Nonlinear Bifurcation Research Articles

Data-driven computation of adjoint sensitivities without adjoint solvers: An application to thermoacoustics

Adjoint methods have been the pillar of gradient-based optimization for decades. They enable the accurate computation of a gradient (sensitivity) of a quantity of interest with respect to all system parameters in one calculation. When the gradient is embedded in an optimization routine, the quantity of interest can be optimized for the system to have the desired behavior. Adjoint methods, however, require the system's governing equations and their Jacobian. In this paper, we propose a computational strategy to infer the adjoint sensitivities from data when the governing equations might be unknown (or partly unknown), and noise might be present. The key component of this strategy is an echo state network, which learns the dynamics of nonlinear regimes with varying parameters, and it evolves dynamically via a hidden state. Although the framework does not make assumptions on the dynamical system, we focus on thermoacoustics, which are governed by nonlinear and time-delayed systems. First, we show that a parameter-aware echo state network (ESN) infers the parametrized dynamics. Second, we derive the adjoint of the ESN to compute two types of sensitivity: (i) parameter sensitivity, which is the gradient of a time-averaged cost functional with respect to physical or design parameters of the system, and (ii) initial condition sensitivity, which is the gradient of a cost functional of the final state with respect to the initial condition. Third, we propose the thermoacoustic echo state network (T-ESN), which embeds the physical knowledge in the network architecture for improved generalization. Fourth, we apply the framework to a variety of nonlinear thermoacoustic regimes of a prototypical system. We show that the T-ESN accurately infers the correct adjoint sensitivities of the acoustic energy with respect to the flame parameters and initial conditions. The results are robust to noisy data, from periodic, through quasiperiodic, to chaotic regimes. The inferred adjoint sensitivities are employed to suppress an instability via steepest descent. We show that a single network predicts the nonlinear bifurcations on unseen scenarios, which allows it to converge to the minimum of the acoustic energy. This work opens new possibilities for gradient-based data-driven design optimization. Published by the American Physical Society 2024

Nonlinear vibrations and static bifurcation of a hyperelastic pipe conveying fluid

This study investigates the free and forced nonlinear vibrations, along with the static bifurcation behavior, of a fluid transmission hyperelastic pipe supported by clamped ends. Both geometrical and material nonlinearities are taken into account. The mathematical model is grounded in Euler–Bernoulli beam theory, incorporating von Kármán nonlinearity to capture the complexities of the system. For the hyperelastic materials, the strain energy function proposed by Yeoh is employed. Considering that the lowest critical velocity of the fluid flow is associated with the first mode, after deriving the governing equation using Hamilton’s principle, Galerkin discretization is applied to obtain the reduced order model in the first mode. Analysis of static bifurcation and stability is performed and critical velocity values of fluid flow are determined. The analysis of free and forced vibrations of the hyperelastic pipe is conducted using the method of multiple time scales, specifically focusing on the sub-critical velocity range of fluid flow, which corresponds to the unbuckled state of the pipe. This approach allows for a comprehensive examination of how fluid flow velocity influences the frequency characteristics of free vibrations, as well as the frequency response in the vicinity of the main resonance. The results obtained from this study can significantly enhance our understanding of the design and application of hyperelastic pipes in various fluid transfer scenarios. By elucidating the dynamic behavior and critical parameters associated with these pipes, the findings can inform engineering practices and contribute to the optimization of systems that utilize hyperelastic materials for fluid conveyance.

Optical control of topological end states via soliton formation in a 1D lattice

Abstract Discrete spatial solitons are self-consistent solutions of the discrete nonlinear Schrödinger equation that maintain their shape during propagation. Here we show, using a pump-probe technique, that soliton formation can be used to optically induce and control a linear topological end state in the bulk of a Su–Schrieffer–Heeger lattice, using evanescently-coupled waveguide arrays. Specifically, we observe an abrupt nonlinearly-induced transition above a certain power threshold due to an inversion symmetry-breaking nonlinear bifurcation. Our results demonstrate all-optical active control of topological states.

