A framework for the investigation of nonlinear free oscillations in FGM beams considering material porosity

Chaotic discretization theorems for forced linear and nonlinear coupled oscillators

Nonlinear oscillations are commonly observed in complex systems far from equilibrium, such as living organisms. These oscillations are essential for sustaining vital processes, like neuronal firing, circadian rhythms, and heartbeats. In such systems, thermodynamic dissipation is necessary to maintain oscillations against noise. However, due to their nonlinear dynamics, it has been challenging to determine how the characteristics of oscillations, such as frequency, amplitude, and coherent patterns across elements, influence dissipation. To resolve this issue, we employ Koopman mode decomposition, which recasts nonlinear dynamics as a linear evolution in a function space. This linearization allows the dynamics to be decomposed into temporal oscillatory modes coherent across elements, with the Koopman eigenvalues determining their frequencies. Using this method, we decompose thermodynamic dissipation caused by nonconservative forces into contributions from oscillatory modes in overdamped nonlinear Langevin dynamics. We show that the dissipation from each mode is proportional to its frequency squared and its intensity, providing an interpretable, mode-by-mode picture. In the noisy FitzHugh-Nagumo model, we demonstrate the effectiveness of this framework in quantifying the impact of oscillatory modes on dissipation during nonlinear phenomena like coherent resonance and bifurcation. For instance, our analysis of coherent resonance reveals that the greatest dissipation at the optimal noise intensity is supported by a broad spectrum of frequencies, whereas at nonoptimal noise levels, dissipation is dominated by specific frequency modes. Our work offers a general approach to connecting oscillations to dissipation in noisy environments and improves our understanding of diverse oscillation phenomena from a nonequilibrium thermodynamic perspective.

We perform a series of numerically accurate tensor-train (TT) simulations of population dynamics and energy transfer in quantum β-Fermi-Pasta-Ulam-Tsingou (FPUT) chains comprising 10 to 30 nonlinear oscillators. The dynamics are propagated over timescales ranging from 18 to 60 fundamental vibrational periods. We demonstrate that the chain anharmonicity opens new energy-transfer channels, accelerates initial ballistic energy propagation, reduces (partial) recurrence periods relative to the harmonic limit, and regularizes oscillatory wavepacket motion. This work serves as the first proof-of-principle demonstration of the high efficiency of TT methods for simulating quantum FPUT dynamics.

Nonlinear oscillator with quadratic damping: moving along semicircular surfaces with friction

Nonlinear oscillations of floating offshore wind turbines under wave excitation

In the ongoing study, by presenting a numerical solution approach, the thermal-induced dynamic response of porous axisymmetric cylindrical shell composed of functionally graded material (FGM) resting on an elastic substrate exposed to a rapid cooling shock is analyzed, which hasn’t been studied hitherto. The cylindrical shell is composed of stainless steel (SUS 304) and low carbon-steel (AISI 1020), whose properties are distributed through the thickness based upon the power-law scheme. By employing the transient heat conduction in the one-dimensional Fourier conformation, the solution of the temperature equation is derived. With the aid of the available laboratory data, the temperature-dependent properties attributed to the FGM cylindrical shell are evaluated. The obtained nonlinear differential equations of motion are achieved incorporating the geometrical nonlinearity in the von Kármán form, the first order shear deformation shell formulations and the Hamilton’s principle. In order to linearize and extract the dynamic response of the equations of motion and the temperature equation, Newton-Raphson, generalized differential quadrature (GDQ) methods and Newmark-beta integration pattern have been used. Validation of the results is done with reliable references and then by presenting a parametric examination, the influences of diverse factors including thermal load rapidity time, power-law index, elastic foundation parameters, and temperature differences on the oscillation feedback and stress distribution of the shell exposed to the rapid cooling shock are examined. It is concluded that through moving from the neutral surface to the top and bottom surfaces of the FGM cylindrical shell, the stress increases, and in a particular surface, the longitudinal stress is more than the circumferential stress.

