The complicated bifurcation of an archetypal self-excited SD oscillator with dry friction

In this paper, we investigate the local and global bifurcation behaviors of an archetypal self-excited smooth and discontinuous oscillator driven by moving belt friction. The belt friction is described in the sense of Stribeck characteristic to formulate the mathematical model of the proposed system. For such a friction characteristic, the complicated bifurcation behaviors of the system are discussed. The bifurcation of the multiple sliding segments for this self-excited system is exhibited by analytically exploring the collision of tangent points. The Hopf bifurcation of this self-excited system with viscous damping is analyzed by making the examination of the eigenvalues at the steady state and discussing the stability of the limit cycles. The bifurcation diagrams and the corresponding phase portraits are depicted to demonstrate the complicated dynamical behaviors of double tangency bifurcation, the bifurcation of sliding homoclinic orbit to a saddle, subcritical Hopf bifurcation and grazing bifurcation for this system.     

Basin boundaries with nested structure in a shallow arch oscillator

The basin boundaries with nested structure are investigated in a shallow arch oscillator. Basin organization is complex yet systematic and it is governed by the ordering of heteroclinic and homoclinic connections of regular saddles. The Wada properties are verified for eight basin boundaries, where five basin boundaries are totally Wada basin boundaries for a given set of parameters. The organization of nested basin boundaries is governed by the order of saddle connections. The term “Wada number” is introduced to describe the nested structure. The partially Wada basin boundaries are investigated by the erodent cells and the remnant cells.     

Wada basin boundaries in switched systems

The Wada basin boundaries of a switched Hénon map have been verified for period-2 switching signals [Zhang in Nonlinear Dyn. 73:2221–2229, 2013]. Based upon the auxiliary dynamical system method, the results are extended to the generally switched systems with time-dependent switching. Under some generic assumptions, some sufficient conditions guaranteeing Wada basin boundaries are presented for the periodic switching signals. The results show that switching signals can give rise to this type of basin structure. It suggests that the unpredictability associated to the Wada property can also occur in the switched system.     

Bi-parametric topology of subharmonics of an asymmetric bubble oscillator at high dissipation rate

The subharmonic topology of a nonlinear, asymmetric bubble oscillator (Keller–Miksis equation) in glycerine is investigated in the parameter space of its external excitation (frequency and pressure amplitude). The bi-parametric investigation revealed that the exoskeleton of the topology can be described as the composition of U-shaped subharmonics of periodic orbits. The fine substructure (higher-order ultra-subharmonic resonances) usually appearing via the well-known period n-tupling phenomenon is completely missing due to the high dissipation rate of the viscous liquid. Moreover, a complex internal structure of the subharmonics has been found, which are composed by interconnected bifurcation blocks (in a zig-zag pattern) each describing the skeleton of a shrimp-shaped domain. The employed numerical techniques are the combination of an in-house initial value problem solver written in C++/CUDA C to harness the high processing power of professional graphics cards, and the boundary value problem solver AUTO to compute periodic orbits directly regardless of their stability.     

The limit case response of the archetypal oscillator for smooth and discontinuous dynamics

In this paper, the limit case of the SD (smooth and discontinuous) oscillator is studied. This system exhibits standard dynamics governed by the hyperbolic structure associated with the stationary state of the double-well. The substantial deviation from the standard dynamics is the non-smoothness of the velocity in crossing from one well to another, caused by the loss of local hyperbolicity due to the discontinuity. Without dissipation, the KAM structure on the Poincaré section is constructed with generic KAM curves and a series of fixed points associated with surrounded islands of quasi-periodic orbits and the chaotic connection orbits. It is found that, for a fixed set of parameters, a special chaotic orbit exits there which fills a finite region and connects a series of islands dominated by different chains of fixed points. As one adds weak dissipation, the periodic solutions in this finite region remain unchanged while the quasi-periodic solutions (isolated islands) are converted to the corresponding periodic solutions. The relevant dynamics for the system with weak dissipation under external excitation is shown having period doubling bifurcation leading to chaos, and multi-stable solutions.     

Fractal basin boundaries in a two-degree-of-freedom nonlinear system

The final state for nonlinear systems with multiple attractors may become unpredictable as a result of homoclinic or heteroclinic bifurcations. The fractal basin boundaries due to such bifurcations for a four-well, two-degree-of-freedom, nonlinear oscillator under sinusoidal forcing have been studied, based on a theory of homoclinic bifurcation inn-dimensional vector space developed by Palmer. Numerical simulation is used as a means of demonstrating the consequences of the system dynamics when the bifurcations occur, and it is shown that the basin boundaries in the configuration space (x, y) become fractal near the critical value of the heteroclinic bifurcations.     

