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This paper introduces an algorithmic approach to the analysis of bifurcation of limit cycles from the centers of nonlinear continuous differential systems via the averaging method. We develop three algorithms to implement the averaging method. The first algorithm allows one to transform the considered differential systems to the normal form of averaging. Here, we restricted the unperturbed term of the normal form of averaging to be identically zero. The second algorithm is used to derive the computational formulae of the averaged functions at any order. The third algorithm is based on the first two algorithms and determines the exact expressions of the averaged functions for the considered differential systems. The proposed approach is implemented in Maple and its effectiveness is shown by several examples. Moreover, we report some incorrect results in published papers on the averaging method.

Bifurcation amplifiers are two-state systems that behave as flip-flops. We propose a two-bit shift register with coupled bidirectional bifurcation amplifiers. We consider a model of the amplifier that is based on a driven nonlinear vibrational system with Duffing nonlinearity. A discussed physical implementation is a nano-mechanical system. The analysis of bifurcation points provides a design framework for driving forces. We use a modulated two-level pulse at the first amplifier and a periodic three-level pulse at the second. We numerically demonstrate that these driving forces allow for the transfer of digital data between each amplifier. Our investigation provides an alternative approach for developing digital shift registers using nonlinear dynamics at the nanoscale.

Study of a food chain model under uncertainty is quite difficult. Because, in an uncertain food chain model, the biological parameters can’t be determined accurately. The aim of this work is to study the stability and local bifurcations (Saddle–node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation) of an imprecise prey–predator system in an uncertain environment. The proposed imprecise model is formulated by considering two more realistic factors: the effect of a fear factor on the growth rate of prey population and non-linear harvesting of predator population. To study the proposed imprecise system mathematically, the dynamical interactions between the imprecise species are presented by the system of governing interval differential equations. And to study the dynamics of the proposed imprecise system theoretically, it is modelled in a precise way by the linear parametric representation of the interval. Then all the theoretical analyses, including Saddle–node bifurcation, Hopf bifurcation and Bogdanov–Takens (BT) bifurcation of the interior equilibrium point of the proposed imprecise model are discussed in parametric form. To verify all the theoretical analyses of the proposed imprecise model, numerical simulations with interval-valued hypothetical data of the imprecise parameters are performed graphically. Finally, the work is concluded with some biological consequences.

In the present work, nonlinear dynamics of a simply-supported fluid-conveying composite pipe subjected to axial tension in sub- and super-critical regimes are investigated. The tension acting on the pipe is assumed to comprise an average component and a harmonically changing disturbance component. More specifically, the nonlinear equation of motion is derived based on the Euler-Bernoulli beam theory considering the von Karman nonlinearity for large deformation assumption, which is then truncated via the Galerkin's approach. In addition, the Kelvin-Voigt model is adopted on the pipe materials with visco-elastic structural damping. Afterwards, parametric studies are performed to evaluate the effects of the axial tension and fiber orientation angle on pre- and post-buckling frequencies, critical fluid velocities and static buckling displacements. The global dynamic behaviors of the pipe are studied through the forms of bifurcation diagrams, phase portraits, Poincare maps, power spectral density (PSD) diagrams and linear instability maps, in both sub- and super-critical regimes. The results show that, based on symmetric layers, pipes with smaller fiber orientation angles have larger critical divergence fluid velocities but smaller static buckling displacements. Some interesting dynamical phenomena, including periodic, multi-periodic, quasi-periodic and chaotic motions, can be observed with different fiber orientation angles. Moreover, the system shows ‘soft spring’ characteristic in super-critical regime, compared with the ‘hard spring’ characteristic in sub-critical case.

The analysis of motion equations of flexible-link manipulators indicates that nonlinear terms and behavioral instability exist in their outcomes. In this system, when the differential equations possess structural time-dependent parameters, there is a higher chance for system instability, especially chaos in comparison to time-invariant ones. It is a point for abnormal behavior that is unpredictable and always accompanied by damages. Therefore, the physical parameters and system inputs that cause chaotic responses should be detected and prevented as much as possible. To this aim, the dynamic equations of flexible link manipulator with revolute-prismatic joints as a case of time-variant dynamic system are obtained by Hamilton’s principle with the assumption of nonlinear strains. The discretized motion equations are solved and the results are presented in the forms of bifurcation diagrams (for variation of torque/force amplitude), Poincare maps, phase-plane portraits, and the largest Lyapunov exponent. Finally, the obtained results are validated with the aid of the experimental setup. The results indicate that although the system response in low torque/force amplitudes is subharmonic (amplitude of 0.01) and quasi-periodic (amplitudes of 0.02-0.03), it becomes quasi-periodic/chaotic with a mild slope with respect to time in high amplitude values. Moreover, since the behavior of modal generalized coordinate is different from the rigid ones, the quasi-periodic and chaotic vibrations do not hurt the joints.