Locomotion in animals such as fish, snakes, inchworms, and octopuses exhibits a remarkable diversity, with each species utilizing distinct body morphologies and movement strategies. Currently, no existing kinematic model is capable of describing the full range of locomotion exhibited by these animals. Addressing this challenge holds important implications for both the study of biomechanics of animals and the development of bioinspired robots. In this work, we propose a general kinematic model that integrates the curvature equation with a nonlinear oscillator. Through parameter adjustments, its morphology can transition between the motions of various animals. It is the most versatile kinematic model to date for describing multimodal locomotion of animals so far as we know. By translating the general kinematic model into a motion control algorithm and combining it with virtual simulation, we create a motion optimization framework that substantially simplifies the complexity of multimodal control for bionic robots with diverse actuation mechanisms, thereby enhancing their maneuverability. Using fish locomotion as an example, we validate the methodology on an untethered multijoint robotic fish, successfully enabling the robotic fish to perform cruising and various fast turn motions, thereby demonstrating its effectiveness in guiding motion control. This work is believed to have laid the foundation for the study of bionic motion and bioinspired robots.

Slow spectral submanifolds (SSMs) are low-dimensional, attracting, invariant surfaces in the phase space of a dynamical system that carry the dominant nonlinear dynamics. Nearby trajectories rapidly converge to such slow SSMs and synchronize with its internal dynamics thereby enabling mathematically rigorous model reduction to the SSM. In general, oblique projections are required for optimally associating full trajectories off the SSM to their SSM-reduced counterparts. In this work, we establish a rigorous mathematical mapping of the SSM onto its tangent space via general oblique projections and develop a data-driven procedure to efficiently construct SSM-based reduced-order models using these projections. Our approach applies irrespective of the SSM dimension and assumes only limited trajectory information. We illustrate the method on numerical and experimental examples, including nonlinear beam oscillations and artificial muscle actuators.

Turbulent reacting flows confined to ducts are plagued by thermoacoustic instability, a state in which a positive feedback between flow, flame, and acoustic perturbations leads to the emergence of catastrophically high-amplitude oscillatory dynamics in the sound and global heat release rate fluctuations. Modeling the interdependence between local interactions and the global emergence of order in such spatially extended complex systems is exacting. Here, we present a novel reduced-order model to capture the influence of the local interactions on distinct variables exhibiting global emergence of order in a turbulent reacting flow system. We represent each variable that exhibits global oscillatory instability as an oscillator with a cubic nonlinearity. The oscillator is driven by a forcing term that represents the holistic influence of the inter-subsystem interactions on the global behavior. The forcing term essentially couples the local interactions and the globally emergent dynamics in the model. Further, the influence of the inter-subsystem interactions on the behavior of each subsystem is different. Therefore, we use different forcing terms for each variable inspired by the physical interactions in the system. The nonlinear oscillators representing the acoustic and the heat release rate oscillations are hence forced using Wiener and Markov-modulated Poisson processes, respectively. Using this approach, we are able to reproduce (i) the multifractal characteristics of acoustic pressure fluctuations during chaotic dynamics, (ii) the loss of multifractality through the experimentally observed scaling law behavior during the transition from chaos to order, and (iii) the emergence of periodicity and bifurcation in heat release rate dynamics.

The cortex generates diverse neural dynamics, ranging from broadband fluctuations to narrowband oscillations at specific frequencies. Here, we investigated whether broadband and oscillatory dynamics play different roles in the encoding and transmission of visual information. We used information-theoretical measures to dissociate neural signals sharing common information (i.e., redundancy) from signals encoding complementary information (i.e., synergy). We analyzed electrocorticography (ECoG) and local field potentials (LFP) in the visual cortex of human and non-human primates (macaque) to investigate the extent to which broadband signals (BB) and narrowband gamma (NBG) oscillations conveyed synergistic or redundant information about images. In both species, the information conveyed by BB signals was highly synergistic within and between visual areas. By contrast, the information carried by NBG was primarily redundant within and between the same visual areas. Finally, the information conveyed by BB signals emerged early after stimulus onset, while NBG sustained information at later time points. These results suggest a potential dual role of BB and NBG cortical dynamics in visual processing, with broadband dynamics supporting nonlinear pattern recognition and oscillations facilitating information maintenance across the cortex.