Synchronization of complex networks with nonhomogeneous Markov jump topology

The problem of synchronization of complex networks with nonhomogeneous Markov jump topology and time-varying coupling delay is investigated in this paper. The Markov process in the underlying complex networks is finite piecewise homogeneous, which is a special case of nonhomogeneous Markov process. Based on the Lyapunov functional approach, an exponential stability condition is derived for the error system in terms of the linear matrix inequality method. Based on the condition, the design method of the desired controller is given. An example is given to show the effectiveness of the proposed technique.     

Behaviour of an oscillator with even non-linear damping

In this short note we prove two theorems on the behaviour of a single degree of freedom oscillator with linear stiffness and even non-linear damping terms.     

Sommerfeld effect in an oscillator with a reciprocating mass

Nonlinear Dynamics - In this paper, the dynamics of a non-ideally driven single-degree-of-freedom vibrating system will be explored in detail. The objective is to describe Sommerfeld effect in a...     

Shape and topology sensitivity analysis for cracks in elastic bodies on boundaries of rigid inclusions

We consider an elastic body with a rigid inclusion and a crack located at the boundary of the inclusion. It is assumed that nonpenetration conditions are imposed at the crack faces which do not allow the opposite crack faces to penetrate each other. We analyze the variational formulation of the problem and provide shape and topology sensitivity analysis of the solution in two and three spatial dimensions. The differentiability of the energy with respect to the crack length, for the crack located at the boundary of rigid inclusion, is established.     

Long-wave instability of an advective flow in an inclined fluid layer with perfectly heat-conducting boundaries

Stability of a steady convective flow in a plane inclined layer with perfectly heat-conducting solid boundaries in the presence of a uniform longitudinal temperature gradient to long-wave perturbations is studied. The boundaries of the domain of stability to long-wave perturbations are found, and the critical Grashof numbers for the most dangerous even helical perturbations are determined.     

Stability of Couette flow of an ideal fluid with free boundaries

The linear and nonlinear stage of development of instability of Couette flow with two free boundaries is studied. It is established that instability occurs only for long waves, and the critical wave number is computed. In the presence of surface-tension forces, instability is preserved only at Weber numbersWe ≤ 1/3. Computer Center, Siberian Division, Russian Academy of Sciences, Krasnoyarsk 660036. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 5, pp. 99–105, September–October, 1998.     

Experimental study of an impact oscillator with viscoelastic and Hertzian contact

Modeling an impact event is often related to the desired outcome of an impact oscillator study. If the only intent is to study the dynamic behavior of the system, numerous researchers have shown that simpler impact models will often suffice. However, when the geometric contours and material properties of the two colliding surfaces are known, it is often of interest to model the contact event at a greater level of complexity. This paper investigates the application of a finite time impact model to the study of a parametrically excited planar pendulum subjected to a motion-dependent discontinuity. Experimental and numerical studies demonstrate the presence of multiple periodic attractors, subharmonics, quasi-periodic motions, and chaotic oscillations.     

Global stability analysis of parametrically excited cylindrical shells through the evolution of basin boundaries

In the present study, the large-amplitude vibrations and stability of a perfect circular cylindrical shell subjected to axial harmonic excitation in the neighborhood of the lowest natural frequencies are investigated. Donnell's shallow shell theory is used and the shell spatial discretization is obtained by the Ritz method. An efficient low-dimensional model presented in previous publications is used to discretize the continuous system. The main purpose of this work is to discuss the use of basins of attraction as a measure of the reliability and safety of the structure. First, the nonlinear behavior of the conservative system is discussed and the basin structure and volume is understood from the topologic structure of the total energy and its evolution as a function of the system parameters. Then, the behavior of the forced oscillations of the harmonically excited shell is analyzed. First the stability boundaries in force control space are obtained and the bifurcation events connected with these boundaries are identified. Based on the bifurcation diagrams, the probability of parametric instability and escape are analyzed through the evolution and erosion of basin boundaries within a prescribed control volume defined by the manifolds. Usually, basin boundaries become fractal. This together with the presence of catastrophic subcritical bifurcations makes the shell very sensitive to initial conditions, uncertainties in system parameters, and initial imperfections. Results show that the analysis of the evolution of safe basins and the derivation of appropriate measures of their robustness is an essential step in the derivation of safe design procedures for multiwell systems.     