Nonlinear harmonic resonant behaviors and bifurcation in a Two Degree-of-Freedom Duffing oscillator coupled system of Tension Leg Platform type Floating Offshore Wind Turbine

The surge-heave coupled motion model of a Tension Leg Platform type Floating Offshore Wind Turbine (TLP FOWT) is first established. By taking the tendons stretching and platform set-down motion into consideration, the nonlinear model is a Duffing oscillator coupled system. We analyze the nonlinear feature in the surge-heave coupled motions by the semi-analytical algorithms in nonlinear dynamics. The harmonic balance method is applied to obtain the analytical solutions of the nonlinear system under wind and wave excitations. The analytical solution verifies the excellent accuracy. To further achieve the heave motion response, we organize the bifurcation analysis and discuss the effects of the dynamic parameters on the heave responses. The results reveal the multi components of heave motion, which consist of constant, primary harmonic resonate and higher order harmonic resonate. In addition, the change of dynamic parameters has various effects on the response. The increment of wave load leads to the expansion of heave response, and an instantaneous expansion appear. Both linear surge and heave damping only affect the amplitudes of heave motion. The nonlinear surge stiffness and heave stiffness can barely affect the primary harmonic resonance amplitudes, but they present different effects on the higher order harmonic resonance component.

Investigation on nonlinear flutter modes competition of a long-span suspension bridge based on full-bridge aeroelastic model tests

Nonlinear flutter has attracted wide attention due to the bottleneck caused by linear flutter theory on flutter-resistant design of super-long-span bridges. The longer the span, the more closely spaced the natural modes, which may cause competition among flutter-modes in a nonlinear flutter, but it is rarely reported so far. To this end, this study experimentally investigates the 3D nonlinear flutter characteristics of a long-span suspension bridge with closely spaced natural modes based on full-bridge aeroelastic model wind tunnel tests. Since there are two unstable flutter-modes within the interested post-critical regime and thus various complex but interesting flutter-modes competition processes were observed, such as continuous modes competition where the vibration amplitude evolution exhibits various different types of “sawtooth” shapes during stable vibration stages. Meanwhile, the complex nonlinear bifurcation behaviors caused by modes competition were also observed. The evolutionary characteristics of flutter-modes competition are analyzed in detail and relevant mechanisms are discussed. As the wind speed increases, the flutter of the studied bridge mainly undergoes a transition from being dominated by the symmetric mode to being dominated by the antisymmetric mode, accompanied by a continuous modes competition zone in between as a transitional zone. The results show that the continuous modes competition will result in a decrease in the maximum amplitude RMS of the full-span, which is beneficial for the structure. But it may also lead to a transient increase in the maximum amplitude of the full-span due to the coupling vibration shape formed by the two significant flutter-modes, which may be bad for the structure.

Discrete mechanical model for nonlinear dynamical analysis of planar guyed towers considering the unilateral contact of cables

Towers are widely used for power line transmission, wind power plants, TV and radio broadcasting, and telecommunications. To enhance their stability, cables are often employed to anchor these towers to the ground. In this study, we investigate the nonlinear static and dynamic responses of a planar guyed tower in which the unilateral constraints on the cables are considered. A representative discrete mechanical model with two degrees of freedom is developed to simulate the central mast of the tower, and the cables are modeled as unilateral springs with linear stiffness. The nonlinear equilibrium equations are derived using an energy approach that incorporates the dissipative forces, total potential, and kinetic energies into the Euler-Lagrange equations. Unilateral cable contact is directly included in the nonlinear equilibrium equation for the guyed tower, allowing for numerical analysis without the need to evaluate the contact point at each time or load step. Several numerical strategies are employed to obtain nonlinear static equilibrium paths, bifurcation diagrams, phase portraits, and Poincaré sections. Our analyses provide novel results for the influence of unilateral cable contact in nonlinear static and dynamic analysis, evaluating the effects of unilateral contact and prestressing on the results. A parametric analysis reveals that cable contact affects nonlinear oscillations, bifurcation, and stability. Our numerical results indicate that unilateral cable contact introduces less structural stiffness compared to bilateral contact, thereby significantly affecting the static and dynamic stability of a planar guyed tower. This is evidenced by a decrease in the static limit load and alterations in the bifurcation diagrams, where unilateral contact destroys the trivial solutions, leading to periodic and quasi-periodic solutions at low levels of vertical load.

