additional system parameters on forced oscillation has also been meticulously evaluated.

Modeling of Human-Induced Dynamic Bouncing Force Using a Self-Sustained Nonlinear Oscillator

This paper proposed a single-degree-of-freedom self-sustained nonlinear oscillator capable of precisely predicting the bouncing force induced by a person during bouncing activity on a flat and rigid surface. A bouncing person produces essential internal energy required to maintain its motion, so it can be modeled as a self-sustained oscillator that can generate (i) the stable limit cycle, (ii) the periodic bouncing force signal, and (iii) the self-sustained motion. A hybrid Van der Pol–Rayleigh oscillator added with two quadratic and one cubic nonlinear terms has been derived to yield desired softening and hardening effects as well as even and odd harmonics, as observed from the analysis of experimental bouncing force data. The force applied on the surface corresponds to its restoring force. The stability analysis of the oscillator has been performed using the energy balance and perturbation methods. The model parameters are extracted from the experimental bouncing force data resulting from a bouncing test on a group of seven subjects with shoe insoles at six different frequencies guided by a metronome. The bootstrapping method has been performed to examine the convergence of mean values of each model parameter by increasing the cardinality of the experimental set. The bouncing force signals produced by the proposed model and experimental results demonstrate an excellent agreement.

Periodic Solutions of Strongly Nonlinear Oscillators Using He’s Frequency Formulation

In this paper, we address several scientific and technological challenges with a novel non-perturbative approach (NPA), simplifying processing time compared to traditional methods. The proposed approach transforms nonlinear ordinary differential equations into linear ones, akin to simple harmonic motion, producing a new frequency. This method yields highly accurate outcomes, surpassing well-known approximate methodologies, as validated through numerical comparisons in mathematical software. The congruence between numerical solution tests and theoretical predictions further supports our findings. While classical perturbation methods rely on Taylor expansions to simplify restoring forces, NPA also enables stability analysis. Thus, for analyzing approximations of highly nonlinear oscillators in mathematical software, NPA serves as a more reliable tool.

Observation of a phase transition from a continuous to a discrete time crystal

Discrete (DTCs) and continuous time crystals (CTCs) are novel dynamical many-body states, that are characterized by robust self-sustained oscillations, emerging via spontaneous breaking of discrete or continuous time translation symmetry. DTCs are periodically driven systems that oscillate with a subharmonic of the external drive, while CTCs are continuously driven and oscillate with a frequency intrinsic to the system. Here, we explore a phase transition from a continuous time crystal to a discrete time crystal. A CTC with a characteristic oscillation frequency ωCTC is prepared in a continuously pumped atom-cavity system. Modulating the pump intensity of the CTC with a frequency ωdr close to 2ωCTC leads to robust locking of ωCTC to ωdr/2 , and hence a DTC arises. This phase transition in a quantum many-body system is related to subharmonic injection locking of non-linear mechanical and electronic oscillators or lasers.

Nonlinear Aeroelastic Oscillations in the Wall of a Flat Channel Filled with Viscous Gas and Resting on a Vibrating Foundation

Nonlinear Aeroelastic Oscillations in the Wall of a Flat Channel Filled with Viscous Gas and Resting on a Vibrating Foundation

Non-linear dynamics and bandgap control in magneto-rheological elastomers metamaterials with inertial amplification

Sandwich plate structures are extensively utilized in engineering due to their favorable stiffness-to-weight ratio. Nonetheless, these lightweight thin-walled structures frequently encounter challenges related to inadequate low-frequency vibration performance, which significantly restricts their applications, particularly in the realm of precision instruments which is sensitive to vibration. This study introduces an innovative design of an active nonlinear metamaterial, to achieve tunable broadband low-frequency bandgaps for sandwich plates. The nonlinear oscillator incorporates an inertia amplification mechanism (IAM), Euler-buckled beams, mass elements, and magneto-rheological elastomers (MREs), which are modulated via external magnetic fields to adjust the material's stiffness dynamically. Employing Hamilton's principles and the plate wave expansion method (PWE), the dispersion relations for the metamaterial plate are derived, elucidating the dispersion surfaces and the band structures within its sandwich-like plates.The dynamical equations of the metamaterial plate are formulated and validated through numerical simulations using the Galerkin method, confirming the theoretical predictions. The results demonstrate effective control over low-frequency and broadband bandgaps under low mass ratio conditions through strategic manipulation of the inertia amplification factor and magnetic flux. The study extensively explores the nonlinear dynamic responses of the metamaterial, highlighting the significant impact of excitation amplitudes on the amplitude-dependent bandgaps.

