Advances in Nonlinear Dynamics, Volume III

We derive a nonlinear equation of motion for a chip-scale pendulum comprising a thick plate suspended from a tensioned nanoribbon. Recently, we explored the use of such a device as a clock gravimeter, exploiting the parametric coupling of its frequency to the local acceleration of gravity and demonstrating micro-g resolution with a silicon nitride prototype. Here we consider the restoring torque arising from the mid-plane stretching of the nanoribbon, finding it is a hardening spring that can be used to counteract the softening of gravitational torques, reducing parametric frequency noise and extending the range of isochronous pendulation. Using the method of multiple scales, we predict that parametric frequency-amplitude coupling can be driven to zero by exploiting fabrication tolerances available using modern nanolithography.

This paper compares two types of inertial electrostatic MEMS methane sensors, one based on a cyclic-fold bifurcation and another based on a primary Hopf bifurcation. We find the Hopf-based sensor is 40% more sensitive than the cyclic-fold sensor and that the detection signal, and therefore the signal-to-noise ratio, of the former is 25% larger than that of the latter.

This study reports on an experimental investigation of modal interactions in a microresonator. The resonator is made of a clamped-clamped curved beam, electrostatically actuated via a side electrode. A two-to-one ratio exists by design between the first anti-symmetric and the second symmetric in-plane bending modes. Because the beam geometry and actuation are symmetric, anti-symmetric modes cannot be directly excited due to the negligible projection of the excitation force onto anti-symmetric modes. This study shows that these modes can be indirectly excited by channeling energy from a symmetric mode to the target anti-symmetric mode. The result is an M-shaped frequency-response curves, with stable trivial and large-amplitude branches. The latter branch combines contributions from both the directly and indirectly excited modes.

Micro compound structures have rich dynamics and are very promising for use in applications. Among the intriguing nonlinear phenomena exhibited by compound structures is the saturation phenomenon due to the 2:1 internal resonance. However, this phenomenon is strongly affected by damping, which is a critical factor in microelectromechanical systems (MEMS), due to variation in the operating pressure inside environmental chambers. Therefore, this work aims to electrostatically control a micro portal frame, modeled as a two-degrees-of-freedom (2DOFs) system, with 2:1 internal resonance between the second (Y-direction motion) and first (X-direction motion) modes. To avoid irregular behaviors under the saturation phenomenon within the internal resonance regime, we propose to use the Linear Quadratic Regulator (LQR) control technique to create an electrostatic force that alleviates the damping effect. A comparison between the efficiency of an ideal control and a limited voltage actuator is highlighted.

The parametric anti-resonance is a well-known phenomenon in which a properly tuned parametric excitation increases the effective damping of the overall system. A parametric anti-resonance occurs for systems with at least two degrees of freedom. This contribution presents an approach to accelerate the mitigation of transient vibrations by applying two parametric anti-resonances at different frequencies. The proposed technique is presented for a potential application in a microelectromechanical system consisting of three flexible beams.

It is often crucial to compute bifurcation diagrams in order to assess the stability and effects of multiple parameters on the overall dynamical properties of nonlinear systems. The case of one-parameter bifurcation diagrams is fairly easy to deal with, but it is more difficult (due to computational requirements) to do so when two or more parameters change at once, as we need fairly advanced hardware and software resources for that purpose [1–4]

In this paper, the modern nonlinear theory (bifurcation and chaos theory) is used to study the dynamics of two well-known phenomena in electric power systems named Voltage Collapse (VC) and Subsynchronous Resonance (SSR). The results showed that at specific control parameter values, the system experienced unstable conditions, which means the electric power system would totally collapse due to some increases in control parameters. So, in addition to finding such unstable conditions, we designed a linear controller to eliminate all such unstable ones.

Echo state networks (ESNs) are known for their simple structure and excellent forecasting ability. In this chapter, we investigate the nonlinear prediction capabilities of ESNs, analyze their structure and principles, and propose a new prediction model based on the output mode, using multiple reservoirs. The proposed model is then tested using Mackey-Glass and Lorenz chaotic systems with different dimensions. Our results demonstrate that the multi-reservoir ESN model can accurately predict longer time series with high precision.

Memristor oscillators have been widely studied in the last years due to their potential applications in several technological areas. Electronic circuits containing a memristor are capable of producing nonlinear periodic and chaotic oscillations, due to their locally active characteristics. In this chapter, we consider a memristive circuit consisting of a locally active memristor, an inductor, and a resistor, which is modeled by a planar three-parameter system of ordinary differential equations. The system presents periodic oscillations, which arise at a Hopf bifurcation. We show that these oscillations evolve into a homoclinic orbit, in a Bogdanov-Takens-type bifurcation scenario. By adding a small time-periodic excitation to the circuit, we obtain complex dynamical behavior, such as quasiperiodic and chaotic oscillations. The system also presents multistability, having periodic oscillations coexisting with chaotic dynamics. As far as we know, it is the first time that time-periodic perturbation is used as the mechanism creation of chaotic dynamics in memristive systems.

