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IN THIS ISSUE

### Guest Editorial

J. Comput. Nonlinear Dynam. 2018;13(9):090301-090301-2. doi:10.1115/1.4040569.
OPEN ACCESS

Over the past few decades, it has been well established that in both natural and engineering systems nonlinearity is one of the primary mechanisms for the generation of complexity. Examples range from nanoscales to geophysical-scales and include chaotic fluid dynamics in environmental settings, biological behavior ranging from cells and viruses to collective dynamics of animals and humans, engineering applications involving interactions of autonomous agents with their environment, as well as large amplitude oscillations, just to name a few. In all of these examples, many of which are considered in the present issue, nonlinearity is manifested through energy transfers between degrees-of-freedom, rapid loss of predictability, nonlocal frequency- and history-dependence, and non-Gaussian statistics with the possible occurrence of extreme transient responses.

Commentary by Dr. Valentin Fuster

### SPECIAL ISSUE PAPERS

J. Comput. Nonlinear Dynam. 2018;13(9):090901-090901-7. doi:10.1115/1.4040410.

We present a new approach to the construction of first integrals for second-order autonomous systems without invoking a Lagrangian or Hamiltonian reformulation. We show and exploit the analogy between integrating factors of first-order equations and their Lie point symmetry and integrating factors of second-order autonomous systems and their dynamical symmetry. We connect intuitive and dynamical symmetry approaches through one-to-one correspondence in the framework proposed for first-order systems. Conditional equations for first integrals are written out, as well as equations determining symmetries. The equations are applied on the simple harmonic oscillator and a class of nonlinear oscillators to yield integrating factors and first integrals.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(9):090902-090902-11. doi:10.1115/1.4038331.

There have been a variety of attempts to model the quasi-static and high energy impact dynamics of vertically aligned carbon nanotube (VACNT) pads. However, very little work has focused on identifying the behavior at the midlevel frequencies that may occur in materials handling or vibration suppression applications. Moreover, the existing models are predominantly very complex, and yet provide only a very rough approximation of the bulk behavior. While several of the existing models make attempts at ascribing physical relevance, an adequate first principles approach has yet to be demonstrated. In this work, a close-fitting continuous model of these midfrequency dynamics is developed utilizing a combination of phenomenological- and identification-based methodologies. First, a set of specially fabricated carbon nanotube pads are preconditioned and subjected to various position-controlled compression experiments. The measured position and force responses are used to develop load–displacement curves, from which several characteristic features are identified. Based on these observations, a preliminary version of the proposed model is introduced. This simplified model is then systematically refined in order to demonstrate completely both the modeling approach and parameter identification scheme. The accuracy of the model is demonstrated through a comparison between the modeled and experimental responses including a normalized vector correlation of >0.998 across all sets of sinusoidal experimental data. A brief analysis utilizing a Lyapunov linearization approach follows, as well as a discussion of the advantages and limitations of the final model.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(9):090903-090903-8. doi:10.1115/1.4039681.

This paper deals with the problem of master-slave synchronization of fractional-order chaotic systems with input saturation. Sufficient stability conditions for achieving the synchronization are derived from the basis of a fractional-order extension of the Lyapunov direct method, a new lemma of the Caputo fractional derivative, and a local sector condition. The stability conditions are formulated in linear matrix inequality (LMI) forms and therefore are readily solved. The fractional-order chaotic Lorenz and hyperchaotic Lü systems with input saturation are utilized as illustrative examples. The feasibility of the proposed synchronization scheme is demonstrated through numerical simulations.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(9):090904-090904-16. doi:10.1115/1.4038643.

We introduce a fractional order model for the human immunodeficiency virus (HIV) dynamics, where time-varying drug exposure and drug resistance are assumed. We derive conditions for the local and global asymptotic stability of the disease-free equilibrium. We find periodic stable endemic states for certain parameter values, for sinusoidal drug efficacies, and when considering a density-dependent decay rate for the T cells. Other classes of periodic drug efficacies are considered and the effect of the phases of these functions on the dynamics of the model is also studied. The order of the fractional derivative plays an important role in the severity of the epidemics.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(9):090905-090905-7. doi:10.1115/1.4037930.

