Bifurcation Graph Research Articles

Nonlinear forced vibration of a moving paper web with varying density

The study investigates the forced nonlinear vibration characteristics of an axially moving printing paper web with variable density in the lateral direction. The nonlinear governing equations of the web can be obtained by the von Karman large deflection thin plate theory. The vibration equations are discretized using the Bubnov–Galerkin method. The fourth-order Runge–Kutta technique is used to solve the differential equations of the nonlinear system. The phase-plane diagrams, time histories, bifurcation graphs, Poincare maps, and power spectrum are employed to analyze the influence of density coefficient, velocity, and aspect ratio on the nonlinear dynamic behavior of the moving paper web. The stable working region and the divergence instability region of the web are obtained. The research provides a theoretical foundation for improving the dynamic stability of a moving paper web.

Chaotic Oscillator Based on Fractional Order Memcapacitor

Many literatures have discussed fractional order memristor and memcapacitor-based chaotic oscillators but the entire oscillator model is considered to be of fractional order. My interest is to propose a nonlinear oscillator with considering only the memcapacitor element of fractional order. Hence, I propose a fractional order memcapacitor (FMC)-based novel chaotic oscillator. The complete mathematical model for the proposed oscillator is derived and presented in this paper. The dimensionless state equations are then analyzed by using the equilibrium points and their stability, Eigen values, Kaplan–Yorke dimensions and Lyapunov exponents. To understand the complete dynamical behavior, bifurcation graphs are obtained and presented. Finally, the proposed fractional memcapacitor oscillator is implemented by using the shelf components.

On fractional–order discrete–time systems: Chaos, stabilization and synchronization

In this paper, we propose three systems, namely the fractional Lozi map, the fractional Lorenz map, and the fractional flow map. We study some of the important dynamics of these maps including the bifurcation graphs. In addition, we propose control laws aimed at stabilizing and synchronizing a combination of these three maps. The convergence of the stabilized states and the synchronization errors is established by means of the linearization method. Numerical simulations are also presented to confirm the main findings of the study.

Nonlinear Parametric Vibration and Chaotic Behaviors of an Axially Accelerating Moving Membrane

Nonlinear vibration characteristics of a moving membrane with variable velocity have been examined. The velocity is presumed as harmonic change that takes place over uniform average speed, and the nonlinear vibration equation of the axially moving membrane is inferred according to the D’Alembert principle and the von Kármán nonlinear thin plate theory. The Galerkin method is employed for discretizing the vibration partial differential equations. However, the solutions concerning to differential equations are determined through the 4th order Runge–Kutta technique. The results of mean velocity, velocity variation amplitude, and aspect ratio on nonlinear vibration of moving membranes are emphasized. The phase‐plane diagrams, time histories, bifurcation graphs, and Poincaré maps are obtained; besides that, the stability regions and chaotic regions of membranes are also obtained. This paper gives a theoretical foundation for enhancing the dynamic behavior and stability of moving membranes.

Bifurcations of twisted heteroclinic loop with resonant eigenvalues

In a small tubular neighborhood of the heteroclinic orbits, we establish a local coordinate system by using the foundational solutions of the linear variational equation of the unperturbed system along the heteroclinic orbits. We study the bifurcation problems of twisted heteroclinic loop with resonant eigenvalues. Under the twisted conditions and some transversal conditions, we obtain the existence, the number, the coexistence and non-coexistence problem of 1-heteroclinic loop, 1-homoclinic loop, 1-periodic orbit, double 1-periodic orbit, and 2-heteroclinic loop, 2-homoclinic loop, 2-periodic orbit. Moreover, the relative bifurcation surfaces and the existence regions are given, and the corresponding bifurcation graphs are drawn.

Complicated dynamics and control of a hyperchaotic complex nonlinear autonomous Lü model with complex parameters

In this work, we consider the dynamics and complicated properties of an independent Lu model with complex nonlinear conditions and parameters. We study the influence of complex parameters on the dynamics and behaviors of nonlinear hyperchaotic models. The complex parameters such as generalized Hamiltonian, symmetry, dispersal, equilibria and their stability, Lyapunov exponents, Lyapunov dimension, bifurcation graphs, and hyperchaotic satisfaction are considered. Furthermore, we analyze the stability of the trivial points and limit the conditions under which the complex nonlinear conditions with complex parameters have negative, zero, or positive Lyapunov exponents; we also focus on the chaos, hyperchaos, periodic, and quasiperiodic attractors for an extensive range of parameter values. Moreover, we verify the control of hyperchaotic arrangements of the autonomous Lu model with complex nonlinear conditions and complex parameters. We propose a method to transform the model from its hyperchaotic state to an unstable equilibrium point using a Lyapunov stability hypothesis. Finally, with the use of complex periodic driving, we demonstrate that the model can be transformed from hyperchaotic to quasiperiodic motions, thus resulting in a correct periodic arrangement in which the amplitude and frequency depend on the parameters of the model. Because the obtained arrangement is steady for an extensive range of parameter values, it may be used to control the model by entraining it with the connected periodic compelling term.