This chapter addresses the reliability analysis of a dynamical system attractor, provided its dynamic-integrity measure has been previously assessed in terms of a meaningful parameter for which the probability density function is known. The probability that the dynamic-integrity measure should be equal or larger than a prescribed safe reference value, for the attractor to be considered “reliable,” is sought. Although the ideas addressed here are applicable to any dynamical system, the structural stability case is focused herewith for the sake of an illustration, aiming at characterizing the safe loading threshold in an archetypal model liable to elastic buckling. By assuming that the buckling strength is an input Gaussian random variable, the basic statistical properties of the output variable, namely, the dynamic-integrity measure, are estimated, from which a simple procedure to obtain a reliability appraisal is proposed. Another procedure is also addressed, considering a more formal way to obtain the output statistical properties. It is expected that both the simplified and the more formal procedures may help the assimilation of reliability analysis within current structural engineering design practices.

Several dynamic behaviors exhibited in a representative model of a Petrobras offshore oil production plant were studied, considering both open-loop operation and closed-loop operation when using a feedback PI control scheme. The present work evaluated three different SISO control scenarios, taking into account three controlled variables in order to determine the suppression effect of the oscillatory dynamic behavior raised due to the slugging flow phenomenon. It was observed that in the PI anti-slug control scheme, when the pressure at the top of the riser was considered the controlled variable, the dynamics exhibited chaotic oscillations as a function of the PI tuning parameters. The chaotic behavior was characterized, and it was determined the control scenarios and tuning conditions that favor the appearance or impression of complex dynamics behaviors.

The Lorenz model is widely considered as the first dynamical system exhibiting a chaotic attractor, the shape of which is the famous butterfly. This similarity led Lorenz to name the sensitivity to initial conditions inherent to such chaotic systems the butterfly effect, making its model a paradigm of chaos. Nearly 30 years ago, Stefan J. Linz presented in a very interesting paper an “exact transformation” enabling to obtain the jerk form of the Lorenz model and a nonlinear transformation “simplifying its jerky dynamics.” Unfortunately, the third-order nonlinear differential equation he finally obtained precluded any mathematical analysis and made difficult numerical investigations since it contained exponential functions. In this work, we provide in the simplest way the jerk form of the Lorenz model. Then, a stability analysis of the jerk dynamics of Lorenz model proves that fixed points and their stability, eigenvalues, Lyapunov characteristic exponents, and of course attractor shape are exactly the same as those of the original Lorenz model.

This study investigates the dynamic behavior of the spiral bevel gears (SBGs) by developing two degrees of freedom dynamic model (2 DOF) to four degrees of freedom (4 DOF), which involves the rotational shaft stiffness. The governing equations of motion are derived based on a nonlinear time-varying model. The nonlinearity and time dependency emanate from the backlash and contact ratio of the pinion and the gear, respectively. Depending on the working conditions, the system could experience a backside contact, which is an undesirable phenomenon in gear systems. A comparison between two systems, i.e., 2 DOF and 4 DOF, is done to understand what kind of phenomena are neglected by decreasing the DOF. The root mean square (RMS) diagrams and bifurcation diagrams are employed to analyze the vibration response of the system. The interesting point is that the simplification of the dynamic model could lead to a different dynamic response with respect to reality.

Multistability is a phenomenon that refers to the coexistence of several possible eventual stable states for a given set of parameters. The final stable states converge to an attractor according to the given initial conditions. The phenomenon of multistability has been found in almost all areas of science and nature [1]. A pioneer study, which also coined the term multistability, was devoted to visual perception. Multistability has been identified in different classes of systems such as weakly dissipative systems, coupled systems, delayed feedback systems, parametrically excited systems, and stochastic systems [2].

This paper proposes a new modified sine chaotic map and implements both the conventional and proposed modified maps on hardware. The proposed modified sine map exhibits continuous chaotic behavior for all values of the main system parameter. This helps generate “pseudo-random” numbers and encrypt systems by making it easier to design system keys and ensuring chaos happens. The chaotic behavior of both maps is validated using time series and bifurcation diagrams. A reconfigurable CORDIC hardware block is used to compute the transcendental mathematical functions by employing shift-add and suboperations. While the sin function is computed in circular rotation mode, the multiplication operation is computed in linear vectoring mode. The proposed hardware architecture is multiplier-less, and hence, it does not consume any DSPs. The sine and modified sine maps are realized on Xilinx Artix-7 FFPGA board using Verilog HDL, in which the maximum frequencies are 5.276 and 5.043 MHz, respectively.