Nonlinear oscillation of elastic sagged cable using refined mode truncation: A comparative study

A nonlinear sagged cable, due to its initial curvature, leads to various challenges of empirical mode truncation used by routine Galerkin method when constructing reduced-order model. It is recently elucidated that ( Guo and Rega, 2023a ), the key for refined mode truncation (and thus for correct nonlinear dynamics prediction) is to first eliminate low-order nonlinear terms of spatial continuous structures. This paper focuses on refined truncation of nonlinear sagged cable by leveraging the recent low-order elimination perspective, which is realized by a normal form development. Further comparative studies for both primary resonant and two-to-one internally resonant dynamics of the sagged cable, including nonlinear frequency responses, backbone curves, and Poincaré mapping, demonstrate notable differences between the two different types of models built by either routine or refined truncation, which confirms necessity of the refined mode truncation used for geometrically nonlinear structures like sagged cables.

Investigation into the DNA's conformations and their conversions from the phase transition theory

The functions and transformation mechanisms of nucleic acids in their various states are of great importance in physiology and pathology. Based on the Landau model, this study reveals the mechanism of the existence of chirality and the transformation rules between different chiral conformations by studying the Duffing oscillation response of the local gauge potential on the DNA chain. The research results show that normal chiral DNA and DNA configuration transition behaviors exhibit regular nonlinear periodic oscillations. Moreover, the gauge potential value approaches zero at the B-Z junction site, which may add a criterion for detecting Z-forming sites. In addition, external force and damping play a key role in the chiral gauge potential of nucleic acids, and they can influence, regulate and even control the configuration transition of nucleic acids. Finally, we verified the rationality of our theory by combining x-ray crystal diffraction data of multiple configurations of DNA. This study provides new insights into the behavior and function of chiral DNA in organisms, and provides new possibilities for regulating DNA conformational transition in the future.

Adaptive Flying Assistance Controller Design to Suppress Nonlinear Pilot-Induced Oscillations

Nonlinear pilot-induced oscillations (PIOs) pose a threat to the safety of piloted aircraft in flight. This study aimed to address this safety concern. A novel method for designing an assistance controller based on incremental model predictive control is proposed to aid pilots in suppressing categories 2 and 3 PIOs. In the assistance controller design, the characteristics of the full pilot–aircraft system are considered through a predictive model that integrates the dynamics of the pilot, actuator, and aircraft. This ensures that the assistance controller coordinates with both the pilot and aircraft. Additionally, an adaptive logic for the pilot–controller authority assignment is proposed to automatically adjust the role of the assistance controller. This adaptive logic adopts a model predictive technique to determine whether the pilot’s independent control will cause actuator saturation over the receding prediction horizon. By doing so, the assistance controller is restricted to work only when the actuator saturation constraints are activated. The pilot model-based simulations and the human-in-the-loop experiments demonstrated that the proposed assistance controller can effectively assist pilots in dealing with factors causing nonlinear PIOs while maintaining a harmonious cooperative relationship with the pilot.

Combined Impact of Multidelay Feedback Number and Interval: A Novel Mechanism for Controlling the Stability of Stochastic Duffing Systems

In this paper, we study a class of nonlinear stochastic Duffing oscillators with multidelay feedback. We propose an effective reduction approach with the help of center manifold theory and stochastic averaging method. Taking the initial time-delay [Formula: see text] as the parameter, we reduce the original system to a one-dimensional averaged Itô equation. Our analysis reveals that the original system exhibits stochastic bifurcations, including stochastic D and P bifurcations. Once we have a clear understanding of the bifurcation structure, we can use this knowledge to choose appropriate system parameters and place the system in the desired state. For instance, by adjusting the initial time-delay [Formula: see text] of the control system, we can stabilize the system and achieve the desired outcome. Numerical simulations also verify the theoretical results. With appropriate parameter choices, multiple time delays can destabilize the equilibrium and promote chaotic behaviors, and can also lead to more stable dynamical behavior. Remarkably, we discovered that increasing the interval of time delays and feedback numbers can enhance system stability. It may potentially serve as a novel mechanism for stabilizing stochastic systems. The study provides a solid theoretical foundation for exploring stochastic systems subject to complex time-delay feedback control, and offers a valuable framework for related fields.

