Bifurcation Modes Research Articles

Edge mode bifurcations of two-dimensional topological lasers.

Topological lasers are of growing interest as a way to achieve disorder-robust single-mode lasing using arrays of coupled resonators. We study lasing in a two-dimensional coupled resonator lattice exhibiting transitions between trivial and topological phases, which allows us to systematically characterize the lasing modes throughout a topological phase. We show that, unlike conventional topological robustness that requires a sufficiently large bulk band gap, bifurcations in topological edge mode lasing can occur even when the band gap is maximized. We show that linear mode bifurcations from single-mode to multi-mode lasing can occur deep within the topological phase, sensitive to both the pump shape and lattice geometry. We suggest ways to suppress these bifurcations and preserve single-edge mode lasing.

Bifurcation Analysis for a Simple Dual-Rotor System with Nonlinear Intershaft Bearing Based on the Singularity Method

This paper aims to classify bifurcation modes for two interrelated primary resonances of a simple dual-rotor system under double frequency excitations. The four degree-of-freedom (4DOF) dynamic equations of the system considering the nonlinearity of the intershaft bearing can be obtained by using the assumed mode method (AMM) and Lagrange’s equation. A simplified method for dynamic equations is developed due to the symmetry of rotors, based on which the amplitude frequency equations for two interrelated primary resonances are obtained by using the multiple scales method. Furthermore, the validity of the simplified method for dynamic equations and the amplitude frequency equations solved by the multiple scales method are confirmed by numerical verification. Afterwards, the bifurcation analysis for two interrelated primary resonances is carried out according to the two-state-variable singularity method. There exist a total of three different types of bifurcation modes because of double frequency excitations of the dual-rotor system and the nonlinearity of the intershaft bearing. The second primary resonance is more prone to have nonlinear dynamic characteristics than the first primary resonance. This discovery indicates that two interrelated primary resonances of the dual-rotor system may have different bifurcation modes under the same dynamic parameters.

Multistability and coexisting transient chaos in a simple memcapacitive system**Project supported by the National Natural Science Foundation of China (Grant No. 51377124) and the Science Fund for New Star of Youth Science and Technology of Shaanxi Province, China (Grant No. 2016KJXX-40).

The self-excited attractors and hidden attractors in a memcapacitive system which has three elements are studied in this paper. The critical parameter of stable and unstable states is calculated by identifying the eigenvalues of Jacobian matrix. Besides, complex dynamical behaviors are investigated in the system, such as coexisting attractors, hidden attractors, coexisting bifurcation modes, intermittent chaos, and multistability. From the theoretical analyses and numerical simulations, it is found that there are four different kinds of transient transition behaviors in the memcapacitive system. Finally, field programmable gate array (FPGA) is used to implement the proposed chaotic system.

Bifurcation of nonlinear normal modes of a cantilever beam under harmonic excitation

Bifurcation analysis of the nonlinear vibration of an inextensible cantilever beam is analyzed by using the nonlinear normal mode concept. Two flexural modes of the cantilever beam, one in each transverse plane is considered. Two degrees-of-freedom nonlinear model for the vibration in the transverse direction is obtained by the discretization of the governing equation using Galerkins method based on the eigenmodes in each direction. The method of multiple scales is used to derive two first-order nonlinear ordinary differential equations governing the modulation of the amplitude and the phase of the dominant mode for the case of 1:1 internal resonance. The bifurcation diagrams are computed considering the frequency of excitation and the magnitude of the excitation as the control parameters. The stability of the fixed point is determined by examining the eigenvalues of the Jacobian matrix. The results show that a saddle-node-type bifurcation of the solution can occur under certain parameter conditions.

Periodic Orbits and Bifurcations of the Vibrational Modes of the Ozone Molecule at High Energies

The study of quantum states near the dissociation threshold is necessary both to understand the formation of molecules and to explore accurately chemical reactions. Quantum calculations of highly excited states can be complemented by methods of nonlinear classical mechanics that help revealing the stable modes of molecular vibrations and their transformations (bifurcations) during excitation of a molecule. The classical approach is particularly efficient at high energies where due to their high density, strong state mixing could occur, and simple assignment in terms of the standard normal modes becomes impossible. In this work we present first results for periodic orbits of the main isotopologue of the ozone molecule (16O3) using recent accurate ab initio potential energy surface. Along with the principle families of the periodic orbits corresponding to the symmetric, antisymmetric, and bending types of molecular vibrations, we have located bifurcations corresponding to the transition to local modes as well as resonance orbits that are responsible for new types of vibrational motions in the high energy range. The classical trajectories of nuclei and their correspondence to the wave functions of the quantum states are also discussed.