In this paper, a fixed-time adaptive neural control (FTANC) for a helicopter-like Twin Rotor MIMO System (TRMS) is presented. The proposed controller has been developed to achieve finite-time convergence of the system dynamics independently of the initial conditions. The control of the TRMS system is a challenging problem due to the significant non-linearities and the cross-coupling between the main and tail rotors. In this study, we proposed an adaptive radial basis function neural networks (RBFNNs) to estimate all unknown nonlinear functions and disturbances. The RBFNN is combined with the backstepping technique to guarantee the trajectory tracking and the overall closed-loop stability. The effectiveness of the proposed control strategy is demonstrated through simulation tests in Matlab/Simulink software environment.

In this chapter, a new key generator is presented, which is constructed by an auto-switched numerical resolution of multiple three dimensional continuous chaotic systems (Lorenz, Rôssler, Chen) that excite a discreet chaotic system (Henon Map). The designed chaotic system provides complex chaotic attractors and can change its behavior automatically via a chaotic switching rule. The originality of the proposed architecture is that it allows to solve the problem of finite precision due to digital implementation while providing a good compromise between high security, performance, and hardware resources (low power and cost). Hardware digital implementation and FPGA circuit experimental results of this generator demonstrate that this promising technique can be applied in efficient embedded ciphering communication systems. Moreover, the proposed chaotic system should be very useful for reducing negative influence of dynamical degradation in real-time embedded applications.

Synchronization of interacting elements is ubiquitous in nature and has been widely investigated in many physical and biological systems, such as flashing fireflies, neurons in the brain, electric power grids, and Josephson junction arrays [1, 2]. Kuramoto introduced an analytically solvable model of coupled oscillators and thus inspired extensive studies on phase synchronization research since the 1980s [3–5]. In spite of its mature age, the theory of synchronization is still full of surprises, applications, and new features [6–8].

This chapter presents the investigation of energy harvesting in a portal frame coupled with a nonlinear electromagnetic energy sink (NES), excited by an unbalanced DC motor, representing a nonideal excitation source. The energy harvesting is performed by two sources: a piezoelectric material and the energy generated by the electromagnetic energy sink. The coupling of the structure with the piezoelectric material and the electromagnetic energy sink resulted in a nonlinear electromechanical coupling model. The main advantage of nonlinear energy harvesters is the conversion of energy over a wider range of frequencies of vibrations. For dynamics, the analysis of the system considers bifurcation diagrams, phase portraits, power spectral densities, and 0-1 tests. Numerical simulations show the existence of chaotic behavior for some regions of the parameter space. Additionally, to control the vibration amplitudes of the structure and improve energy production, an adjustment of the parameters of the nonlinear electromagnetic energy sink is proposed.

In ocean engineering applications, parametric resonance is normally detrimental for the stability of large structures, so the vast majority of effort in the literature is toward preventing and reducing it; similarly, parametric excitation is usually undesired for wave energy extraction too, since it often reduces the conversion ability. Conversely, this chapter investigates the possibility to purposely introduce a 2:1 parametric resonance into a pitching wave energy harvester in order to inherently increase the energy absorption capabilities. Such a change in perspective is enabled by the use of a computationally efficient nonlinear hydrodynamic model (nonlinear Froude-Krylov force), which is able to articulate such a parametric instability at an early-development stage in a design-oriented simulation framework. The introduced 2:1 instability is found to be promising, since a significant amplification is obtained in the 2:1 region, where the oscillation amplitude is similar or even higher than in the 1:1 region.

The hypothesis that a bi-stable nonlinear restoring element will help reduce the sensitivity of a point wave energy absorber’s response to variations in the frequency and amplitude of the incident waves is supported by many previous theoretical investigations. In this study, the hypothesis is investigated experimentally by designing, fabricating, and assessing the response behavior of a prototype bi-stable wave energy absorber excited by harmonic incident waves in an experimental wave flume. Results demonstrate that bi-stability does not seem to improve the size of the effective bandwidth over the linear design. Nonetheless, a key advantage remains in the ability of the bi-stable restoring force to push the effective bandwidth of the wave energy absorber toward lower frequencies, where most of the wave energy is trapped.

Simultaneous vibration absorption and energy harvesting (EH) from a tuned mass damper and a variant of nonlinear energy sink (NES) with linear and nonlinear stiffness elements subjected to harmonic excitation and random excitation respectively are considered. Response statistics of the vibrating system and EH mechanism are obtained by the equivalent linearization method. Multi-objective optimization with a dynamic weighted aggregation (DWA) technique is adopted to obtain optimum design parameters with the twin objectives of vibration control and EH. The particle swarm optimization algorithm is adopted to obtain the 2-D Pareto optimal fronts.