Phase-locked laser arrays have been extensively investigated in terms of their stability and nonlinear dynamics. Specifically, enhancing the phase-locking stability allows laser arrays to generate high-power and steerable coherent optical beams for a plethora of applications, including remote sensing and optical communications. Compared to other coupling architectures, laterally coupled lasers are especially desirable since they allow for denser integration and simpler fabrication process. Here, we present the theoretical effects of varying the spontaneous emission factor $\beta $ , an important parameter for micro- and nanoscale lasers, on the stability conditions of phase-locking for two laterally coupled semiconductor lasers. Through bifurcation analyses, we observe that increasing $\beta $ contributes to the expanding of the in-phase stability region under all scenarios considered, including complex coupling coefficients, varying pump rates, and frequency detuning. Moreover, the effect is more pronounced for $\beta $ approaching 1, thus underlining the significant advantages of implementing nanolasers with intrinsically high $\beta $ in phase-locked laser arrays for high-power generation. We also show that the steady-state phase differences can be widely tuned – up to $\pi $ radians – by asymmetrically pumping high- $\beta $ coupled lasers. This demonstrates the potential of high- $\beta $ nanolasers in building next-generation optical phased arrays requiring wide scanning angles with ultra-high resolution.

The aims of this work are a detailed consideration in a geometrically nonlinear formulation of the stages of the equilibrium behavior of a compressed stiffened plate, taking into account the interaction of the general form of buckling and local forms of wave formation in the plate or in the reinforcing ribs, comparison of the results of the semi-analytical solution of the system of nonlinear equations with the results of the numerical solution on the Patran-Nastran FEM complex of the problem of subcritical and postcritical equilibrium of a compressed stiffened plate. Methods. Geometrically-nonlinear analysis of displacement fields, deformations and stresses, calculation of eigenforms of buckling and construction of bifurcation solutions and solutions for equilibrium curves with limit points depending on the initial imperfections. An original method is proposed for determining critical states and obtaining bilateral estimates of critical loads at limiting points. Results. An algorithm for studying the equilibrium states of a stiffened plate near critical points is described in detail and illustrated by examples, using the first nonlinear (cubic terms) terms of the potential energy expansion, the coordinates of bifurcation points and limit points, as well as the corresponding values of critical loads. The curves of the critical load sensitivity are plotted depending on the value of the initial imperfections of the total deflection. Equilibrium curves with characteristic bifurcation points of local wave formation are constructed using a numerical solution. For the case of action of two initial imperfections, an algorithm is proposed for obtaining two-sided estimates of critical loads at limiting points.

In this paper, we study the global bifurcation of infinity of a class of nonlinear eigenvalue problems for fourth-order ordinary differential equations with nondifferentiable nonlinearity. We prove the existence of two families of unbounded continuance of solutions bifurcating at infinity and corresponding to the usual nodal properties near bifurcation intervals.

Aiming at the problem of pitch error of helical gear pair in engineering practice, the influence of pitch error on vibration, bifurcation, and chaos characteristics of the helical gear pair system is mainly studied. Due to the periodic time-varying nature of pitch error, a method of simulating the pitch error as a sine function is proposed to calculate pitch error. A nonlinear dynamic model of bending-torsion-shaft coupling of the helical gear pair system is established considering the effect of pitch error. The influence of pitch error on the vibration, bifurcation, and chaos characteristics of the system is analyzed by the Runge–Kutta numerical integration method. The research results show that the introduction of pitch error has the most significant impact on the torsional vibration of the system. With the increase in pitch error, the system exhibits rich bifurcation and chaos characteristics in the torsional direction. Moreover, it is also found that the vibration response in the torsional orientation of the system increases or decreases to the same degree when the system is in a periodic motion state, and the pitch error varies by the same extent. Therefore, the impact of pitch error on the dynamic performance of the helical gear pair system should be considered in engineering practice.

In this paper, the nonlinear dynamics of the biological system modeled by the fractional incommensurate order Van der Pol equations are investigated. The stability of the proposed fractional non-autonomous system is analyzed by varying both the fractional order derivative and system parameters. Moreover, very interesting phenomena such as symmetry, multi-stability and coexistence of attractors are discovered in the considered biological system. Numerical simulations are performed by considering the Caputo fractional derivative and results are reported by means of bifurcation diagrams, computation of the largest Lyapunov exponent, phase portraits in 2D and 3D projections.

