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Spectral submanifolds (SSMs) have recently been shown to provide exact and unique reduced-order models for nonlinear unforced mechanical vibrations. Here, we extend these results to periodically or quasi-periodically forced mechanical systems, obtaining analytic expressions for forced responses and backbone curves on modal (i.e. two dimensional) time-dependent SSMs. A judicious choice of the parametrization of these SSMs allows us to simplify the reduced dynamics considerably. We demonstrate our analytical formulae on three numerical examples and compare them to results obtained from available normal-form methods.

This volume contains the proceedings of the AMS Special Session on Nonstandard Finite-Difference Discretizations and Nonlinear Oscillations, in honor of Ronald Mickens's 70th birthday, held January 9-10, 2013, in San Diego, CA. Included are papers on design and analysis of discrete-time and continuous-time dynamical systems arising in the natural and engineering sciences, in particular, the design of robust nonstandard finite-difference methods for solving continuous-time ordinary and partial differential equation models, the analytical and numerical study of models that undergo nonlinear oscillations, as well as the design of deterministic and stochastic models for epidemiological and ecological processes. Some of the specific topics covered in the book include the analysis of deterministic and stochastic SIR-type models, the assessment of cost-effectiveness of vaccination problems, finite-difference methods for oscillatory dynamical systems (including the Schrodinger equation and Brusselator system), the design of exact and elementary stable finite-difference methods, the study of a two-patch model with Allee effects and disease-modified fitness, the study of the delay differential equation model with application to circadian rhythm and the application of some special functions in the solutions of some problems arising in the natural and engineering sciences. A notable feature of the book is the collection of some relevant open problems, intended to help guide the direction of future research in the area.

This research focuses on solving the nonlinear second-order jet engine vibration equation utilizing a hybrid analytical technique named the least square homotopy perturbation method (LSHPM). The numerical and graphical comparison of the solutions obtained using the homotopy perturbation method (HPM), LSHPM, and the MATLAB built-in solver bvp5c is presented across four distinct cases. Additionally, a comparative analysis between the solutions derived from LSHPM and those reported in previous literature is also presented. The tabular and graphical representation of the solutions, along with the numerical validation through residual error analysis, are given. Furthermore, the convergence analysis of the LSHPM for its stability and solution reliability is provided. The graphical and numerical representation of the residual error analysis reveals that LSHPM exhibits superiority over HPM in terms of rapid convergence and accuracy. The strong agreement between the results obtained from HPM and bvp5c with those of LSHPM demonstrates that LSHPM offers a more efficient, reliable, and fast convergent solution of the initial and boundary value problem.

We investigate analytically and numerically a basic model of driven Brownian motion with a velocity-dependent friction coefficient in nonlinear viscoelastic media featured by a stress plateau at intermediate shear velocities and profound memory effects. For constant force driving, we show that nonlinear oscillations of a microparticle velocity and position emerge by a Hopf bifurcation at a small critical force (first dynamical phase transition), where the friction’s nonlinearity seems to be wholly negligible. They also disappear by a second Hopf bifurcation at a much larger force value (second dynamical phase transition). The bifurcation diagram is found in an analytical form confirmed by numerics. Surprisingly, the particles’ inertial and the medium’s nonlinear properties remain crucial even in a parameter regime where they were earlier considered entirely negligible. Depending on the force and other parameters, the amplitude of oscillations can significantly exceed the size of the particles, and their period can span several time decades, primarily determined by the memory time of the medium. Such oscillations can also be thermally excited near the edges of dynamical phase transitions. The second dynamical phase transition combined with thermally induced stochastic limit cycle oscillations leads to a giant enhancement of diffusion over the limit of vast driving forces, where an effective linearization of stochastic dynamics occurs.

This paper analyses the evolution of nonlinear oscillation control methods and presents an innovative approach known as analytical design of aggregated oscillation controllers (ADACO). This method is based on the principles of synergetic control theory and focuses on the integration of self-organization and control processes to synthesize energy-efficient control laws for nonlinear oscillating systems. The authors elaborate on the theoretical foundations of ADACO, which extends the previous analytical design of aggregated controllers (ADAC) method by incorporating energy invariants and integrals of motion into the synthesis of control laws. This approach demonstrates significant advantages over traditional methods, offering a versatile framework for the design of energy-efficient control systems for a wide range of nonlinear oscillating systems in various fields such as aerospace, robotics, vibromechanical systems, and objects with chaotic dynamics. The aim of the paper is to establish a unified approach to the control of nonlinear oscillations, solving both the problems of generation of stable oscillations and suppression of unwanted perturbations. The application of synergetic control principles in the framework of ADACO opens prospects for further development of nonlinear control theory.