A quasi-zero stiffness (QZS) two degree-of-freedom (DOF) galloping piezoelectric energy harvester (GPEH) is proposed for efficient energy harvesting in the ultra-low wind speed range. At low wind speeds, traditional 1DOF and 2DOF linear harvesters are not capable of generating enough voltage to power microelectronics; moreover, the bistable harvester tends to be trapped in intrawell motion, which is unfavorable for wind energy harvesting. With the feature of a low dynamic frequency, the QZS nonlinear mechanism is applied to remarkably decrease the onset galloping wind speed and boost the power generation. A coupled aero-electro-mechanical model is built to evaluate the characteristics of the harvester. The simulation result shows that the 2DOF QZS-GPEH has the best output performance at low wind speeds, outperforming its 1DOF linear, 2DOF linear, and 2DOF bistable counterparts. This work opens new opportunities for efficiently harvesting wind energy at the ultra-low wind speed region.

In this chapter, we provide results on the qualitative modeling of landslide dynamics under the assumption of delayed failure and the effect of background noise. Results obtained indicate that the examined time series, which represent the actual recordings of background noise, belong to a group of stationary linear stochastic processes with Gaussian inputs. Such noise is introduced as the additive term in the system of delay differential equations governing the dynamics of a spring-block model composed of n units with delayed failure. The friction law assumed in the model represents the cubic friction force. Results of the performed research, using mean-field approximation and numerical computation, indicate the conditions for the occurrence of Andronov-Hopf direct supercritical bifurcation. In particular, it is shown that small-amplitude background noise could contribute to the onset of instability if the system under study is on the verge of stability.

The aim of this chapter is to develop a methodology to quantify the nonlinear response of MREs under LAOA loadings using Chebyshev polynomials of the first kind. This permitted determination of the compression and tension elastic moduli at the minimum, zero, and maximum strain, as alternatives to equivalent linear first harmonic moduli extracted using Fourier transform. The proposed local measures can provide the interpretation of inter- and intra-cycles nonlinearities without observing the stress-strain hysteresis response of MREs under LAOA loadings.

In this chapter, a stimuli-responsive silicone filament embedded with composite microspheres of alginate and magnetite is produced and then studied in terms of modal and damping response under the influence of a magnetic field. The interaction of the filament with the magnetic field results in a shift in the resonance frequency and a decrease in the damping ratio, especially under the application of increasing loads. The latter behavior is observed to oppose that of the unexposed filament.

Modeling asymmetric hysteresis is challenging, especially when computational efficiency is sought. This study continues with an effort to capture asymmetric hysteretic restoring force as motivated, inspired, and validated by a set of carefully designed and acquired laboratory experimental data at Sapienza University of Rome. With improved testing apparatus, richer datasets deliver more comprehensive views of the asymmetric hysteresis behaviors, making it possible to further advance the proposed generalized extended Masing model involving the virgin loading curve and minor loops.

This chapter proposes a novel approach to simulate the hysteretic behavior of a pressurized sand damper. In particular, the typical symmetric and pinched force-displacement hysteresis loops exhibited by the tested device are first illustrated. Subsequently, the proposed hysteresis model, denominated Vaiana-Rosati model, is briefly described and adopted to reproduce the device experimental responses.

The purpose of this study is to present a force-based macroelement for the static and dynamic analysis of the in-plane response of masonry panels. The nonlinear behavior of masonry is described by a constitutive model based on the Bouc-Wen hysteretic formulation modified with the introduction of damage and flexibility increase by means of two scalar variables that regulate the rate and type of degradation. Damage is considered as a reduction of the hysteretic force, while flexibility increase is modeled through an amplification of the elastic displacement, both depending on the dissipated energy. The aim is to give a more accurate representation of the strength and stiffness decay masonry walls undergo when subjected to cyclic loadings and to better represent the loading and unloading branches of the response curves. To investigate the effect of the degradation in the dynamic field, the behavior of a slender wall is analyzed under harmonic excitations.

As an important structure of micro-robots, micro-beams play an increasingly important role in daily production and life, especially in the biological and medical fields. During the use of micro-beam instruments, vibrations occur due to the unevenness of the skin, which affect the accuracy and stability of precision instruments. To analyze this problem, this chapter studies the free vibration of beams with spring at arbitrary positions and nonlinear spring foundations. Through the Laplace transform and the principle of linear superposition, the constrained Green’s function is obtained. Numerical calculations are performed to validate the present solutions and the effects of various important physical parameters are investigated. It was found that the mode and deflection of the beams were changed by the springs and foundations.

The experimental characterization of a helical wire rope isolator and the identification of its asymmetric hysteretic behavior along the shear direction is illustrated. The restoring force time histories and the asymmetric force-displacement hysteresis loops are identified by using two different models: a differential one, obtained by a generalization the Bouc–Wen model, and an exponential one, denominated Vaiana–Rosati model. The accuracy of the identifications are quantified according to the mean square error between the experimental and the simulated curves. Finally, the frequency response curves of a one degree of freedom oscillator connected to the ground with the restoring forces provided by the two models are computed showing the dynamical features of the device and paving the way to a future experimental validation.