Bifurcation from chaos to periodic states in bidirectional interconnected Lorenz systems by the variation of the coupling strengths

Abstract After analyzing the resulting states of two bidirectionally linear interconnected Lorenz systems by the variation of the coupling strengths a bifurcation from chaos to period have been detected. The systems lose asymptotical stability as the coupling strengths takes specific values. These coupling strengths have been adjusted in order to be opposite in sign (i.e., one positive and one negative), and can be considered as a bifurcation parameter on the states of the resulting trajectories. The coupled system has been studied by means of bifurcation analysis, the distance between the resulting trajectories and the standard deviation along the iterated time. Numerical examples of the phase states of the coupled systems along with the bifurcation graphs are presented in order to visualize the resulting chaotic or periodic trajectories as well as the synchronized states.

Bifurcations of Heteroclinic Loop with Twisted Conditions

The bifurcation problems of twisted heteroclinic loop with two hyperbolic critical points are studied for the case [Formula: see text], [Formula: see text], [Formula: see text], where [Formula: see text] and [Formula: see text] are the pair of principal eigenvalues of unperturbed system at the critical point [Formula: see text], [Formula: see text]. Under the transversal conditions, the authors obtained some results of the existence and the number of 1-homoclinic loop, 1-periodic orbit, double 1-periodic orbit, 2-homoclinic loop and 2-periodic orbit. Moreover, the relative bifurcation surfaces and the existence regions are given, and the corresponding bifurcation graphs are drawn.

Carbody elastic vibrations of high-speed vehicles caused by bogie hunting instability

ABSTRACTIn particular locations of the high-speed track, the worn wheel profile matched up with the worn rail profile will lead to an extremely high-conicity wheel–rail contact. Consequently, the bogie hunting instability arises, which further results in the so-called carbody shaking phenomenon. In this paper, the carbody elastic vibrations of a high-speed vehicle in service are firstly introduced. Modal tests are conducted to identity the elastic modes of the carbody. The ride comfort and running safety indices for the tested vehicle are evaluated. The rigid–flexible coupling dynamic model for the high-speed passenger car is then developed by using the FE and MBS coupling approach. The rail profiles in those particular locations are measured and further integrated into the simulation model to reproduce the bogie hunting and carbody elastic vibrations. The effects of wheel and rail wear on the vehicle system response, e.g. wheelset bifurcation graph and carbody vibrations, are studied. Two improvement measures, including the wheel profile modification and rail grinding, are proposed to provide possible solutions. It is found that the wheel–rail contact conicity can be lowered by decreasing wheel flange thickness or grinding rail corner, which is expected to improve the bogie hunting stability under worn rail and worn wheel conditions. The carbody elastic vibrations caused by bogie hunting instability can be further restrained.

Nonlinear Dynamic Response of Cylindrical Roller Bearing–Rotor System with 9 Degree of Freedom Model Having a Combined Localized Defect at Inner–Outer Races of Bearing

ABSTRACTThis article presents a nonlinear dynamic model for a cylindrical roller bearing–rotor system with interaction forces between the inner race, outer race, and roller. Roller–race contacts are modeled predicting nonlinear stiffness (Hertz contact theory) and nonlinear damping for a rotor–cylindrical roller bearing system. Here a shaft–rotor bearing system is modeled with 9 degrees of freedom with one defect on the inner race and one defect on the outer race for a case of combined localized defects. In the mathematical formulation, contacts between rolling elements and inner and outer races are considered as nonlinear springs and nonlinear damping is taken into consideration. Contact force calculations with nonlinearity are solved using the Newton-Raphson method for n unknown nonlinear simultaneous equation. The Newmark-β implicit integration technique coupled with the Newton-Raphson method is used to solve the differential equations. The results are obtained in the form of a time domain plot, frequency domain plot, and phase plot/Poincare map. The validity of the proposed model is compared with experimental results. A bifurcation graph of speed versus peak amplitude predicts the behavior of the system.