This paper is devoted to investigating the relation between Hadamard-type fractional derivatives and finite part integrals in Hadamard sense; that is to say, the Hadamard-type fractional derivative of a given function can be expressed by the finite part integral of a strongly singular integral, which actually does not exist. Besides, our results also cover some fundamental properties on absolutely continuous functions, and the logarithmic series expansion formulas at the right end point of interval for functions in certain absolutely continuous spaces.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(9):090906-090906-11. doi:10.1115/1.4039841.

This paper proposes new fractional-order (FO) models of seven nonequilibrium and stable equilibrium systems and investigates the existence of chaos and hyperchaos in them. It thereby challenges the conventional generation of chaos that involves starting the orbits from the vicinity of unstable manifold. This is followed by the discovery of coexisting hidden attractors in fractional dynamics. All the seven newly proposed fractional-order chaotic/hyperchaotic systems (FOCSs/FOHSs) ranging from minimum fractional dimension ($nf$) of 2.76 to 4.95, exhibit multiple hidden attractors, such as periodic orbits, stable foci, and strange attractors, often coexisting together. To the best of the our knowledge, this phenomenon of prevalence of FO coexisting hidden attractors in FOCSs is reported for the first time. These findings have significant practical relevance, because the attractors are discovered in real-life physical systems such as the FO homopolar disc dynamo, FO memristive system, FO model of the modulation instability in a dissipative medium, etc., as analyzed in this work. Numerical simulation results confirm the theoretical analyses and comply with the fact that multistability of hidden attractors does exist in the proposed FO models.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(9):090907-090907-6. doi:10.1115/1.4038450.

We investigate exact enlarged controllability (EEC) for time fractional diffusion systems of Riemann–Liouville type. The Hilbert uniqueness method (HUM) is used to prove EEC for both cases of zone and pointwise actuators. A penalization method is given and the minimum energy control is characterized.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(9):090908-090908-9. doi:10.1115/1.4039980.

By using a simple state feedback control technique and introducing two new nonlinear functions into a modified Sprott B system, a novel four-dimensional (4D) no-equilibrium hyper-chaotic system with grid multiwing hyper-chaotic hidden attractors is proposed in this paper. One remarkable feature of the new presented system is that it has no equilibrium points and therefore, Shil'nikov theorem is not suitable to demonstrate the existence of chaos for lacking of hetero-clinic or homo-clinic trajectory. But grid multiwing hyper-chaotic hidden attractors can be obtained from this new system. The complex hidden dynamic behaviors of this system are analyzed by phase portraits, the time domain waveform, Lyapunov exponent spectra, and the Kaplan–York dimension. In particular, the Lyapunov exponent spectra are investigated in detail. Interestingly, when changing the newly introduced nonlinear functions of the new hyper-chaotic system, the number of wings increases. And with the number of wings increasing, the region of the hyper-chaos is getting larger, which proves that this novel proposed hyper-chaotic system has very rich and complicated hidden dynamic properties. Furthermore, a corresponding improved module-based electronic circuit is designed and simulated via multisim software. Finally, the obtained experimental results are presented, which are in agreement with the numerical simulations of the same system on the matlab platform.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(9):090909-090909-6. doi:10.1115/1.4037995.

Planar dynamics of a rotor supported by long hydrodynamic journal bearing is investigated theoretically. An analytical model of the long journal bearing system is numerically integrated for analysis of fixed point and periodic oscillations. The nonlinearities in the system arise due to a nonlinear fluid film force acting on the journal. The influences of three dimensionless parameters, viz. bearing parameter, unbalance, and rotor speed, on the dynamic behavior of the rotor bearing system is studied and compared with the short journal bearing. For the same bearing parameter, short bearing has large operating speed compared to a long bearing. The results are presented in the form of a bifurcation diagram and Poincaré map of the oscillations based on numerical computation. The considered unbalanced system shows periodic, multiperiodic, and quasi-periodic motion in different speed range. Jumping phenomenon is also observed for a high value of bearing parameter with unbalance.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(9):090910-090910-12. doi:10.1115/1.4040239.