The levitation force in the high temperature superconducting (HTS) maglev system has obvious nonlinear characteristics. When the excitation frequency and amplitude meet certain conditions, nonlinear vibration phenomena such as period-doubling bifurcation would appear. Nonlinear vibration will bring difficulties to dynamic analysis and even affect the safety and comfort of driving. While an electromagnetic shunt damper (EMSD) can effectively reduce the vibration amplitude. This paper analyzes the influence of EMSD on the nonlinear vibration of HTS maglev systems from the perspective of experiments and simulations. Firstly, free vibration and forced vibrations in the resonance interval are tested. Secondly, based on numerical simulations, the vibration response under various excitation was analyzed. The results show that the EMSD can effectively suppress its nonlinear vibration. This research has reference value for the detailed understanding of the dynamic characteristics of the HTS maglev and further vehicle design.

In this paper, bifurcation and adaptive synchronization of a new fractional-order system are investigated. Firstly, bifurcations for the system with the variation of a system parameter and a derivative order are studied and presented in a three-dimensional space. The dynamics of the system changes from regular to chaos with an increase of derivative orders. It is found that three derivative orders have different contributions to a critical total order for the system to change from regular to chaos, meaning that the nonlinearity in the system downgrades the critical order for transitions to chaos. By the analysis of bifurcations in a two-parameter space, we obtain that the minimal order is 2.62 for the system to remain chaos. Secondly, the entropy is analyzed to measure the level of chaos present in the fractional-order system. Finally, based on the stability theory of fractional-order systems, synchronization of the fractional-order system with fully uncertain parameters is realized by designing appropriate adaptive controllers and estimation laws. Numerical simulations are implemented to demonstrate the effectiveness and flexibility of the synchronization controllers and the estimation laws for the unknown parameters.

A loss of ground directional stability can trigger a high-speed Unmanned Aerial Vehicle (UAV) to veer off the runway. In order to investigate the combined effects of the key structural and operational parameters on the UAV ground directional stability from a global perspective, a fully parameterized mathematical high-speed UAV ground nonlinear dynamic model is developed considering several nonlinear factors. The bifurcation analysis procedure of a UAV ground steering system is introduced, following which the simulation efficiency is greatly improved comparing with the time-domain simulation method. Then the numerical continuation method is employed to investigate the influence of the nose wheel steering angle and the global stability region is obtained. The bifurcation parameter plane is divided into several parts with different stability properties by the saddle nodes and the Hopf bifurcation points. We find that the UAV motion states will never cross the bifurcation curve in the nonlinear system. Also, the dual-parameter bifurcation analyses are presented to give a complete description of the possible steering performance. It is also found that BT bifurcation appears when the UAV initial rectilinear velocity and the tire frictional coefficient vary. In addition, results indicate that the influence of tire frictional coefficient has an opposite trend to the influence of initial rectilinear velocity. Overall, using bifurcation analysis method to identify the parameter regions of a UAV nonlinear ground dynamic system helps to improve the development efficiency and quality during UAV designing phase.

This paper deals with the parametrically excited vibrations and mode transitions of a nonlinear damped triple-well oscillator in detail. The multiple timescale structure of the triple-well oscillator with resonant frequency is revealed. We present the correct predictions of the region of parameter space where the periodic orbits alternate between epochs of fast and slow motions in such excited vibrations. Typically, one can treats the cosinoidal part of the perturbation as a singular variable of the whole system, which only remains autonomous form. We study the critical manifold and bifurcation structure of this system. Since the attracting manifold terminated at a fold leading to the change of local stability, the corresponding singular orbit would jump to another stable branch. Characteristic features of this method are folded singularities and structure of the critical manifold. Parametrically excited vibrations with resonant frequency could reflect multiple timescale structure. A singular periodic orbit can be constructed, which consists of distinct segments. Starting along the critical manifold with small damped oscillations, then there is a fast transition that flows to another attracting branch for the folded singularity. As the fifth stiffness value varies, mixed-mode oscillations with more jumps are produced. Moreover, the jump-doubling behaviors can be presented under the flip of excitation frequency ratio. As we show in the paper, damping and external excitations with resonant frequency ratio in nonlinear vibration system could construct singular periodic orbits. It is possible for the trajectory to jump away from the folded singularity with finite speed. Then, it performs small damped oscillations about one attracting branch of stable foci. By taking advantage of time scale separation, we demonstrate the origin of such bursting behaviors.

The special issue entitled “Mathematical modelling, nonlinear dynamics, bifurcation, synchronization and control of mechanisms driven by power supply,” published by The European Physical Journal: Special Topics (EPJST), aims to bring into discussion works involving nonlinear and dynamic and chaotic behaviors, which are frequently found in mechanical and electromechanical systems, nonlinear circuits, and biologically inspired systems, stimulating the integration of findings on the nonlinear dynamics that can be observed in various systems in the physical sciences and engineering. This special issue presents 27 papers concerning the dynamics and control of flexible structure systems, nonideal systems, energy harvesting problems, and vibration absorbers. In addition, the papers address problems involving chaos control of nonlinear oscillations, phase-locked loops, synchronization of nonlinear oscillators, dynamics of bio-inspired and biostructural systems, nonlinear methods, and nonlinear circuits.