Engineering structures with viscoelastic materials are generally modelled by a fractional-order system. The reliability problem of relative structural vibration under random excitations is always a hot issue in the field of the stochastic dynamical systems. Consider a generalized Van der Pol system with fractional derivatives excited by a white Gaussian noise. Firstly, a generalized harmonic transformation is used to get an approximated expression for fractional derivative by converting the fast-varying variables to the slow-varying variables and then applying stochastic averaging methods with energy envelopes to obtain the Ito differential equations and obtaining the Kolmogorov backward equations (KBE) related to the system energy. Then, combining Monte Carlo sampling to perform data-driven and neural network, a new algorithm is obtained to solve the reliability function that satisfies KBE, which is the innovation of this paper. The algorithm does not need boundary conditions and reduces the need for data volume in high-dimensional problems.

A fractional-order dynamic model for microresonator is analyzed in this paper. The model is transformed from the integer-order dynamic model by setting the second-order derivative in the integer-order dynamic model as fractional-order p1 and the first-order derivative as fractional-order p2 according to the definition of Caputo fractional-order derivative. The results of simulation show that it is efferent to predict the behavior of the fractional-order dynamic model using the rules from integer-order dynamics provided that the difference between the fractional-order and integer-order is in a small interval. Additionally, the variations of both p1 and p2 certainly cause changes in the motion state of microresonator, while the effect of p1 is more significant than that of p2.

This paper reports a novel six-dimensional nonlinear dynamical system with the tangent hyperbolic memristor circuit and amalgamated image encryption. The dynamical system is analyzed using standard tools, including phase portraits, equilibrium points, Eigenvalues, and Lyapunov exponents. The analysis suggests that the developed system is chaotic in nature and has exciting new 2D trajectories. The system generates two different trajectories of bird-like phase portrait. The chaotic system is numerically solved with Caputo fractional order derivative for different values of fractional order (q). The fractional order circuits are designed for q = 1 and q = 0.99 utilizing the approximation fractional order technique of transfer function, showing a great deal of agreement with the numerical results. Lastly, the random number generated from the chaotic system is utilized to scramble the image via the scheme of Amalgamated Image Encryption. The scrambled image is tested using a different image security test algorithm to support the idea that the chaotic system and image can together form an advantageous key.

This paper introduces a generalized fractional-order complex logistic map and the FPGA realization of a corresponding fractal generation application. The chaotic properties of the proposed map are studied through the bifurcation behavior and maximum Lyapunov exponent (MLE). A concise fractal generation process is presented, which results in designing and implementing an optimized hardware architecture. An efficient FPGA implementation of the fractal behavior is validated experimentally on Artix-7 FPGA board. An example of fractal implementation is verified, yielding a frequency of 24.34 MHz and a throughput of 0.292 Gbit/s. Compared to recent related works, the proposed implementation demonstrates its efficient hardware utilization and suitability for potential applications.

There are many approaches to assess control quality, starting from the mean-square error or the variance, through model-based or model-free approaches, to fractal or entropy measures. These indicators can be used for a variety of control systems. Nonlinear industrial applications pose new challenges. The indicator must be robust, reliable, and informative. It must cope with disturbances and uncertainties. Process complexity and its nonlinearities, mutual correlations, variable delays, and outlying anomalies or human influence should not limit it. Fractional calculus can meet these demands. The fractional order of the ARFIMA filter represents persistence of time series and it is a potential control quality index. The research shows that the Geweke-Porter-Hudak (GPH) fractional order estimator can assess control system quality. The research is validated using laboratory nonlinear mechanical servomechanism.

Nonlinear system identification based on output-only data is challenging since the stochastic approaches require the structure to be excited by random input with a uniform Gaussian distribution. This chapter applies a deterministic output-only approach to the parameter estimation of a linear multi-story specimen with an amplitude-dependent geometrical nonlinearity. The approach is independent of the input type, value, and number but requires the excitation to be applied away from the nonlinearity. The vibration responses to high-amplitude excitations are taken into a subspace-based identification algorithm that simultaneously yields both nonlinear and underlying linear parameters. The process is verified by comparing the underlying linear parameters with the linear modal parameters of the structure under low-amplitude excitation. The results indicate a superior accuracy of the estimated parameters in the simulation and an acceptable confidence range for the experimental test.

Monitoring potential bolt faults is very necessary and meaningful to keep structures in healthy operation. For this purpose, an improved approach to monitor bolt faults in two-dimensional structures without reference is proposed. In the new method, the nonlinear dynamic behaviors of the structure to be monitored are studied by a general multi-degree-of-freedom (MDOF) mode with nonlinear elements. By exciting the structure many times with the same excitation and the local structural modification method, only nonlinear features from the structure to be monitored are defined. Based on these features, a novel fault index and corresponding improved approach are proposed and explained. With some numerical examples on a two-dimensional structure, the effectiveness and reliability of the method are fully verified.