A Novel Chaos Control Strategy for a Single-Phase Photovoltaic Energy Storage Inverter

The single-phase photovoltaic energy storage inverter represents a pivotal component within photovoltaic energy storage systems. Its operational dynamics are often intricate due to its inherent characteristics and the prevalent usage of nonlinear switching elements, leading to nonlinear characteristic bifurcation such as bifurcation and chaos. In this paper, a deep investigation of a single-phase H-bridge photovoltaic energy storage inverter under proportional–integral (PI) control is made, and a sinusoidal delayed feedback control (SDFC) strategy to mitigate the nonlinear characteristics is proposed. A frequency domain mapping model of the system is established, then, by analyzing the Jacobian matrix and equilibrium points, the bifurcation diagram is formed, and finally, the stable operational domains are determined under double and triple bifurcation parameters. Through simulation experiments, the efficacy of this strategy is validated. The findings show that through the control strategy, the stable operational envelope of the inverter can be greatly expanded and the nonlinear dynamic phenomena can be notably suppressed.

Advanced vibrant controller results of an energetic framework structure

Abstract This research shows the influence of a new active controller technique on a parametrically energized cantilever beam (PECB) with a tip mass model. This article remains primarily concerned with regulating the system’s response using a novel control mechanism. This study describes a novel control mechanism called the nonlinear proportional-derivative cubic velocity feedback controller (NPDCVFC). The motivation of this article is to design a novel control algorithm in order to mitigate the nonlinear vibrations of a parametrically energized cantilever beam with a tip mass model. The proposed controller NPDCVFC incorporates nonlinearly second- and first-order filters into the system. The system is governed by one nonlinear differential equation having both quadratic and cubic nonlinearities within the parametric force. The controller’s efficiency in reducing framework vibrations, managing nonlinear bifurcations, and calming unstable motion is evaluated using numerical simulations of instantaneous vibrations. The perturbation technique is beneficial for solving the current model under the proposed worst resonance case ( Ω ˆ p = 2 ω ˆ 0 ) \text{(}{\hat{{\Omega }}}_{\text{p}}=2{\hat{{\omega }}}_{0}) . In order to choose the optimal controller, we have also added three more controller approaches to the configuration. Integral resonant control, positive position feedback, and nonlinear integral positive position feedback are the three controller approaches that are applied to the structure under consideration. We determine that the NPDCVFC as a new controller is the most effective for lowering the high vibration amplitudes. Over the investigated model, all numerical results were performed using the MATLAB 18.0 programmer software. The stability analysis and the effects of various elements on the controlled structure have been investigated. A comparison with recently published works of a comparable model has also been prepared. Experiment capacities for a PECB with a tip mass are obtainable to validate the results, and they demonstrate good agreement with analytical and numerical results.

Fractional-Order Modeling and Nonlinear Dynamic Analysis of Forward Converter

To accurately investigate the nonlinear dynamic characteristics of a forward converter, a fractional-order state-space averaged model of a forward converter in continuous conduction mode (CCM) is established based on the fractional calculus theory. And nonlinear dynamical bifurcation maps which use PI controller parameters and a reference current as bifurcation parameters are obtained. The nonlinear dynamic behavior is analyzed and compared with that of an integral-order forward converter. The results show that under certain operating conditions, the fractional-order forward converter exhibits bifurcations characterized by low-frequency oscillations and period-doubling as certain circuit and control parameters change. Under the same circuit conditions, there is a difference in the stable parameter region between the fractional and integral-order models of the forward converter. The stable zone of the fractional-order forward converter is larger than that of the integral-order one. Therefore, the circuit struggles to enter states of bifurcation and chaos. The stability domain for low-frequency oscillations and period-doubling bifurcations can be accurately predicted by using a small signal model and a predictive correction model of the fractional-order forward converter, respectively. Finally, by performing circuit simulations and hardware-in-the-loop experiments, the rationality and correctness of the theoretical analysis are verified.