Piston impacts against the cylinder liner are the most significant sources of mechanical noise in internal combustion (IC) engines. Traditionally, the severity of impacts is reduced through the modification of physical and geometrical characteristics of components in the piston assembly. These methods effectively reduce power losses at certain engine operating conditions. Frictional losses and piston impact noise are inversely proportional. Hence, the reduction in power loss leads to louder piston impact noise. An alternative method that is robust to fluctuations in the engine operating conditions is anticipated to improve the engine's noise, vibration and harshness (NVH) performance, while exacerbation in power loss remains within the limits of conventional methods. The concept of targeted energy transfer (TET) through the use of nonlinear energy sink (NES) is relatively new and its application in automotive powertrains has not been demonstrated yet. In this paper, a TET device is conceptually designed and optimized through a series of parametric studies. The dynamic response and power loss of a piston model equipped with this nonlinear energy sink is investigated. Numerical studies have shown a potential in reducing the severity of impact dynamics by controlling the piston's secondary motion.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(9):090911-090911-11. doi:10.1115/1.4040261.

Electrostriction is a recent actuation mechanism which is being explored for a variety of new micro- and millimeter scale devices along with macroscale applications such as artificial muscles. The general characteristics of these materials and the nature of actuation lend itself to possible production of very rich nonlinear dynamic behavior. In this work, principal parametric resonance of the second mode in in-plane vibrations of appropriately designed electrostrictive plates is investigated. The plates are made of an electrostrictive polymer whose mechanical response can be approximated by Mooney Rivlin model, and the induced strain is assumed to have quadratic dependence on the applied electric field. A finite element model (FEM) formulation is used to develop mode shapes of the linearized structure whose lowest two natural frequencies are designed to be close to be in 1:2 ratio. Using these two structural modes and the complete Lagrangian, a nonlinear two-mode model of the electrostrictive plate structure is developed. Application of a harmonic electric field results in in-plane parametric oscillations. The nonlinear response of the structure is studied using averaging on the two-mode model. The structure exhibits 1:2 internal resonance and large amplitude vibrations through the route of parametric excitation. The principal parametric resonance of the second mode is investigated in detail, and the time response of the averaged system is also computed at few frequencies to demonstrate stability of branches. Some results for the case of principal parametric resonance of the first mode are also presented.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(9):090912-090912-8. doi:10.1115/1.4038204.

In this paper, the analytical solutions of periodic evolutions of the periodically diffused Brusselator are obtained through the generalized harmonic balanced method. Stable and unstable solutions of period-1 and period-2 evolutions in the Brusselator are presented. Stability and bifurcations of the periodic evolution are determined by the eigenvalue analysis, and the corresponding Hopf bifurcations are presented on the analytical bifurcation tree of the periodic motions. Numerical simulations of stable period-1 and period-2 motions of Brusselator are completed. The harmonic amplitude spectra show harmonic effects on periodic motions, and the corresponding accuracy of approximate analytical solutions can be prescribed specifically.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(9):090913-090913-3. doi:10.1115/1.4038895.

For stochastic systems, the Fokker–Planck equation (FPE) is used to describe the system dynamics. The FPE is a partial differential equation, which is a function of all the variables in state space and of time. To solve the FPE, several methods are used, including finite elements, moment neglect methods, and cumulant neglect methods. This paper will study the cumulant neglect equations, which are derived from the FPE. It will be shown that the cumulant neglect method, while being a useful and popular tool for studying the system response, introduces several nonphysical artifacts.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2018;13(9):090914-090914-12. doi:10.1115/1.4039309.

We characterize the complex, heavy-tailed probability density functions (pdfs) describing the response and its local extrema for structural systems subject to random forcing that includes extreme events. Our approach is based on recent probabilistic decomposition-synthesis (PDS) technique (Mohamad, M. A., Cousins, W., and Sapsis, T. P., 2016, “A Probabilistic Decomposition-Synthesis Method for the Quantification of Rare Events Due to Internal Instabilities,” J. Comput. Phys., 322, pp. 288–308), where we decouple rare event regimes from background fluctuations. The result of the analysis has the form of a semi-analytical approximation formula for the pdf of the response (displacement, velocity, and acceleration) and the pdf of the local extrema. For special limiting cases (lightly damped or heavily damped systems), our analysis provides fully analytical approximations. We also demonstrate how the method can be applied to high dimensional structural systems through a two-degrees-of-freedom (TDOF) example system undergoing extreme events due to intermittent forcing. The derived formulas can be evaluated with very small computational cost and are shown to accurately capture the complicated heavy-tailed and asymmetrical features in the probability distribution many standard deviations away from the mean, through comparisons with expensive Monte Carlo simulations.

Commentary by Dr. Valentin Fuster

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