We numerically study solitary waves in the coupled nonlinear Schrodinger equations. We detect pitchfork bifurcations of the fundamental solitary wave and compute eigenvalues and eigenfunctions of the corresponding eigenvalue problems to determine the spectral stability of solitary waves born at the pitchfork bifurcations. Our numerical results demonstrate the theoretical ones which the authors obtained recently. We also compute generalized eigenfunctions associated with the zero eigenvalue for the bifurcated solitary wave exhibiting a saddle-node bifurcation, and show that it does not change its stability type at the saddle-node bifurcation point.

Numerical continuation and bifurcation methods can be used to explore the set of steady and time–periodic solutions of parameter dependent nonlinear ODEs or PDEs. For PDEs, a basic idea is to first convert the PDE into a system of algebraic equations or ODEs via a spatial discretization. However, the large class of possible PDE bifurcation problems makes developing a general and user–friendly software a challenge, and the often needed large number of degrees of freedom, and the typically large set of solutions, often require adapted methods. Here we review some of these methods, and illustrate the approach by application of the package to some advanced pattern formation problems, including the interaction of Hopf and Turing modes, patterns on disks, and an experimental setting of dead core pattern formation.

The aim of this work is to analytically investigate the nonlinear dynamic behaviors of a proposed reduced Lorenz system based on q-deformations. The effects of varying the new q-deformation parameter on the dynamical behaviors of the system along with the induced bifurcations of fixed points are explored. In particular, the codimension-one bifurcation analysis is carried out at interior fixed point of the q-deformed system. Explicit conditions for the existence of pitchfork and Neimark–Sacker bifurcations are obtained. Numerical simulations are performed to confirm stability and bifurcation analysis in addition to investigate the effects of variations in system parameters. The changes in system dynamics are explored via the bifurcation diagrams, phase portraits and time series diagrams. Moreover, the quantification of system complex behaviors is depicted through the maximal Lyapunov exponent plots. A cascaded version of the model is proposed to boost its complex dynamics. Then, a chaos-based image encryption scheme, relying on a proposed key-distribution algorithm, is introduced as an application. Finally, several aspects of security analysis are examined for the encryption system to prove its efficiency and reliability against possible attacks.

The increase in renewable energy sources connected to the grid has increased flexibility requirements in the operation of thermal power plants. Because of severe nonlinearity and various disturbances, the dynamic behavior of the boiler-turbine unit will change significantly at different operating conditions. Further, modeling uncertainties cofound definitive knowledge about the changes, exacerbating efforts to control. In this paper, a general disturbed model of the boiler-turbine unit for control design is determined through the nonlinear dynamic analysis on the concepts of bifurcation and limit cycles behavior rather than given by an artificial chosen. Then, a dual-mode fuzzy predictive control strategy is proposed on discrete-time Takagi-Sugeno fuzzy model. Online optimal problem of the approach is solved to accomplish the fast-tracking task; to achieve the anti-bifurcation control objective, a novel minimal robust positively invariant set corresponding with a local controller is offline obtained in order to change the undesired nonlinear behavior into stable limit cycles. With the input-to-state stability theory, the proposed control strategy is theoretically proved and given in terms of linear matrix inequalities. Simulation carried out on the boiler-turbine unit demonstrates that the proposed dual-mode control strategy achieves the disturbance suppression ability and the improvement of the nonlinear dynamic behavior.

This study investigates the nonlinear dynamic response behavior of a rotating ring that forms an essential element of MEMS (Micro Electro Mechanical Systems) ring-based vibratory gyroscopes that utilize oscillatory nonlinear electrostatic forces. For this purpose, the dynamic behavior due to nonlinear system characteristics and nonlinear external forces was studied in detail. The partial differential equations that represent the ring dynamics are reduced to coupled nonlinear ordinary differential equations by suitable addition of nonlinear mode functions and application of Galerkin’s procedure. Understanding the effects of nonlinear actuator dynamics is essential for characterizing the dynamic behavior of such devices. For this purpose, a suitable theoretical model to generate a nonlinear electrostatic force acting on the MEMS ring structure is formulated. Nonlinear dynamic responses in the driving and sensing directions are examined via time response, phase diagram, and Poincare’s map when the input angular motion and nonlinear electrostatic force are considered simultaneously. The analysis is envisaged to aid ongoing research associated with the fabrication of this type of device and provide design improvements in MEMS ring-based gyroscopes.