Approximate analytical solutions of nonlinear vibration and bifurcation of dielectric elastomer balloon

Dielectric elastomers (DE), a novel class of actuators, are increasingly employed in fields such as soft robotics and vibration control. Most existing studies investigating the dynamic characteristics of DE structures neglect the viscoelasticity and strain-stiffening effects of materials, with only a few offering approximate analytical solutions. This paper establishes the governing equation for the dynamics of a DE balloon, utilizing the principle of virtual work and considering material viscoelasticity and strain-stiffening effects through linear damping and the Gent hyperelastic model, respectively. The nonlinear governing equations are simplified using the Chebyshev polynomial approximation. The multiple scales method is applied to derive approximate solutions for the structure's free vibration under static voltage excitation and parametrically excited vibration under harmonic voltage excitation. The accuracy of these approximate analytical solutions is validated by comparison with numerical solutions. Additionally, the paper analyzes stability and bifurcation phenomena through time-domain response curves, amplitude-frequency curves, and phase trajectories. Furthermore, the influence of the stretching limit, voltage, and damping on the nonlinear vibration of the balloon structure is studied, providing a valuable theoretical foundation for the future design and utilization of dielectric elastomers.

An asymmetric rotor model under external, parametric, and mixed excitations: Nonlinear bifurcation, active control, and rub-impact effect

This article explores the nonlinear dynamical analysis and control of a [Formula: see text] DOF system that emulates the lateral vibration of an asymmetric rotor model under external, multi-parametric, and mixed excitation. The linear integral resonant controller ([Formula: see text]) has been coupled to the rotor as an active damper through a magnetic actuator. The complete mathematical model, governing the nonlinear interaction among the rotor, controller, and actuator, is derived based on electromagnetic theory and the principle of solid mechanics. This results in a discontinuous [Formula: see text] DOF system coupled with two [Formula: see text] DOF systems, incorporating the rub-impact effect between the rotor and stator. The complicated mathematical model is investigated using analytical techniques, employing the perturbation method, and validated numerically through time response, basins of attraction, bifurcation diagrams, [Formula: see text] chaotic test, and Poincaré return map. The main findings indicate that the asymmetric system model may exhibit nonzero bistable forward whirling motion under external excitation. Additionally, it can whirl either forward or backward under multi-parametric excitation, besides the trivial stable solution. Furthermore, in the case of mixed excitation, the rotor displays nontrivial tristable solutions, with two corresponding to forward whirling orbits and the other one corresponding to backward whirling oscillation. These findings are validated through the establishment of different basins of attraction. Finally, the performance of the [Formula: see text] in mitigating rotor vibrations and averting nonlinear catastrophic bifurcations under various excitation conditions. Furthermore, the rotor’s dynamical behavior and stability are explored in the event of an abrupt failure of one of the connected controllers. The outcomes demonstrate that the proposed [Formula: see text] effectively eliminates dangerous nonlinearities, steering the system to respond akin to a linear system with controllable oscillation amplitudes. However, the sudden controller failure induces a local rub-impact effect, leading to a nonlocal quasiperiodic oscillation and restoring the dominance of the nonlinearities on the system’s response.

Lambert W Functions in the Analysis of Nonlinear Dynamics and Bifurcations of a 2D γ-Ricker Population Model

The aim of this paper is to study the use of Lambert W functions in the analysis of nonlinear dynamics and bifurcations of a new two-dimensional γ-Ricker population model. Through the use of such transcendental functions, it is possible to study the fixed points and the respective eigenvalues of this exponential diffeomorphism as analytical expressions. Consequently, the maximum number of fixed points is proved, depending on whether the Allee effect parameter γ is even or odd. In addition, the analysis of the bifurcation structure of this γ-Ricker diffeomorphism, also taking into account the parity of the Allee effect parameter, demonstrates the results established using the Lambert W functions. Numerical studies are included to illustrate the theoretical results.