The paper investigates the nonlinear transversal vibrations of a cantilever beam structure in the primary resonance case. A time-delayed position-velocity control is suggested to reduce the nonlinear vibrations of the structure under consideration. A non-perturbative method (NPM) is used to get an equivalent analogous linear differential equation (DE) to the original nonlinear one. For the benefit of the readers, a comprehensive description of the NPM method is provided. The theoretical findings are validated through a numerical comparison carried out by employed the Mathematica Software. Both the numerical solutions and the theoretical outcomes showed excellent agreement. As well-known, all classic perturbation techniques use Taylor expansion, when the restoring forces are present, to expand these forces and therefore lessen the difficulty of the given problem. Under the NPM, this weakness is no longer present. Furthermore, one may examine the stability examination of the issue with the NPM something that was not possible with prior traditional techniques. The controlled linear equivalent model is examined using the multiple-scales homotopy method. The amplitude-phase modulation equations which control the dynamics of the structure at the various resonance circumstances are established. The loop-delay stability diagrams are analyzed. It is looked at how the different controller parameters impact the oscillation behaviors of the system. The obtained theoretical outcomes showed that the loop delay has an important impact on the effectiveness of the control. Therefore, the ideal loop-delay values are given and used to develop the enactment of the organized control. The completed analytical results are also numerically validated, to reveal their good correlation with the achieved theoretical new results.

Pendulum oscillators study harmonic motion, energy conservation, and nonlinear dynamics, providing insights into mechanical vibrations, wave phenomena, weather patterns, and quantum mechanics, with real-world applications in engineering, seismology, and clock mechanisms. The present study addresses three distinct issues related to SPs; a charged magnetic spherical simple pendulum (SP), and a SP composed of heavy cylinders that roll freely in a horizontal plane, and a nonlinear model depicting the motion of a damped SP in a fluid flow. The SPs are analyzed via an innovative technology known as the non-perturbative approach (NPA), which is based on He’s frequency formula (HFF). This advanced approach linearizes a nonlinear ordinary differential equation (ODE), enabling more straightforward analysis and solution. As-well known, implementing the NPA has several advantages, chief among them the removal of the constraints associated with managing Taylor expansions. Consequently, there have been no augmentations to the current restorative forces. Secondly, the novel method enables us to assess the stability criteria of the system away from the traditional perturbation techniques. The numerical comparison of nonlinear ODEs into linear ones using Mathematica Software (MS) is conducted to validate this innovative method. An analysis of the two responses demonstrates a strong concordance, underscoring the necessity of precision of the methodology. Furthermore, to demonstrate the influence of the components on motion behavior, the time history of the calculated solution and the corresponding phase plane plots are accumulated. The use of multiple phase portraits aims to explore stability and instability near equilibrium points by examining the interaction between expanded and cyclotron frequencies, modulated by the magnetic field, for varying azimuthal angular velocities.

An analytical technique based on the certain odd polynomial functions is introduced for solving nonlinear oscillatory problems, applicable to both autonomous and non-autonomous cases. A set of algebraic equations are found when the proposed technique is applied. The algebraic equations are linear and relatively straightforward for determining the unknown coefficients associated with the solution. An excellent agreement is found between approximated and numerical solutions, which prove that the technique is very efficient and easy to implement. The technique allows to obtain any desired order of approximate solutions. The results are shown in figures.

The internal resonances between the longitudinal and transversal oscillations of a forced Timoshenko beam with an axial end spring are studied in depth. In the linear regime, the loci of occurrence of 1 :  ir , i r ∈ N , internal resonances in the parameters space are identified. Then, by means of the multiple time scales method, the 1 : 2 case is investigated in the nonlinear regime, and the frequency response functions and backbone curves are obtained analytically, and investigated thoroughly. They are also compared with finite element numerical simulations, to prove their reliability. Attention is paid to the system response obtained by varying the stiffness of the end spring, and it is shown that the nonlinear behaviour instantaneously jumps from hardening to softening by crossing the exact internal resonance value, in contrast to the singular (i.e. tending to infinity) behaviour of the nonlinear correction coefficient previously observed (without properly taking the internal resonance into account).

The performance of the nonlinear energy sink (NES) that composed of a small mass and essentially nonlinear coupling stiffness with a linear structure is considerably enhanced here by including the negative linear and nonlinear coupling stiffness components. These negative linear and nonlinear stiffness components in the NES are realized here through the geometric nonlinearity of the transverse linear springs. By considering these components in the NES, very intersecting results for passive targeted energy transfer (TET) are obtained. The performance of this modified NES is found here to be much improved than that of all existing NESs studied up to date in the literature. Moreover, nearly 99 % of the input shock energy induced by impulse into the linear structures considered here has been found to be rapidly transferred and locally dissipated by the modified NES. In addition, this modified NES maintains its high performance of shock mitigation in a broadband fashion of the input initial energies where it keeps its high performance even for sever input energies. This is found to be achieved by an immediate cascade of several resonance captures at low- and high- nonlinear normal modes frequencies. The findings obtained here by including the negative linear and nonlinear stiffness components are expected to significantly enrich the application of these stiffness components in the TET field of such nonlinear oscillators.