Bistable nonlinear energy sinks have attracted extensive attention due to their efficient broad-band targeted energy transfer over a wide range of input energy levels. The precise identification of local bistability is of significance for predicting and controlling the system performance of vibration energy harvesting and absorption. This paper proposes a new physics-informed sparse identification method for parameter estimation of a three-degree-of-freedom bistable nonlinear energy sink. The restoring force surface is constructed on the local bistable structure, and the nonlinear elastic force trajectory is intercepted by assuming two quasi-zero velocity planes. Furthermore, the candidate functions of the nonlinear elastic force can be physically informed in the sparse identification algorithm. Numerical simulation shows that the proposed method not only gives sparse identification physics information but also improves accuracy by 1.61% under the noise level of 30 dB. Experimental verifications are performed on a three-story beam-type bistable energy sink. The result shows that the identified nonlinear elastic force has a good agreement with the measured one, and the NMSE is only 0.97%.

We present a novel algorithm for accurately estimating frequency-amplitude backbone curves of weakly damped mechanical systems with weak stiffness-based nonlinearities. Our smooth-coordinate-decomposition-based algorithm decomposes multivariate decay responses into sets of single-degree-of-freedom responses and rejected noisy coordinates. By applying local and global techniques on orthogonal smooth coordinate pairs, our algorithm estimates each mode’s instantaneous frequency and amplitude, allowing for simultaneous estimation of multiple backbone curves for all excited modes.

In this contribution, a novel approach to optimize multilayer perceptron artificial neural networks (MLP-ANN) devoted to approximate quaternion valued functions is presented. The approach is based on the definition of proper auxiliary networks devoted to predict the trends of the main network weights during the learning phase, thus reducing the number of epochs needed to reach a suitable abstraction level. The approach is specifically designed for MLP-ANN devoted to the approximation of complex valued functions, characterized by more than one imaginary part, a framework typical of application-oriented problems.

Wire rope isolators are devices with non-linear behavior used to reduce vibrations during transportation and operation of machineries. Their hysteretic behavior produces a relevant damping, which guarantees a good mitigation performance without needing further damping devices. Nevertheless, the linear spring-damper model is not able to describe the component behavior. Additionally, mechanical properties provided by the manufacturers are generally partial and limited to equivalent stiffness and damping coefficients. These parameters are good to predict the quasi-static behavior of the spring, such as the deflection under component weight. Nevertheless, they are not suitable to predict the vibrating behavior, to describe the hysteresis cycle, and to predict the actual filtering capabilities. In this chapter, a procedure to experimentally characterize the component behavior is proposed. The obtained experimental data were used to fit a phenomenological model, which is a development of the Bouc-Wen model. The proposed approach was experimentally validated.

Shape-shifting materials are entering into many disciplines from engineering to medicine where passive sensors are needed. Within this field, having materials which could automatically and cyclically reconfigure their shape in response to the surrounding environment while maintaining their performance during their in-service life is of great interest and challenging. Here, an experimental study is performed to analyze the cyclic morphing capabilities and the dynamic response of a unique multilayer system that changes shape in a pre-defined and, indeed, tunable temperature range which is triggered by an unconventional training. The material is formed by a composite of polyethylene (PE) film reinforced with polyethylene terephthalate (PET) fibers sandwiched with an aluminum layer. The results highlight the great capability of the material to spontaneously morph from a flat to a rolled shape when exposed to pre-set thermal gradients while maintaining a nearly constant frequency response. This result is due to a mutual compensation of material, structural, and geometric properties acquired in different configurations.

Since bolted complex structures are easily subjected to faults like fatigue crack/bolt loosening during their service, monitoring faults is very meaningful and helpful for them. Therefore, a novel vibration response-based approach for monitoring faults in bolted complex structures is proposed in this chapter. In this new approach, bolted complex structures are simplified as some discrete substructures, whose nonlinear dynamics are studied by a nonlinear multi-degree-of-freedom (MDOF) mode. By stimulating the structure many times with different magnitudes, nonlinear features from the substructure to be monitored only are defined, and a novel fault index and corresponding approach are proposed accordingly. With some experimental studies on a lab-bolted complex structure, the effectiveness of the proposed approach is fully vindicated.

In this work, we investigate the capabilities of the sparse identification of nonlinear dynamics method for time-delay identification. A possible solution is shown how delayed terms can be introduced into the method. We test the robustness and effectiveness of the method through data generated by simulation of different reference systems with known time delay. Through our test examples, we investigate the effect of noise and the delay distribution in the candidate terms. We also test the method in the presence of multiple delays. It is shown that by iterating through a range of threshold values with the STLSQ algorithm, the delayed terms can be identified in a robust manner.