Chaos and Synchronization in Complex Fractional-Order Chua’s System

In this paper, chaos in complex-order Chua’s system and its chaotic synchronization for secure communication are studied based on the fractional derivative investigated. Chaos in complex-order Chua’s system is illustrated by presenting its waveform graphs, states diagrams and bifurcation graphs. The dynamic behaviors in the complex-order Chua’s system and time-delayed Chua’s system are compared. In addition, the largest Lyapunov exponents of normal and time-delayed Chua’s systems are calculated and verified by numerical simulations. The experiment indicates that the chaos in such Chua’s system can achieve two channel communications.

Stability and boundary equilibrium bifurcations of modified Chua’s circuit with smooth degree of 3

Chua’s circuit is a well-known nonlinear electronic model, having complicated nonsmooth dynamic behaviors. The stability and boundary equilibrium bifurcations for a modified Chua’s circuit system with the smooth degree of 3 are studied. The parametric areas of stability are specified in detail. It is found that the bifurcation graphs of the supercritical and irregular pitchfork bifurcations are similar to those of the piecewise-smooth continuous (PWSC) systems caused by piecewise smoothness. However, the bifurcation graph of the supercritical Hopf bifurcation is similar to those of smooth systems. Therefore, the boundary equilibrium bifurcations of the non-smooth systems with the smooth degree of 3 should receive more attention due to their special features.

The nonlinear bifurcation and chaos of coupled heave and pitch motions of a truss spar platform

This paper presents the results from a numerical study on the nonlinear dynamic behaviors including bifurcation and chaos of a truss spar platform. In view of the mutual influences between the heave and the pitch modes, the coupled heave and pitch motion equations of the spar platform hull were established in the regular waves. In order to analyze the nonlinear motions of the platform, three-dimensional maximum Lyapunov exponent graphs and the bifurcation graphs were constructed, the Poincare maps and the power spectrums of the platform response were calculated. It was found that the platform motions are sensitive to wave frequency. With changing wave frequency, the platform undergoes complicated nonlinear motions, including 1/2 sub-harmonic motion, quasi-periodic motion and chaotic motion. When the wave frequency approaches the natural frequency of the heave mode of the platform, the platform moves with quasi-periodic motion and chaotic motional ternately. For a certain range of wave frequencies, the platform moves with totally chaotic motion. The range of wave frequencies which leads to chaotic motion of the platform increases with increasing wave height. The three-dimensional maximum Lyapunov exponent graphs and the bifurcation graphs reveal the nonlinear motions of the spar platform under different wave conditions.

FFT Bifurcation Analysis of Routes to Chaos via Quasiperiodic Solutions

The dynamics of a ring of seven unidirectionally coupled nonlinear Duffing oscillators is studied. We show that the FFT analysis presented in form of a bifurcation graph, that is, frequency distribution versus a control parameter, can provide a valuable and helpful complement to the corresponding typical bifurcation diagram and the course of Lyapunov exponents, especially in context of detailed identification of the observed attractors. As an example, bifurcation analysis of routes to chaos via 2-frequency and 3-frequency quasiperiodicity is demonstrated.

Study on the Stability of the High-speed Train with Semi-active Suspension

The stability of the high-speed train(HST) with semi-active suspension is studied comprehensive using the nonlinear bifurcation theory. The high-speed train model, semi-active damper model and the semi-active controller model are set up. Using those models, the co-simulation model system of the semi-active high-speed train is built. The nonlinearities of the wheel/rail contact and the yaw damper are analyzed. The hunting bifurcation graph of the high-speed train with the semi-active is calculated using the brute-force method. The influence of the semi-active suspension on the stability of the HST which equipped the damping type yaw damper and the high conicity thread is less. It reduces the critical speed of the HST about 15 km/h, which equipped the friction type yaw damper and the high conicity thread. However, if the HST equipped the friction type yaw damper and the worn low conicity tread, the semi-active suspension would worsen dramatically the stability. The semi-active suspension should be chosen carefully as designing the HST suspension system, when the low conicity thread and friction type yaw damper have been adopted. The influence of the wear of tread on the stability semi-active suspension HST should be concerned in the process of service.