We analyze the stability of elastic systems composed of the shells of revolution with variable curvature and complex structures in the field of conservative axisymmetric loads of different nature. Within the framework of classical and refined theories of shells, we determine the limit and bifurcation critical values of the acting loads based on the geometrically nonlinear statement of the problem and a criterion of dynamic stability. To solve the corresponding nonlinear and eigenvalue problems, we propose to use a numerical-analytic approach based on their rational reduction to one-dimensional linear boundaryvalue problems in the meridional coordinate and their numerical solution by the discrete-orthogonalization method. We present test examples that confirm the applicability of the proposed procedure to the analyzed class of problems. The limit and bifurcation values of the critical loads in the shell system are analyzed depending on its geometric parameters.

The work is aimed at the construction of an algorithm for studying the equilibrium states of a reinforced plate near critical points, using the first (cubic terms) nonlinear terms of the potential energy expansion. Using geometrically nonlinear analysis of displacement, deformation and stress fields, the Eigenforms of buckling were calculated and bifurcation solutions and solutions for equilibrium curves with limit points were constructed depending on the initial imperfections.

Conventional remote operation of an excavator has low work efficiency comparing with on-site operations. This is because it is difficult for an operator to recognize the excavator status and surrounding environments. Moreover, there are restrictions such as a limitation on the amount of information that can be transmitted, delays in communication, and harsh environments that cause sensor failure. Therefore, we have developed a semi-autonomous control system that consists of autonomy and human manipulation. This paper proposes (1) a bilateral control system based on variable admittance and (2) an autonomous control using nonlinear dynamics that has attractors with stagnation and trajectory-bifurcation. (1) In the proposed method, the admittance parameters are changed based on the difference between the leader and follower position. We implement the proposed method into a prototype of excavator and verify the operation with two types of external environmental forces: free movement and contact with a rock. (2) The attractor’s stagnation is designed as a sink of vector field converging to an arbitrary point on the trajectory. A stagnation is placed at a trajectory’s bifurcation point, and the operator selects the next operation by adding a force to the leader system, which is measured through a force sensor. We design two types of bifurcating attractors and conduct an experimental verification of the proposed semi-autonomous control system.

Acoustically excited microbubbles (MBs) have shown to exhibit rich dynamics, enabling them to be employed in various applications ranging from chemistry to medicine. Exploiting the full potential of MBs for applications requires a good understanding of their complex dynamics. Improved understanding of MB oscillations can lead to further enhancement in optimizing their efficacy in many applications and also invent new ones. Oscillating MBs have been shown to generate secondary pressure waves that modify the dynamics of the MBs in their proximity. A modified Keller-Miksis equation is used to account for inter-bubble interactions. The oscillatory dynamics of each MB within clusters was computed by numerically solving the resulting system of coupled nonlinear second order differential equations in potential fluid flow. Frequency response analysis and bifurcation diagrams were employed to track the dynamics of interacting MBs. We start with investigating the effect of inter-bubble interactions for cases of three and four MBs over a wide range of acoustic and geometric parameters. Emergent collective behavior was observed which are dominated by the dynamics of the largest MB within the cluster. The emergent dynamics of smaller MBs within clusters can be characterized by constructive and destructive inter-bubble interactions. In constructive interactions, the radial oscillations of smaller MBs matched those of the largest MB and their oscillations are amplified. In destructive interactions, the oscillations of smaller bubbles are suppressed so that their oscillations match those of the largest MB. Furthermore, a special case of constructive interactions is presented where dominant MB (largest) can force smaller MBs into period doubling and subharmonic oscillations. The collective behavior is further investigated in large MB cluster and it is shown that largest MBs, even in small numbers can force smaller ones into period doubling and subharmonic oscillations.

In light of the previous recent studies by Jaume Llibre et al. that dealt with the finite cycles of generalized differential Kukles polynomial systems using the first- and second-order mean theorem such as (Nonlinear Anal., 74, 1261–1271, 2011) and (J. Dyn. Control Syst., vol. 21, 189–192, 2015), in this work, we provide upper bounds for the maximum number of limit cycles bifurcating from the periodic orbits of Hamiltonian system using the averaging theory of first order.

In this paper, a modified incremental harmonic balance (IHB) method combined with Tikhonov regularization has been proposed to achieve the semi-analytical solution for the periodic nonlinear system. To the best of our knowledge, the convergence of the traditional IHB method is bound up with the iterative initial values of harmonic coefficients, especially near the bifurcation point. To this end, the Tikhonov regularization is introduced into the linear incremental equation to tackle the ill-posed situation in the iteration. To this end, the convergence performance of the traditional IHB method has been improved significantly. Moreover, convergence proof of the proposed method also has been given in this paper. Finally, a van der Pol–Duffing oscillator with external excitation and a cubic nonlinear airfoil system with the external store are adopted as numerical examples to illustrate the efficiency and the performance of the presented modified IHB method. The numerical examples show that the results achieved by the proposed method are in excellent agreement with the Runge–Kutta method, and the accuracy is not significantly reduced compared with the traditional IHB method. Especially, the modified IHB method also can converge to the exact solution from the initial values that the traditional IHB method cannot converge in both examples.