Dynamics and non-integrability of the double spring pendulum

This paper investigates the dynamics and integrability of the double spring pendulum, which has great importance in studying nonlinear dynamics, chaos, and bifurcations. Being a Hamiltonian system with three degrees of freedom, its analysis presents a significant challenge. To gain insight into the system’s dynamics, we employ various numerical methods, including Lyapunov exponents spectra, phase-parametric diagrams, and Poincaré cross-sections. The novelty of our work lies in the integration of these three numerical methods into one powerful tool. We provide a comprehensive understanding of the system’s dynamics by identifying parameter values or initial conditions that lead to hyper-chaotic, chaotic, quasi-periodic, and periodic motion, which is a novel contribution in the context of Hamiltonian systems. In the absence of gravitational potential, the system exhibits S1 symmetry, and the presence of an additional first integral was identified using Lyapunov exponents diagrams. We demonstrate the effective utilization of Lyapunov exponents as a potential indicator of first integrals and integrable dynamics. The numerical analysis is complemented by an analytical proof regarding the non-integrability of the system. This proof relies on the analysis of properties of the differential Galois group of variational equations along specific solutions of the system. To facilitate this analysis, we utilized a newly developed extension of the Kovacic algorithm specifically designed for fourth-order differential equations. Overall, our study sheds light on the intricate dynamics and integrability of the double spring pendulum, offering new insights and methodologies for further research in this field.

Nonlinear vibration analysis of thin plate with periodically distributed tunable-stiffness nonlinear oscillators in subsonic flow

This paper provides a new passive control method for nonlinear vibration of subsonic thin plates. The control method uses periodically distributed tunable-stiffness nonlinear oscillators (NLO). By using von Karman’s nonlinear plate theory and subsonic aerodynamic model, the nonlinear dynamic model of thin plate is established based on Hamilton’s principle. Through analyzing the system’s nonlinear dynamics bifurcation, the impact of periodic distribution on the stability is discussed. The amplitude-frequency responses are discussed under different NLO installation modes and design parameters by using Matcont software. The results show that the periodically distributed tunable-stiffness nonlinear oscillators can improve the stability of the system and have a remarkable suppression effect. The optimal control scheme for vibration suppression by periodically distributed tunable-stiffness nonlinear oscillators is obtained. The results obtained from this research can offer a valuable understanding of nonlinear vibration control, thereby enabling the identification of various control approaches that can be implemented in practical applications during future design endeavors.

Hand Vibration Reduction Using Nonlinear Vibration Absorber for the Vibro-Impact Hammer Model

Abstract This work investigates the effectiveness of a passive vibration control device to minimize the amplitude of vibrations from a hand-held impact machine (HHIM) transmitted to a hand-arm system (HAS). For this purpose, we employ a simplified lumped parameter model of an HHIM and the hand-arm system consisting of lumped masses, springs, and viscous dampers. For a better understanding of the system dynamics, we use a vibro-impact model for a more accurate representation of HHIM–ground interactions. We then examine the effectiveness of a cubic nonlinear vibration absorber in minimizing these undesirable vibrations via a nonlinear bifurcation analysis of the system. A parametric study on the numerical bifurcation analysis of the system reveals the criticality of different design parameters of the proposed nonlinear vibration absorber on the system dynamics. Our findings demonstrate that an appropriate selection of absorber parameters could significantly change the amplitude and the periodicity of vibrations transmitted to the hand. Furthermore, we observe the existence of complex motions such as period-2, period-4, and chaotic attractors in the system for the given values of the absorber and operating parameters.

Nonlinear vibrations, bifurcations and chaos of piezoelectric composite lattice sandwich plate with four simply supported edges

This paper is devoted to investigate the nonlinear dynamics, bifurcation and chaos of the piezoelectric composite lattice sandwich plate with 1:3 internal resonance. Galerkin method is used to discretize the nonlinear partial differential governing equations of motion into the ordinary differential equations. The multi-scale method is used to obtain the modulation equations of the piezoelectric composite lattice sandwich plate in the polar coordinates and Cartesian coordinates in the primary and parametric resonances, respectively. Using Newton-Raphson method, the bifurcation diagrams of the steady-state equilibrium solution are obtained for the modulation equation with the change of the system parameters. The stability of the steady-state equilibrium solution is analyzed for the modulation equation. The existence of the static bifurcations, such as the saddle-node bifurcation, pitchfork bifurcation and Hopf dynamic bifurcations are found. The complex nonlinear jump phenomena caused by the existence of the multiple nonlinear steady-state solution regions are analyzed in detail. Fourth-order Runge-Kutta method is used to continue tracing the nonlinear periodic solution of the modulation equation in Cartesian coordinates. The time-history diagram, phase portrait and Poincaré map are obtained for the piezoelectric composite lattice sandwich plate in the nonlinear resonance. The dynamic process of the nonlinear periodic solution is described for the modulation equation from the limit cycle oscillation to the chaotic attractor in detail.