Effect of nonlinear impact damping with non-integer compliance exponent on gear dynamic characteristics

A nonlinear impact damping model of single-degree-of-freedom spur cylindrical gear with backlash and time-varying stiffness was established. Systematic analyses of the dynamic responses were performed. First, the nonlinear damping coefficient was considered as a constant parameter with two types of compliance exponent, meanwhile, dynamic factors were adopted to depict the dynamic characteristics. Second, the bifurcation graphs were plotted, where the damping coefficient was obtained along with the impact velocity and coefficient of restitution. The results show that light and heavy load conditions have an effect on the responses when the compliance exponent is integer. On the contrary, when the compliance exponent is non-integer, the dynamic responses are slightly affected, namely the system is more stable than the former situation.

Chaos Synchronization of the Modified Sprott E System

In this paper, chaos synchronization of the modified Sprott E system is investigated. Firstly, the chaotic attractors on different phase planes of the system are got by means of numerical simulation. When , the abundance dynamical behavior of the system is presented by the global bifurcation graph. Then, the chaos synchronization of the system by the full state hybrid projective synchronization (FSHPS) method is realized.

Habitability and Multistability in Earth‐like Planets

AbstractIn this paper we explore the potential multistability of the climate for a planet around the habitable zone. We focus on conditions reminiscent to those of the Earth system, but our investigation has more general relevance and aims at presenting a general methodology for dealing with exoplanets. We describe a formalism able to provide a thorough analysis of the non‐equilibrium thermodynamical properties of the climate system and explore, using a flexible climate model, how such properties depend on the energy input of the parent star, on the infrared atmospheric opacity, and on the rotation rate of the planet. We first show that it is possible to reproduce the multi‐stability properties reminiscent of the paleoclimatologically relevant snowball (SB)–warm (W) conditions. We then characterise the thermodynamics of the simulated Wand SB states, clarifying the central role of the hydrological cycle in shaping the irreversibility and the efficiency of the W states, and emphasizing the extreme diversity of the SB states, where dry conditions are realized. Thermodynamics provides the clue for studying the tipping points of the system and leads us to constructing empirical parametrizations allowing for expressing the main thermodynamic properties as functions of the emission temperature of the planet only. Such empirical functions are shown to be rather robust with respect to changing the rotation rate of the planet from the current terrestrial one to half of it. Furthermore, we explore the dynamical range where the length of the day and the length of the year are comparable. We clearly find that there is a critical rotation rate below which the multi‐stability properties are lost, and the ice‐albedo feedback responsible for the presence of SB and W conditions is damped. The bifurcation graph of the system suggests the presence of a phase transition in the planetary system. Such critical rotation rate corresponds roughly to the phase‐lock 2:1 condition. Therefore, if an Earth‐like planet is 1:1 phase‐locked with respect to the parent star, only one climatic state would be compatible with a given set of astronomical and astro‐physical parameters. These results have relevance for the general theory of planetary circulation and for the definition of necessary and sufficient conditions for habitability. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Bifurcation and Strange Behavior for a Nonlinear Viscoelastic Rod System

Longitudinal vibration of a nonlinear viscoelastic rod system with one end fixed and another end subjected to an axial periodical excitation was studied under the consideration of transverse inertia. By using Galerkin method, a combined Parametric and Forcing Excited cubic nonlinear dynamic system is derived for hard stiffness nonlinear material. Furthermore, arc-length technique is used for an accurate integral procedure, and numeric results are given detailedly. The process of the system evolved from stable periodic motion to chaos is illustrated in a period-doubling bifurcation graph in a parameter space, and the Lyapunov exponent spectrum is also given that is perfectly consistent with bifurcation process. The strange attractor obtained from Poincaré Map is present, which has different fractal dimension from Duffing’s one, so it may be a new chaotic attractor.

Stability Analysis of Mode Coupling Vibration Systems with Coulomb Friction

In the article, the stability analysis of mode coupling systems with Coulomb friction is performed. A two-degree-of-freedom (2-DOF) vibration system with Coulomb friction is constructed. Taking the friction coefficient as a bifurcation parameter, a Hopf bifurcation graph is obtained using the Routh criterion. It is illustrated that Coulomb friction can induce self-excited oscillation in the mode coupling system. The analytic expressions of the limit cycle amplitudes of the oscillation system are achieved based on the extended harmonic balance method. Moreover, the effects of some parameters on the size of limit cycles are investigated. All the research method and results presented can provide a valuable reference for the structure optimal design of mechanical systems and vibration reduction.