Hybrid vibro-impact (HVI) oscillations is a strongly nonlinear dynamical regime that involves both linear oscillations and collisions under periodic, impulsive, or stochastic excitation. This regime arises in various engineering systems, such as mechanical components under tight rigid constraints, seismic-induced sloshing in partially-filled liquid storage tanks, and more. The adaptive nonlinearity of the HVI oscillator is used by the HVI-nonlinear energy sink as an effective vibration mitigation solution for broad energy and frequency range. Due to the extreme nonlinearity of this regime, traditional analytical methods are inapplicable for the description of its transient dynamics. In the current work, we model the HVI oscillator by a forced particle in a truncated quadratic potential well with infinite depth. The slow flow dynamics of the system in the vicinity of primary resonance is described by canonical transformation to action-angle (AA) variables and the corresponding reduced resonance manifold (RM). Two types of bifurcation are examined. The former is associated with transition between linear oscillations and the HVI-regime and vice versa, and the latter with reaching a chosen maximal transient energy level. The transition boundaries on the forcing parameters plane associated with both bifurcation types are obtained analytically. The maximal transient energy level obtained for any given set of forcing parameters is described analytically as well. The energy jumps associated with the bifurcation of type I and crossing the corresponding transition boundary are obtained. Two underlying dynamical mechanisms that govern the occurrence of bifurcations are identified. They correspond to two distinct scenarios: in the first scenario, the energy of the slow flow gradually reaches the threshold energy level and is thus referred to as the ”maximum” mechanism. The second, potentially more dangerous scenario, involves abrupt transitions of the system’s energy response from a relatively small value to the threshold energy level. This pattern is related to the passage of the slow-flow phase trajectory through the saddle point of the RM, and thus is referred to as the “saddle” mechanism. Both mechanisms are universal for systems that undergo escape from a potential well. All theoretical results are in complete agreement with full-scale numerical simulations.

In this article, we consider the global behavior of components of positive nonconstant radial solutions for the Neumann problem with indefinite weight −Δu=λh(x)f(u),inB,∂νu=0,on∂B, where λ>0 is a p...

A new 3D chaotic system with only one nonlinear term is constructed in this paper, which has conservative and dissipative features respectively by choosing different parameters. The complex dynamic behaviors of the system are analyzed theoretically and numerically by using nonlinear analytical tools, such as Lyapunov exponents, phase portraits, and bifurcation diagrams. In a certain range of parameters, the new system is a non-Hamiltonian conservative chaotic system, which can produce infinitely many coexisting conservative flows. With the variation of parameters, the divergence of the system is less than zero, and the system can also generate hidden and self-excited attractors with multi-scroll. To verify the feasibility of the new system, it has been implemented by field-programmable gate array; phase diagrams obtained in the experiment are consistent with the simulation of the data.

The coupling system of brake disc and brake sub-system responds in a variety of nonlinear dynamics manner under low-speed braking condition. In this article we aim to obtain the bifurcation and chaotic behavior of vehicle brake system under this condition. We established a pure brake condition 4-DOF differential equations of torsional vibration in this paper based on Crowther's model, which predicts the existence of chaotic stick-slip vibration. Based on this model, by means of numerical methods, we have calculated firstly the low-speed bifurcation characteristics of the brake disc and brake sub-system angular velocity difference relative to vehicle speed, and secondly the effects of system parameters on the bifurcation. Metal type, semi-metallic type, non-metallic type friction material. The results showed that: brake disc and brake sub-system angular velocity difference exhibits chaotic vibration in the low-speed zone. With the increase of inertia ratio of brake sub-system and brake disc, system vibrations changed from periodic doubling bifurcation to chaos, then inversed back to doubling period and single period stick-slip motions. With the increase of ratio of the kinetic friction coefficient and the static friction coefficient, system vibrations changed from chaotic stick-slip motions to single period stick-slip motions. With the increase of brake pressure, the system stick-slip vibrations changed from doubling bifurcation to chaos, which worsened the vehicle chatter. The results can provide theoretical guidance for parameter design, which can optimize the chaos interval, but we need further experiments to explore and study.