Bifurcations and dynamics of a plant disease system under non-smooth control strategy

Mathematical models and analyses can assist in designing the control strategies to prevent the spread of infectious disease. The present paper investigates the bifurcations and dynamics of a plant disease system under non-smooth control strategy. The generalized Lyapunov approach is employed to perform the analysis of the plant disease model with non-smooth control. It is found that the controlled disease system can have three types of equilibria. The globally asymptotically attractor for each of three types of equilibria is determined by constructing Lyapunov functions and using Green’s Theorem. It is shown that the disease system can exhibit rich dynamic behaviors including globally stable equilibrium, stable pseudo-equilibrium and sliding mode bifurcations. The solution of the disease system can converge to the disease-free equilibrium, endemic equilibrium or sliding equilibrium on discontinuous surfaces. Biological implications of the obtained results are discussed for implementing the control strategies to the infectious plant diseases.

Modal analysis and imperfection sensitivity of the post-buckling behaviour of cylindrical steel panels under in-plane bending

This paper presents and discusses elastic geometrically non-linear Generalised Beam Theory (GBT) results on simply supported cylindrical steel panels under in-plane bending stresses, extending the knowledge on curved panels under uniform compression recently reported by the authors. Due to its inherent modal nature, GBT enables the acquisition of in-depth knowledge on the behaviour of these complex structural elements, which cannot be obtained with standard shell finite element analysis. In particular, a modal analysis investigation is conducted to assess the imperfection sensitivity of curved panels characterised by distinct curvatures and by considering (i) four distinct loading conditions (including the pure bending case), (ii) two distinct critical-mode initial geometrical imperfection shapes (first two bifurcation modes), and (iii) three distinct amplitudes – whenever relevant, three “positive” and “negative” amplitudes are considered. The work begins by the GBT buckling analysis of the selected panels followed by an in-depth investigation on the corresponding post-buckling behaviour. These results provide the evolution, along the equilibrium paths, of relevant modal displacement profiles, modal participation diagrams and deformed configurations. For comparison and validation purposes, ABAQUS shell finite element results are also reported.

Relation between Dynamic Characteristics and Parameters of a Two-degree-of-freedom Vibro-impact System with Multi-elastic Constraints

A mechanical model of mechanical vibration system with multi-elastic constraints was founded. Based on the multiple parameter and multiple objective cooperative simulation calculation, the three-dimensional parameter space was constructed by the key parameters such as constrained stiffness, excitating frequency and clearance threshold. The large data onto two-dimensional parameter bifurcation diagrams reflecting the correlation between dynamic properties and parameters of the model are obtained. The mode types, existing regions and bifurcation characteristics of periodic impact vibration of the system on the parameter plane of frequency and clearance are described with the change of constrained stiffness. Similarly, the three-dimensional parameter spaced consisting of mass ratio, or damping ratio and excitating frequency and clearance threshold are analyzed. The results show that the high mass ratio excites rich and wonderful periodic vibration, and different damping ratio determines, the bifurcation characteristics and the number of chattering impacts groups in the small frequency domain. The mode types and transition rules of the fundamental periodic impact vibration in small frequency domain and subharmonic impact vibration in high frequency domain induced by different parameters of the system are revealed.

Bifurcation Modes of Periodic Solution in a Duffing System Under Constant Force as Well as Harmonic Excitation

This paper focuses on the classification of the bifurcation modes of a Duffing system under the combined excitations of constant force and harmonic excitation. The Harmonic Balance method combined with the arc-length continuation is used to obtain the periodic solutions of the system, and the Floquet theory is employed to analyze the stability of the corresponding solutions. Accordingly, the frequency-response curves affected respectively by the constant force and the magnitude of the harmonic excitation are analyzed to show the basic dynamical properties of the system. Afterwards, the bifurcation investigations are carried out with the aid of the two-state variable singularity method. It is derived that there are a total of six different types of bifurcation modes due to the effects of the constant force and the magnitude of the harmonic excitation. At last, the effects of the nonlinearity parameter and the damping ratio on the bifurcation modes of the system are also discussed. The results obtained in this paper extend the findings in reference that the system can have markedly three types of frequency-response curves: with only one solution, or with maximum three or five solutions for a certain excitation frequency, and contribute to a better understanding of the significant influence of the constant force.

Bifurcations of autoresonant modes in oscillating systems with combined excitation

AbstractA mathematical model describing the capture of nonlinear systems into the autoresonance by a combined parametric and external periodic slowly varying perturbation is considered. The autoresonance phenomenon is associated with solutions having an unboundedly growing amplitude and a limited phase mismatch. The paper investigates the behavior of such solutions when the parameters of the excitation take bifurcation values. In particular, the stability of different autoresonant modes is analyzed and the asymptotic approximations of autoresonant solutions on asymptotically long time intervals are proposed by a modified averaging method with using the constructed Lyapunov functions.