This paper proposes a hybrid physics-machine learning method for probabilistic parameter estimation of a nonlinear dynamic system. The ability of this method to quantify the uncertainty of estimations is utilized at different levels of the hybrid method. In this method, a set of physics-based features are introduced to amplify the information content of initial observations. With this objective, the perturbation method is applied to obtain the asymptotic solution and frequency response of the nonlinear system in physics-based modeling. Extracted mathematical relationships provide for the identification of root causes of changes in frequency response. Subsequently, topological changes are quantified to be used as the inputs of the machine learning model. A Gaussian process regression (GPR) model is developed as a probabilistic estimator which uses the above physics-based features. The method is demonstrated using the case study of a linearly coupled Duffing oscillator system. The effectiveness and robustness of the hybrid method are demonstrated by estimating the coupling coefficient under strong nonlinear and uncertain parameter situations.

The identification of the nonlinear governing equations of a single degree of freedom oscillator under harmonic excitation, including Coulomb friction damping, is investigated. The so-called RK4-SINDy approach is enhanced here by incorporating part of the known physics to handle the non-smooth friction law since state-of-the-art approaches fail to capture such nonlinearities. This simple, yet representative, case study is examined using both artificially generated noisy data and data obtained from an experimental setup, in both cases for continuous motion. The obtained results highlight the potential of this framework in nonlinear system identification, given sparsely collected corrupted data, and the benefits of incorporating already known system information.

This contribution summarizes some preliminary results obtained in the development of a platform based on recycle Smart Devices to monitor the vibrations of buildings and infrastructure. The platform, designed within the framework of a project funded by the Italian Ministry of Economic Development, exploits the accelerometer, the other sensors, and communication boards included in Smart Devices that have reached their end of life. The idea is to realize a fully autonomous setup capable of acquiring information provided by the different sensors, store the datasets in a cloud server that can be accessed either during the monitoring phase or the post-processing phase, and perform locally a real-time nonlinear analysis to extract features leading to alarms.

We consider the forced Kadomtsev-Petviashvili I (KPI) equation derived in a recent work on magnetosonic waves excited by space debris objects of Acharya et al. [Adv. Space Res. 69, 4045–4057 (2022)] for further analysis in this work. For first time, we have derived a special exact distorted lump wave solution of the forced KPI equation for a specific localized form of the forcing function. The reason for choosing such a typical forcing function has been discussed in detail in the context of orbital motions of charged space debris objects in ionospheric plasma. Such exotic magnetosonic lump wave structures showing characteristic distortions resulted by orbital charged space debris objects can have potential implications in their indirect detection methods.

In the nonlinear dynamics of periodic microstructured systems, the amplitude-dependent dispersion properties of mechanical metamaterials are attracting increasing interest. The present paper investigates the free propagation of harmonic and superharmonic waves through a non-dissipative one-dimensional metamaterial with pantographic mass amplification. The effects of the quadratic and cubic inertial nonlinearities on the dispersion curve are analyzed. A perturbation strategy based on the Fourier series decomposition is adopted to determine the superharmonic amplitudes of the propagating periodic waves.

This chapter presents an acceleration technique for the fast numerical solution to nonlinear shallow water system with the help of a regular personal computer. This chapter helps with the timely evaluation of potential threats from tsunami waves at a particular part of the coast. High performance is achieved by using hardware acceleration which is a specialized Calculator based on the Field Programmable Gates Array (FPGA) microchip. Precision of the obtained numerical solutions was proved by comparing them to the available exact solutions. The achieved performance is comparable to the one of a supercomputer while it takes valuably less energy. The proposed way makes it possible to calculate the wave parameters along the coast (in a water area of several hundred kilometers wide) within a minute, provided that the initial sea surface displacement at tsunami source is given. In case of a major seismic event offshore Japan, it takes nearly 20 min for tsunami waves to reach the nearest coast. The proposed approach can provide tsunami warning centers with decision support information on evacuation measures, industry shutdown, vital goods supply, and other actions before the wave reaches the coast.

This article is a sequel to [3, 4] on explicit solutions of the

The present work investigates the chirped optical solitons in a medium of fiber Bragg gratings (BGs) with dispersive reflectivity. BGs is considered here with polynomial law of nonlinear refractive index. The model of coupled nonlinear Schrödinger equations is analyzed and reduced to an integrable form under specific conditions. The results are obtained with the aid of the soliton ansatz technique. Different structures of solutions including W-shaped, bright, dark, kink, and anti-kink solitons are retrieved. The behaviors of chirped optical solitons are illustrated graphically and described physically which may enhance the applications of fiber BGs.