The influence of the electromagnetic parameters on the torsional dynamics of the electric vehicle powertrain is studied by considering the electromechanical coupling effect. By adding the electromagnetic torque on the drive side, the powertrain is simplified as nonlinear drive-shaft model. The number, stability, and bifurcation conditions of the equilibrium points of the nonlinear drive-shaft model are deduced. Based on the averaged equations and the amplitude-frequency response equation, the stability and bifurcation conditions, such as fold bifurcation and Hopf bifurcation, of the resonance curve are discussed. The influence of electromagnetic parameters on the torsional dynamics is studied by simulation. It is shown that with the change of the parameters, the number as well as the stability of the equilibrium points may be changed which is affected by fold bifurcation. It is also shown that the resonance curve may lose its stability when fold bifurcation happens. By limiting the parameters in the region without fold bifurcation, the unstable dynamics of the resonance curve can be controlled.

The existence of periodic windows in the parameter space is a common feature of nonlinear systems capable to produce chaotic behavior. Detection of positions of periodic windows is important both from the theoretical and practical points of view. In this work, a systematic method to detect periodic windows in bifurcation diagrams is proposed. The search method is a combination of the trajectory monitoring approach to find unstable periodic orbits and the continuation method to calculate positions of periodic windows. The method is applied to the Chua’s circuit with a smooth nonlinearity. It is shown that the proposed method outperforms standard methods to find periodic windows in bifurcation diagrams.

Studies on the nonlinear dynamics of a modified Van der Pol oscillator based electronic circuit has been reported. It has been found that inclusion of an active RC section (or filter) in the Van der Pol oscillator is able to generate rich nonlinear behavior. Effect of a low pass filter as well as an all pass filter has been reported. The behavior of the proposed system has been explained analytically and numerically with respect to its parameters by exploiting standard nonlinear analysis tools such as bifurcation diagrams, Lyapunov exponent, phase portraits, Poincare sections, and basins of attraction. Occurrence of some fascinating features like antimonotonicity, hysteresis and coexisting attractors have been found in the modified system. The analytical and numerical studies has been verified experimentally with a prototype experiment.

In gear transmissions, vibration causes noise and malfunction. In actual applications, misalignments contribute to intensifying the destructive effect of vibrations. In this paper, the nonlinear dynamics of a spiral bevel gear pair, with small helix angle, considering different misalignments, are deeply investigated. Axial misalignment, radial misalignment, and the combination of these two types are considered in this study. The governing equation is numerically solved through an implicit Runge–Kutta scheme. Since the main goal of this study is the analysis of the dynamic scenario, the mesh stiffness of the gear pair is obtained from the literature. The dynamical system is nonlinear and time-varying, it is analyzed through time responses, phase portraits, Poincaré maps, and bifurcation diagrams. Results show that, among the considered three cases with different types of misalignments, the spiral bevel gear with axial misalignment is the worst destructive case, aperiodic, subharmonic, and multiperiod responses are observable for this case. It is interesting that the chaotic responses for the case, having both types of misalignments, are less likely for the case with axial misalignment, only.

This paper primarily investigates five novel compound bursting patterns in the parametrically amplified Mathieu-Duffing nonlinear energy harvesters (PAMDNEH). The five new compound bursting patterns, i.e. compound “subHopf/LPC-delayed subHopf/LPC-subHopf/LPC” bursting, compound “subHopf/LPC-subHopf/LPC” bursting, compound “subHopf/LPC-subHopf/LPC” bursting via “subHopf/subHopf” hysteresis loop, compound “subHopf/supHopf-supHopf/LPC” bursting via “subHopf/subHopf” hysteresis loop and “subHopf/LPC” bursting via “subHopf/subHopf” hysteresis loop, are explored. In addition, two normal bursting patterns, i.e. “subHopf/subHopf” bursting and “fold/fold” bursting, are also investigated. By using the two-bifurcation analysis method, the compound “subHopf/LPC-delayed subHopf/LPC-subHopf/LPC” bursting and its dynamical evolutions is investigated. The correctness of the results is verified by the numerical simulation. We find that the parameter plane can be divided into six different areas according to the types and numbers of the Hopf bifurcation points, and different bifurcation structures and different patterns of the clusters will be created, which lead to different bursting oscillations, including compound bursting types and normal bursting patterns. Because bursting oscillations is created when the variables switch between/among different attractors, the bursting oscillation amplitude is big enough to surmount the potential obstacles of the energy harvesters. Therefore, bursting oscillations may have a very important application in energy harvesting.

In this article, periodic solutions from a fine focus $ U = 0 $, are accomplished for several classes. Some classes have polynomial coefficients, while the remaining classes $ C_{14, 7} $, $ C_{16, 8} $ and $ C_{5, 5}, $ $ C_{6, 6} $ have non-homogeneous and homogenous trigonometric coefficients accordingly. By adopting a systematic procedure of bifurcation that occurs under perturbation of the coefficients, we have succeeded to find the highest known multiplicity $ 10 $ as an upper bound for the class $ C_{9, 4} $, $ C_{11, 3} $ with algebraic and $ C_{5, 5}, $ $ C_{6, 6} $ with trigonometric coefficients. Polynomials of different degrees with various coefficients have been discussed using symbolic computation in Maple 18. All of the results are executed and validated by using past and present theory, and they were found to be novel and authentic in their respective domains.