Invariant torus and its destruction for an oscillator with dry friction

Nonlinear Dynamics - Mechanical systems with dry friction are typical Filippov systems. Such class of systems have complicated dynamical behaviors due to the existence of sliding motion. In this...     

Dynamic stabilization of an asymmetric nonlinear bubble oscillator

The principal objective of this study is to present a new numerical scheme based on a combination of q-homotopy analysis approach and Laplace transform approach to examine the Fitzhugh–Nagumo (F–N) equation of fractional order. The F–N equation describes the transmission of nerve impulses. In order to handle the nonlinear terms, the homotopy polynomials are employed. To validate the results derived by employing the used scheme, we study the F–N equation of arbitrary order by using the fractional reduced differential transform scheme. The error analysis of the proposed approach is also discussed. The outcomes are shown through the graphs and tables that elucidate that the used schemes are very fantastic and accurate.     

On the dynamic behavior of an elastoplastic oscillator

Summary We describe an experimental research on the behavior of an elastoplastic oscillator with one degree of freedom and we point out some differences with respect to the results of studies on a bilinear isteretical model. Therefore a mechanical scheme of an elastoplastic oscillator endowed with a continuous skeleton intrinsec curve is studied and the results of the theory agree with the experimental ones.

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Sommario Si descrivono le esperienze condotte su un oscillatore elastoplastico a un grado di libertà. Si mettono in evidenza alcune differenze del comportamento reale con le risultanze teoriche, queste vengono spiegate attraverso lo studio di un modello teorico elastoplastico con caratteristica intrinseca curvilinea anzichè bilineare.

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The present investigation has been promoted and financed by the Consiglio Nazionale delle Ricerche (C.N.R.) at the Istituto di Scienza e Tecnica delle Costruzioni del Politecnico di Milano.     

Impact of node dynamics parameters on topology identification of complex dynamical networks

This paper aims at investigating the topology identification problem of complex dynamical networks with varying node dynamics parameters and fixed inner coupling matrices. In particular, by employing the unified chaotic system as node dynamics, this work further explores the influence of continuously changing node dynamics parameters on topology identification of complex dynamical networks with different coupling strengths. Results show that for sufficiently small or large coupling strengths, the performance of topology identification is not affected by the change of node parameters. Specifically, for small enough coupling strengths, the topological structure can be completely identified regardless of the change of node parameters, while for sufficiently large coupling strengths, the connectivity (presence and absence of connections) cannot be successfully identified. Furthermore, for certain coupling strengths, with the increase of node dynamics parameters, the topology identification varies from completely unidentifiable to partially or event completely identifiable. Therefore, the synchronization-based topology identification depends on node dynamics. Even for the same node dynamical model, different parameters can have a significant impact on identification results. Furthermore, for networks consisting of chaotic oscillators defining node dynamics, small coupling strengths are conducive to topology identification. A broader conclusion is that projective synchronization, rather than just complete synchronization, is an obstacle to the network topology identification. The findings in this paper will add to our understanding of conditions for identifying topologies of complex networks.     

Complex and chaotic response of a non-linear oscillator with an isothermal gas spring

In this paper we study the dynamics of a non-linear one-degree-of-freedom system subjected to an external harmonic excitation, representing a simplified model for the synchronous hydraulic oscillations that can occur in the draft tube of Francis turbines at partial loads. The application of different typical numerical techniques has shown the existence of multiple coexisting periodic solutions, and the non-periodic bounded solutions which exhibit deterministic chaotic behaviour. The relevant strange attractor has been defined and the loss of memory associated with an exponential divergence in time of close initial conditions resulting in chaotic dynamics have been found and measured. A partial classification of qualitatively different dynamical behaviours for the system has been outlined in the control parameter space.

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Sommario In questo articolo viene studiata la dinamica di un sistema non-lineare ad un singolo grado di liberta' soggetto ad una forzante armonica esterna, rappresentante un modello semplificato per le oscillazioni idrauliche sincrone che hanno luogo nei diffusori delle turbine tipo Francis a carico parziale. Applicando differenti tecniche numeriche, viene mostrata l'esistenza di soluzioni periodiche multiple, oltre che soluzioni non-periodiche limitate con tipico comportamento caotico deterministico. L'attrattore strano corrispondente e' stato definito e caratterizzato: la perdita di memoria associata alla divergenza esponenziale di orbite inizialmente vicine, tipica della dinamica caotica, e' stata individuata e calcolata numericamente. Una prima parziale classificazione dei vari comportamenti dinamici per il sistema viene evidenziata attraverso la rappresentazione nello spazio parametrico.