Damped and driven oscillators are generally modeled with a nonautonomous second-order nonlinear ordinary differential equation including a sinusoidal driving forcing term, such as the forced Duffing equation and the forced Holmes–Rand equation. These equations have been extensively studied during the last century and the last two decades. In the early 1990s, Abarbanel, Rabinovich and Sushchik proposed replacing the sinusoidal forcing term with a “force controlled by the movements of the oscillator itself”, i.e. by the product of two variables: the first being the solution of the oscillator itself, while the second is the solution of a first-order nonlinear ordinary differential equation. They referred to the resulting autonomous dynamical system of two coupled nonlinear ordinary differential equations as a “controlled nonlinear oscillator”. To that end, they introduced a change of variables and parameters to transform the “controlled nonlinear oscillator” that corresponds to a particular case of the forced Duffing equation into the Lorenz system. The aim of this work is to show that their idea can be further generalized and applied to many other dynamical systems, including the forced Holmes–Rand equation, Chua’s cubic circuit, Chen’s system and the forced Helmholtz oscillator. It is proved that a certain class of three-dimensional dynamical systems can be rewritten into the form of “generalized controlled nonlinear oscillators”, which can then be transformed into various Lorenz-like systems. Such a transformation could be very useful for the study of intermittent chaos.

ABSTRACT Non-stationary and non-linear processes govern thermo-acoustic instability in a combustion or propulsion device, which involves coupling between the chamber acoustics and the heat-release oscillations. In this study, we demonstrate the efficacy of combining Empirical Mode Decomposition (EMD), Hilbert-Huang Transform (HHT), and non-linear time-series analysis to reveal the underlying oscillatory modes and the evolution of system dynamics. The experimental configuration consisted of a confined, partially premixed methane-air flame, in which the influence of swirl strength on the flame characteristics and naturally excited thermo-acoustic instabilities was investigated. The objective is to understand the swirl-flame combustion characteristics and the thermo-acoustic aspects of the system at different swirl intensities. The EMD-based analysis reveals that the flame characteristics, the temporal nature of the heat and pressure oscillations, and their coupling are closely linked to the strength of the swirling flow. For a fixed global equivalence ratio ( ϕ g ), the burner exhibited self-excited acoustic instability, accompanied by reduced combustion efficiency and a shorter flame standoff distance, at high swirl strengths. With changing swirl strength, the dynamics of the acoustic pressure changed from limit cycle oscillations and intermittency to a quasi-stable state. The study shows that analyzing thermo-acoustic instability using a combination of EMD and HHT can successfully isolate and reveal valuable information about the temporal features of the instability from non-stationary, non-linear acoustic pressure and heat release oscillation signals. The proposed EMD–Hilbert framework can be extended to practical systems, such as gas turbines and rocket combustors for early detection and characterization of thermo-acoustic instabilities.

Stochastic Vibration Analysis of Nonlinear Oscillator with Fractional Derivative Damping Under Gaussian White Noise

Time-reversible synchronization (TRS) of nonlinear oscillators is a recently proposed technique that ensures super-exponential convergence of dynamics between master and slave systems, which is beneficial in many real-time applications. Nevertheless, this approach has not been demonstrated in any real-time embedded system to practically verify it and quantitatively estimate its advantages. Furthermore, previous studies did not consider the application of time-reversible synchronization to a wide, practically relevant class of chaotic systems with piecewise-linear nonlinearity. To fill these gaps, in this work, we developed an FPGA-based time-reversible synchronization controller for the analog Chua circuit and its digital counterpart. To achieve complete synchronization, we first reconstructed dynamical equations of the circuit. Then, we performed a rigorous theoretical analysis of synchronization possibility between analog and digital systems by each single variable. Next, we implemented the digital model of the Chua circuit in the MyRIO-1900 FPGA using the reconstructed dynamical model and showed its capability of digital-to-analog and analog-to-digital conventional Pecora–Carroll (PC) synchronization. Then, an algorithm of time-reversible synchronization on MyRIO-1900 was tested, achieving complete synchronization at the predefined normalized RMSE level of 0.01, requiring an average of 8.0 fewer points and a median of 10.1 fewer points than the PC synchronization. Finally, we implemented a proof-of-concept version of a capacitive sensor based on the analog Chua circuit with an FPGA-based observer using PC synchronization or the TRS algorithm with a heuristic selection of a starting point. Our experiments reveal that when using the TRS algorithm, the time needed to detect a pre-selected 3% level of capacitance change is reduced by a mean factor of 4 and a median factor of 4.9 in comparison with the conventional PC synchronization. This allows for using the developed solution in applications where the synchronization rate is crucial, including chaos-based sensing, communication, and monitoring.