This work brings an overview of the nonlinear dynamics of four-segment tilting pad journal bearings (TPJB) in load-between-pads configuration supported on flexible pivots. We focus primarily on analysing the sub-synchronous motion of statically unloaded pads, also called pad fluttering. In this particular case, the motion deviates between chaotic and periodic due to 1:4, 1:5, 1:6 and 1:7 internal resonances. The response is analysed with bifurcation diagrams and characterised with estimates of the largest Lyapunov exponents. Unlike previous works, we show that pad fluttering disappears at relative eccentricities lower than 0.34. This behaviour is enabled due to a fixed point trajectory that is not predicted by analysing static equilibria. The observation is crucial for high-speed rotating machinery supported on TPJBs, including pinions in multi-stage compressors and microturbines. The analysis is performed using a verified computational model which includes fluid-structure interaction and potential Hertzian contacts between individual parts of the TPJB. This model exploits local and auxiliary coordinate systems for a straightforward description of acting forces. Furthermore, we provide a detailed step-by-step discussion regarding the model formulation employing the finite difference method to solve hydrodynamic lubrication.

Made available in DSpace on 2022-05-01T07:58:44Z (GMT). No. of bitstreams: 0 Previous issue date: 2021-01-01 In this paper, the dynamical behavior of a four-dimensional magnetohydrodynamic model, consisting of a generalized Lorenz model, is investigated. A nonlinear dynamical analysis is performed using Lyapunov exponents and bifurcation diagrams, focusing on the chaotic and hyperchaotic behaviors associated with the bifurcation parameter (k1) that couples the equations of fluid displacement to the induced magnetic field. The State-dependent Riccati Equation (SDRE) and the Optimal Linear Feedback Control (OLFC) techniques are considered to design the state feedback control system that stabilizes the system to a previously defined orbit. The performance of the control systems are compared showing that the OLFC presents better results. Federal Technological University of Paraná School of Engineering São Paulo State University Faculty of Mechanical Engineering Lublin University of Technology Institute of Science and Technology São Paulo State University School of Engineering São Paulo State University Institute of Science and Technology São Paulo State University

The application of multi-degree of freedom nonlinear energy sinks (MDOF-NES) to control the vortex-induced vibrations of a sprung cylinder passively is investigated in this paper. The flow-induced loads on the cylinder are modeled by the wake Van der Pol oscillator. The MDOF-NES consists of three oscillators connected in series, where the first mass is linked to the cylinder structure by a linear spring. The second mass is connected with the first and third masses by pure cubic springs and linear viscous dampers. The cylinder-MDOF-NES assemble model is analyzed in two scenarios involving lower and strong linear stiffness couplings. Numerical investigations of the bifurcation and limit cycle oscillation (LCO) behavior associated with parametric changes are performed. An appropriate design of the MDOF-NES parameters allows mitigating the sub-critical behavior and control LCO amplitudes. The MDOF-NES approach based on the lower linear stiffness coupling has shown to be more effective in controlling the LCO amplitudes. Moreover, it is verified that the MDOF-NES masses have a significant effect on the LCO stability, amplitudes, and synchronization frequency range. Comparison between the MDOF-NES and a classical Type I NES was also performed to ensure the superiority of the MDOF-NES in suppressing the vortex-induced oscillations.

In this paper, we build a simple chaotic circuit by introducing a new magnetic flux-controlled memristor model. The basic dynamics, such as the type of equilibrium point, phase diagram, bifurcation diagram, and Lyapunov exponent spectrum, are studied theoretically and numerically. It is found that this memristor circuit has complex nonlinear behavior, including coexistence bifurcation, transient dynamics, and anti-monotonicity. The salient feature of the memristor chaotic circuit is that the initial conditions of both the memristor and system can realize the modulation of the position and amplitude of the attractor. To verify the chaotic characteristics of the system, an analog circuit is designed and the experimental results provide the consistency with the numerical simulation.

The Hopf bifurcation control of hyperchaotic system has become the focus of bifurcation researches. By setting up the controller for the hyperchaotic system, the critical point of Hopf bifurcation is delayed or advanced,and the stability of the system is changed. In this paper, based on the first Lyapunov coefficient, the Hopf bifurcation characteristics of a chaotic system are studied. The stability of the system is judged by the value of the first Lyapunov coefficient. The dynamic state feedback control method is used to set the nonlinear controller for the system. By adjusting the control parameters, the Hopf bifurcation of the system can be changed, and the unstable range becomes stable. Finally, by the numerical simulations observation, it is shown that the theoretical analysis is correct.