Inspection of Some Extremely Nonlinear Oscillators Using an Inventive Approach

Inspection of Some Extremely Nonlinear Oscillators Using an Inventive Approach

Resonancelike emergence of chaos in complex networks of damped-driven nonlinear systems.

Characterizing the emergence of chaotic dynamics of complex networks is an essential task in nonlinear science with potential important applications in many fields such as neural control engineering, microgrid technologies, and ecological networks. Here, we solve a critical outstanding problem in this multidisciplinary research field: the emergence and persistence of spatiotemporal chaos in complex networks of damped-driven nonlinear oscillators in the significant weak-coupling regime, while they exhibit regular behavior when uncoupled. By developing a comprehensive theory with the aid of standard analytical methods, a hierarchy of lower-dimensional effective models, and extensive numerical simulations, we uncover and characterize the basic physical mechanisms concerning both heterogeneity-induced and impulse-induced emergence, enhancement, and suppression of chaos in starlike and scale-free networks of periodically driven, dissipative nonlinear oscillators.

A modified generalized harmonic function perturbation method and its application in analyzing generalized Duffing–Harmonic–Rayleigh–Liénard oscillator

A modified generalized harmonic function perturbation method is proposed in this paper. Compared with the classical version of this method, the modified version can execute its procedures pure symbolically without the need to assign any system parameters even for some complicated nonlinear oscillators. This means that the relations between amplitude of limit cycles and system parameters can be derived analytically from the proposed method. Meanwhile, the analytical expression of characteristic quantity of limit cycles can be also obtained. Via these analytical expressions, the evolutional process of limit cycles can be studied quantitatively in amplitude domain. It demonstrates the entire live period of each limit cycle from its generation to bifurcation to destination. To show the feasibility of the proposed method, a complicated oscillator named generalized Duffing–Harmonic–Rayleigh–Liénard oscillator is investigated in this paper. First, the two analytical expressions mentioned above are derived and the global evolution of its limit cycles are analyzed quantitatively. Second, the critical value of homoclinic and heteroclinic bifurcation parameters are also predicted via this two analytical expressions. Moreover, the analytical approximate solutions of both limit cycles and homo-heteroclinic orbits are calculated. To prove the accuracy, all the above results obtained via the proposed methods are confirmed by the Runge–Kutta method, which show a good accordance. Therefore, the proposed method can be considered as an effective modification for a classical perturbation method. It provides another feasible and reliable analytical quantitative method for analyzing global dynamics of strongly nonlinear oscillators.

On the stability of prestressed beams undergoing nonlinear flexural free oscillations

We study the nonlinear free undamped motions of a hinged-hinged beam exhibiting geometric stretching-induced nonlinearity and arbitrary initial conditions. We treat the governing integral-partial-differential equation of motion as an infinite dimensional Hamiltonian system. We analytically obtain a quantitative Birkhoff Normal Form via a nonlinear coordinate transformation that yields the reduced (modulation) equations describing the free oscillations to within a certain nonlinear order with an estimate of the reminder. The obtained solutions provide a very precise description of small amplitude oscillations over large time scales. The analytical optimization of the involved estimates yields time stability results obtained for plausible values of the physical quantities and of the perturbation parameter. The role played by internal resonances in determining the time stability of the solution is highlighted and discussed. We show that initial conditions with a finite number of eigenfunctions yield bounded solutions living on invariant subspaces of the involved modes at all times. Conversely, initial conditions comprising the full (infinite) spectrum of eigenfunctions provide solutions for which time stability for all times cannot be stated.