Периодические орбиты и бифуркации колебательных мод молекулы озона при высоких энергиях

Периодические орбиты и бифуркации колебательных мод молекулы озона при высоких энергиях

Wrinkling to crinkling transitions and curvature localization in a magnetoelastic film bonded to a non-magnetic substrate

This work studies experimentally and numerically the post-bifurcation response of a magnetorheological elastomer (MRE) film bonded to a soft non-magnetic (passive) substrate. The film-substrate system is subjected to a combination of an axial mechanical pre-compression and a transverse magnetic field. The non-trivial interaction of the two fields leads to a decrease of the critical magnetic field with applied pre-compression, while the observed wrinkling patterns evolve into crinkles, a bifurcation mode that is defined by the accompanied curvature localization and strong shearing of the side faces of the wrinkled geometry. Using a magneto-elastic variational formulation in a two-dimensional finite element numerical setting, we find that the crinkling is an intrinsic feature of magnetoelasticity and its presence is directly associated with the repulsive magnetic forces of the neighboring wrinkled-crinkled faces. As a result, the presence of the magnetic field prohibits the formation of creases and folds. In an effort to obtain a good quantitative agreement between the numerical and the experimental results, we also introduce an approximate way to model the friction of the lateral film-substrate faces. This analysis reveals the strong effects of friction upon the magneto-mechanical wrinkling modes.

Changes in the North Pacific Current divergence and California Current transport based on HadGEM2-ES CMIP5 projections to the end of the century

This study aims to evaluate the potential changes to the strength of the North Pacific Current (NPC) and its position of divergence, which would alter the variability of the Alaska (AC) and California (CC) currents throughout the century. For this, the volume transports of these currents were evaluated using the historical CMIP5 and RCP4.5 experimental results of the HadGEM2-ES model from 1985 to 2100. In general, the NPC will migrate to the north, in accord to the northwestward expansion of North Pacific High, which would alter the positioning and the amount of flow entering the AC and CC systems by 2100. Along with the bifurcation shift, changes in the two main modes of variability, i.e., the bifurcation and breathing modes, are expected. The bifurcation mode will become responsible for only 11% of the total variability while presenting a positive anomaly throughout the century. These results indicate the role of NPC migration on the southward transport of enriched waters and the possible impact on the primary productivity along the North American coast.

Quasi-period, periodic bursting and bifurcations in memristor-based FitzHugh-Nagumo circuit

A third-order non-autonomous memristor-based FitzHugh-Nagumo (FHN) circuit is proposed through introducing a memristor emulator to realize the nonlinear function of the membrane voltage in the FHN model. This memristive FHN model possesses two slow time-scales, i.e. the external forcing current and the memristor inner state variable, and quasi-periodic, periodic bursting, and periodic behaviors are numerically revealed. The mechanism of the two-slow-time-scale-dependent periodic bursting dynamics is explored through using graphical method. Moreover, with the evolution of time, the equilibrium point of the memristive FHN circuit evolves between no equilibrium and line equilibrium point, and the stabilities of the line equilibrium point are highly related to the memristor initial condition. Consequently, the exhibited dynamics can be adjusted not only by tuning the system parameters, i.e. the forcing amplitude and frequency, but also by changing the memristor initial condition. The coexistence of multiple bifurcation modes are also found in some system parameter regions. Finally, a simulated circuit model is designed and PSIM circuit simulations are performed to confirm the numerical simulations. The presented memristive FHN circuit can enrich the family of memristor-based FHN circuits and their exhibited firing dynamics.

Threshold control strategy for a non-smooth Filippov ecosystem with group defense

A non-smooth Filippov ecosystem with group defense is proposed to investigate the effects of threshold and intermittent control strategy for IPM. The proposed model has been analyzed theoretically using the qualitative analysis techniques related to Filippov non-smooth systems in the first place. In particular, the existence and stability of equilibria are discussed for the sliding mode dynamics, through which it can be shown that the real equilibrium and pseudo-equilibrium can coexist. After that, some relevant topics such as sliding mode bifurcation, boundary-focus bifurcation, grazing bifurcation, crossing bifurcation and buckling bifurcation are investigated using numerical analysis. These complex sliding bifurcations reveal that the proposed Filippov system admits the coexistence of multi-attractors including equilibrium and crossing cycles, thus indicating that there exists a close relationship between the intermittent control strategy and the initial densities of both populations. Similarly, the global stability properties of real equilibrium and pseudo-equilibrium of subsystems show that the intermittent control strategy can effectively control the pests under the prescribed threshold which is the aim of the IPM strategy.