We study a classical nonlinear Duffing oscillator driven by Gaussian white noise by developing a self-consistent Born approximation (SCBA) within a field-theoretic framework. In analogy with particle field theory, we construct a self-consistent modal mean-field solution that renormalizes the oscillator's natural frequency and reproduces the characteristic amplitude-frequency dependence of such systems. At the Hartree level, this mean field captures all static interactions among the noise-activated internal Fourier modes (NAIFMs). By subsequently incorporating the Born approximation, we naturally include dynamic correlations between NAIFMs, which substantially improve the description in the large-amplitude regime where nonlinear effects become prominent. We show that standard perturbative expansions-particularly those at one and two loops-fail to describe observables such as the mean-square displacement (MSD) in this regime, exhibiting a breakdown of the expansion. In contrast, the SCBA accurately reproduces both the MSD and the renormalized frequency over a broad amplitude range, in excellent agreement with numerical simulations. This approach provides a robust analytical framework for nonlinear oscillators under stochastic driving, with direct relevance to micro- and nanomechanical resonators.

Chaotic and dynamic vibration analysis of a time-delayed nonlinear mathieu oscillator via non-perturbative approach.

In genuine nonequilibrium systems under continuous driving, thermodynamic forces are nonconservative and cannot be described by any free energy potential. Nonetheless, we show that such systems can be associated with a derived from a large-deviation variational principle. This variational principle yields a decomposition of fluxes, forces, and entropy production into a conservative part and a nonconservative part, exemplifying an information-geometric Pythagorean theorem. The decomposition is broadly applicable—including to stochastic master equations as well as closed and open deterministic chemical reaction networks—and accessible to thermodynamic inference from short-time trajectory data. We also show that the excess entropy production obeys a thermodynamic speed limit bounding the rate of state evolution and external fluxes. We illustrate the framework on driven Markov jump processes, nonlinear chemical oscillators, and real-world metabolic networks, where we obtain tight dissipation bounds and identify futile metabolic cycles. Connections are drawn to large deviations, Onsager theory, and previous excess/housekeeping decompositions.

Abstract This paper presents a stability analysis of a strongly nonlinear damped cubic-quintic oscillator using three approaches: the classical multiple scales (CMS) method, an enriched multiple scales (EMS) method based on homotopy perturbation, and numerical continuation via MatCont. Comparisons across different nonlinearity regimes reveal that CMS accuracy degrades substantially when the perturbation parameter $$\varepsilon$$ ε is not small. In the cubic-dominant case ( $$\alpha _3 = 10$$ α 3 = 10 , $$\alpha _5 = 1$$ α 5 = 1 , $$\varepsilon = 1$$ ε = 1 ), CMS overestimates peak amplitudes by approximately 60% and mislocates bifurcation points, whereas EMS predictions remain within 1–2% of numerical results. Even under strong cubic nonlinearity ( $$\alpha _3 = 100$$ α 3 = 100 ), EMS maintains agreement within 3% while CMS errors reach 25%. The EMS method also accurately captures both stable and unstable solution branches, with stability boundaries matching Floquet-based numerical detection to within 0.1%. These results suggest that EMS may serve as a useful analytical tool for strongly nonlinear oscillators where traditional perturbation methods lose accuracy.

Clouds are important features of the atmosphere, determining the energy budget by interacting with incoming solar radiation and outgoing thermal radiation, respectively. For pure ice clouds, the net impact of the radiative effect is still unknown. In this study, we develop a simple but physically consistent ice cloud model and analyze it using methods from the theory of dynamical systems. We find that the model constitutes a nonlinear oscillator with two Hopf bifurcations in the relevant parameter regime. In addition to the characterization of the equilibrium states and the occurring limit cycle, we find scaling behaviors of the bifurcations and the limit cycle, reducing the parameter space crucially. Finally, the model shows very good agreement with real measurements, indicating that the main physics is captured and such simple models are helpful tools for investigating ice clouds.