Cardiac dysfunctions and arrhythmias are nonlinear and complex phenomena and can be monitored using electrocardiogram (ECG) recordings. ECG signals, and their underlying signal generation mechanisms, have strong nonlinear characteristics and, in some cases, present rich dynamic responses. In this paper we aim to characterize the abnormalities found in patients with arrhythmias through a novel signal processing procedure applied to ECG signals, and by characterizing them in the time and frequency domains. Specifically, we propose use of the wavelet-based Synchroextracting transform (WSET), an emerging method for time-frequency analysis (TFA). The central idea of WSET is to increase the concentration of energy in the time-frequency representation (TFR) and capture variations of the instantaneous frequency (IF) of the original, weak signal, which enables better characterization of anomalies in the frequency domain. In this study, using a public arrythmia database, WSET is employed to extract nonlinear and complex features of pathologies present in the ECG signals, thus facilitating characterization and diagnosis of subtle anomalies in the patient’s heart. The initial results obtained, based on the analysis of signals obtained from the MIT-BIH Arrhythmia dataset, demonstrated the ability of the new signal processing technique to detect short transients and subtle changes in the frequency spectrum of ECG signals.

Bio-electric activity of the heart is modeled by the Barrio–Varea–Aragon–Maini (BVAM) model that covers normal rhythm and several arrhythmia that lie in the chaotic region and exhibits several bifurcations. In this chapter, we develop the analytic solution of the BVAM model and identify the Hopf bifurcation. The center manifold reduction is applied to the governing equations to reduce the order of the system. The method of multiple scales is employed to develop the normal form of the Hopf bifurcation for the center manifolds. These are then transformed back into original coordinates where the analytical solution is compared with the numerical solution.

It is of a great interest that staking of young seedlings is done properly, ensuring oscillations of the trunk that will stimulate the root growth and proper anchoring to the ground. Stability of forced oscillations of double and triple staking methods is analyzed using a modification of a previously developed model of a young seedling staked with one stake. A previously developed model is in the form of a complex discrete cantilever coupled with a nonlinear spring. Stability and instability of the complex structure regarding the number of nonlinear springs to which the structure is coupled is investigated. The geometric nonlinearity of the system is introduced by a spring with cubic nonlinear properties that oscillates in the horizontal plane.Two systems of five non-linear differential equations of biodynamic complex and discrete structure alignment in two orthogonal directions were derived for all three models of spring coupling. It is shown that in the third model of coupling with three springs (tensioners), there is an interaction of the external force in one direction with oscillations in the original direction

Coral reef ecosystems are most vulnerable to changes in sea surface temperature (SST), a key environmental factor critical to reef-building growth. Elevated SST reduces the ability of corals to produce their calcium carbonate skeletons. Prolonged high SST results in coral bleaching, owing to the uncoupling of symbiosis among corals and microalgae. Corals have narrow temperature tolerances. The skeletal growth rate of corals falls sharply to zero even at a slight increase of SST above its temperature tolerance level. Corals are also vulnerable to macroalgal toxicity. Several benthic macroalgae species are known to bring about allelopathic chemical compounds that are very harmful to corals. The toxic macroalgae produce allelochemicals for which the survivability and settlement of coral larvae are highly affected. Toxic macroalgae species damage coral tissues when in contact by transferring hydrophobic allelochemicals present on macroalgal surfaces, leading to a reduction of corals and even coral mortality. The abundance of toxic macroalgae changes the community structure toward a macroalgae-dominated reef ecosystem. We use a continuous-time model to investigate coral-macroalgal phase shifts in the presence of elevated SST and macroalgal toxicity. We have derived the conditions for locally asymptotic stability of steady states. Computer simulations have been carried out to illustrate different analytical results.

In this study, it is assumed that harvesting is permitted only when the ratio of prey to predator is below a threshold value. The Filippov type dynamical system consisting of two smooth subsystems is investigated. The existence of harvesting factor within an interval dependent on biologically parameters of the system would ensure the existence and stability of interior equilibrium state for the harvested system. The boundary that splits the two subsystems is classified into sliding and crossing region. It is also proved that there does not exist any escaping region on the boundary. Different equilibrium points such as boundary points, tangent points, pseudo-equilibrium points are found out for the boundary. The visibility condition for these tangent points is determined. Boundary bifurcation and sliding bifurcation of the Filippov system are demonstrated numerically. The half saturation rate of predators plays a crucial role in determining whether the system to lie in specific sub-regions or is oscillating in between the two regions.

We propose a mathematical model to study the dynamics of dengue fever in a susceptible population. Our main goal is to assess the effect of four different control strategies, namely, the existence of sterile male mosquitoes, the use of pesticides for larvae, the use of pesticide for adult mosquitoes, and vaccination, in the disease propagation. We discuss the results of numerical simulations for the four different control strategies from an epidemiological point of view. We also design a vaccination protocol via an optimal control scheme in order to reduce the number of infected individuals. The results suggest that our model is well-posed, and inferences are made to help define health policy measures.