A third-order memristive Wien-bridge circuit and its integrable deformation

In this paper, a novel passive memristor model and its equivalent circuit model are designed, analysed and realised to investigate the memristor characteristics and their applications in nonlinear circuits. By employing this memristor model, a new third-order memristive Wien bridge is set up. Dynamical behaviours of the system are studied in detail, and multiscroll attractors, coexisting bifurcation modes, coexisting attractors, antimonotonicity and transient chaotic bursting are observed in this system by using theoretical analysis, simulation analysis and circuit experiment. Integrable deformation of the memristive Wien-bridge system is analysed. The circuit experiment is performed by replacing the memristor with its equivalent circuit model in the proposed memristor-based Wien-bridge circuit.

Bifurcation of an Oscillatory Mode under a Periodic Perturbation of a Special Oscillator

We study a bifurcation from the zero solution of the differential equation ẍ + xp/q = 0, where p > q > 1 are odd coprime numbers, under periodic (in particular, time-invariant) perturbations depending on a small positive parameter e. The motion separation method is used to derive the bifurcation equation. To each positive root of this equation, there corresponds an invariant two-dimensional torus (a closed trajectory in the time-invariant case) shrinking to the equilibrium position x = 0 as e → 0. The proofs use methods of the Krylov-Bogolyubov theory to study time-periodic perturbations and the implicit function theorem in the case of time-invari ant perturbations.

Effects of pre-stretch, compressibility and material constitution on the period-doubling secondary bifurcation of a film/substrate bilayer

We refine a previously proposed semi-analytical method, and use it to study the effects of pre-stretch, compressibility and material constitution on the period-doubling secondary bifurcation of a uni-axially compressed film/substrate bilayer structure. It is found that compared with the case of incompressible neo-Hookean materials for which the critical strain is approximately 0.17 when the thin layer is much stiffer than the substrate, the critical strain when the Gent materials are used is a monotonically increasing function of the constant Jm that characterizes material extensibility, becoming as small as 0.12 when Jm is equal to 1, whereas for compressible neo-Hookean materials the critical strain is a monotonically decreasing function of Poisson’s ratio; the period-doubling secondary bifurcation seems to become impossible when Poisson’s ratio is approximately equal to 0.307. The latter result may indicate that when Poisson’s ratio is small enough there are other preferred secondary bifurcations — an example is given where a secondary bifurcation mode with 7∕4 times the original period occurs at a lower strain value. The effect of a pre-stretch (compression or extension) in the substrate is not monotonic, giving rise to a critical strain that varies between 0.15 and 0.22.

Self-Propelled Motion of a Coumarin Disk Characteristically Changed in Couple with Hydrolysis on an Aqueous Phase.

In this study, a coumarin disk was examined as a simple self-propelled object under a chemical reaction. A coumarin disk placed on an aqueous phase containing Na3PO4 as a base exhibited continuous and oscillatory motion at lower and higher initial concentrations of Na3PO4, [Na3PO4]0, respectively. In addition, the period of the oscillation between rest and motion increased with increasing [Na3PO4]0. The mechanism of mode bifurcation between continuous and oscillatory motion and a change in the period of oscillation were discussed in terms of hydrolysis of coumarin and the surface tension of the aqueous solution as a driving force. A reduced mathematical model based on the reaction kinetics of coumarin around the air/aqueous interface, which adequately reproduced the experimental observation, was constructed. These results suggest that the characteristics of the self-propelled motion were determined by the kinetics of hydrolysis.

Dual-Band (28,38) GHz Coupled Quarter-Mode Substrate-Integrated Waveguide Antenna Array for Next-Generation Wireless Systems

A novel dual-band substrate-integrated waveguide (SIW) antenna array topology is proposed for operation in the 28 and 38 GHz frequency bands. Four miniaturized quarter-mode SIW cavities are tightly coupled, causing mode bifurcation, and yielding an antenna topology with four distinct resonance frequencies. A pair of resonances is assigned to both the 28 and 38 GHz band, achieving wideband operation in both frequency ranges. Moreover, owing to the exploited miniaturization technique, an extremely compact array topology is obtained, facilitating easy and straightforward integration. The computer-aided design process yields a four-element antenna array that entirely covers the 28 GHz band (27.5–29.5 GHz) and 38 GHz band (37.0–38.6 GHz) with a measured impedance bandwidth of 3.65 and 2.19 GHz, respectively. A measured broadside gain of 10.1 dBi, a radiation efficiency of 75.75% and a 3 dB beamwidth of 25° are achieved in the 28 GHz band. Moreover, in the 38 GHz band, the measured broadside gain amounts to 10.2 dBi, a radiation efficiency of 70.65% is achieved, and the 3 dB beamwidth is 20°.