Impact of Chaos on MOSFET Thermal Stress and Lifetime

The reliability of power electronic switching components is of great concern for many researchers. For their usage in many mission profiles, it is crucial for them to perform for the duration of their intended lifetime; however, they can fail because of thermal stress. Thus, it is essential to analyze their thermal performance. Non-linear switching action, bifurcation and chaotic events may occur in DC-DC power converters. Consequently, they show different behaviors when their parameters change. However, this is an opportunity to study these bifurcation phenomena and the existence of chaos, e.g., in boost converters, on their performance as the effects of load variations (mission profiles) on the system’s behavior. These variations generate many non-linear phenomena such as periodic behavior, repeated period-doubling bifurcations and chaos in the MOSFET drain-source current. Thus, we propose, for the first time, an analysis of the influence of chaos on the junction temperature. First of all, this paper provides a step-by-step procedure to establish an electrothermal model of a C2M0080120D MOSFET with integrated power loss. Then, the junction temperature is estimated by computing the power losses and a thermal impedance model of the switch. Additionally, this model is used to investigate the bifurcation and chaotic behavior of the MOSFET junction temperature. The paper contributes by providing a mathematical model to calculate several coefficients based on experimental data and thermal oscillations. Estimation of the number of cycles to failure is given by the Coffin–Manson equation, while temperature cycles are counted using the rainflow counting algorithm. Further, the accumulated damage results are calculated using the Miner’s model. Finally, a comparison is made between the damage accumulated during different mission profiles: significant degradation of the MOSFET’s lifetime is pointed out for chaotic currents compared to periodic ones.

Nonlinear dynamic response and bifurcation of variable thickness sandwich conical shell with internal resonance

Nonlinear dynamic response and bifurcation of variable thickness sandwich conical shell with internal resonance

Enhanced sensitivity with nonlinearity-induced exceptional points degeneracy lifting

Bifurcation of exceptional points (EPs), particularly higher-order EPs, can offer applications in metrology by amplifying sensitivity, but this method suffers from a tradeoff between sensitivity and robustness. To break this constraint, we experimentally introduce nonlinearity into the EP degeneracy lifting at the coupled electric resonators and observe a sixth-order nonlinear bifurcation which amplifies the sensitivity elevenfold compared to the conventional EP-based approach operating in the linear regime, while maintaining the degrees of freedom, thereby without cost in robustness. Moreover, we discover a chaotic dynamics near the EP due to the nonlinear contribution, which constitutes a distinct difference from the EP degeneracy lifting in the linear regime with random noise. Our study expands the scope of EP degeneracy lifting into nonlinearity, providing a paradigm to exploit the benefit of EPs.

Non-linear vibration and bifurcation analysis of Euler-Bernoulli beam under parametric excitation

This paper presents an analysis of the non-linear vibrations of beams, which play a crucial role in various industrial and construction structures. Understanding the transverse vibrations of beams and accurately determining their frequency response is essential for achieving optimal design and structural performance. The novelty of this study lies in conducting a transverse non-linear vibration analysis of a three-dimensional beam while considering the effect of mid-plane elongation. By incorporating this aspect into the analysis, the study aims to provide deeper insights into the dynamic behavior of beams subjected to non-linear effects. A multiple-time scale approach has been adopted to conduct this research. To verify the accuracy of the method as well as the accuracy of the outcomes gained from this method, a contrast has been made with the 4th-order Runge-Kutta technique, which indicates that the results obtained are acceptable. The frequency response of the beam indicates the presence of a phenomenon of splitting into two non-linear branches during the three-dimensional vibrations of the beam, as well as a hardening state in the frequency response as a result of stretching the middle plane of the beam. Furthermore, a parametric study was conducted in which different parameters were examined to determine the starting point of non-linear bifurcation. As a result, the damping coefficient and resonance deviation parameter are two factors that affect the preference for critical bifurcation over safe bifurcation. Furthermore, the stretching of the middle plane results in a higher non-linear term coefficient in the vibration equations of the beam, which increases the oscillation frequency of the beam.