Bifurcation Route Research Articles

Some bifurcation routes to chaos of thermocapillary convection in two-dimensional liquid layers of finite extent

As a part of the preliminary studies for the future space experiment (Zona-K) in the Russian module of the International Space Station, some bifurcation routes to chaos of thermocapillary convection in two-dimensional liquid layers filled with 10 cSt silicone oil have been numerically studied in this paper. As the laterally applied temperature difference is raised, variations in the spatial structure and temporal evolution of the thermocapillary convection and a complex sequence of transitions are observed. The results show that the finite extent of the liquid layer significantly influences the tempo-spatial evolution of the thermocapillary convection. Moreover, the bifurcation route of the thermocapillary convection changes very sensitively by the aspect ratio of the liquid layer. With the increasing Reynolds number (applied temperature difference), the steady thermocapillary convection experiences two consecutive transitions from periodic oscillatory state to quasi-periodic oscillatory state with frequency-locking before emergence of chaotic convection in a liquid layer of aspect ratio 14.25, and the thermocapillary convection undergoes period-doubling cascades leading to chaotic convection in a liquid layer of aspect ratio 13.0.

Propagation of envelope bright breather coupled wave modes in Ablowitz–Ladik chains

We investigate a model of a two coupled Ablowitz–Ladik (AL) lattices which admits the integrable AL model, when it is reduced to symmetric states. We compute the discrete modulational instability gain analytically and verified numerically. The play-role of the coupling parameter ϵ which is the back-bone of the two coupled AL chain is analyzed. We perform molecular dynamics (MD) simulations and elucidate the short time as well as the long time instability of AL chain. We demonstrate that the AL chain supports the formation of bright breather coupled wave modes spanning over 36 lattice sites on either side of the bond-center. We explore a kink solitary profile of solutions for the AL chains by invoking the extended tangent hyperbolic function method (ETHF) embedded with symbolic computation. Spectral stability of the exact solutions is determined by the eigenvalues of the discrete spectrum. We demonstrate the onset of chaos that leads the quasi-periodic dynamics of AL chains through the period-doubling bifurcation route to chaos.

Periodic solution and bifurcation of a suspension vibration system by incremental harmonic balance and continuation method

A compressed air generator hang under vehicle is simplified as a suspension mass connected to a vertical spring and two horizontal springs. It is configured generally as a geometrical negative stiffness to reduce dynamic stiffness. The periodic motion, chaotic motion and bifurcation of the compressed air generator model are investigated using the incremental harmonic balance method in combination with arc length continuation technique. The stability and bifurcation route are also distinguished with Floquet theory. The system exhibits a period doubling bifurcation route to chaos in different regions of excitation frequency. The stiffness ratio of the vertical spring and the horizontal spring has a significant influence on the dynamic response. When the vertical stiffness is close to the stiffness at horizontal direction, resonance occurs with the emergence of the chaotic motion. The dynamic response of the vibration system can be improved by reducing the stiffness in the horizontal direction to increase the stiffness ratio.

Complexity analysis of remanufacturing duopoly game with different competition strategies and heterogeneous players

In this paper, we consider a repeated remanufacturing duopoly game with different competition strategies and heterogeneous players by assuming that the original equipment manufacturer is boundedly rational and takes profit maximization as his business objective, and the third-party remanufacturer is adaptive and takes share maximization on the basis of obtaining a certain profit as his business objective. We study the equilibrium points and their stability in two situations: Consumer’s willingness-to-pay (WTP) is higher and lower. Numerical simulations show complex dynamic behaviors of this two-dimensional discrete system. We find that in the first situation, the stability of the system has nothing to do with the adjustment speed of the third-party remanufacturer. Under different conditions, the WTP has different impact on the stability region of the conditional equilibrium point. The system can lose stability through two different bifurcation routes (flip and Neimark–Sacker bifurcations). In the second situation, the stability region of the Nash equilibrium becomes large with WTP increasing. A higher WTP can enhance the stability of the Nash equilibrium. The third-party remanufacturer’s business objective can expand the stability region. A fiercer competition can have a stronger stabilization effect to the Nash equilibrium. The stability of the system can be lost only through flip bifurcation. The results of system performance measuring show that unstable behavior is not all harmful to both manufacturers.

Bifurcation Analysis of a Belousov–Zhabotinsky Reaction Model

We investigate the bifurcation phenomena in a Belousov–Zhabotinsky reaction model by applying Hopf bifurcation theory in frequency domain and harmonic balance method. The high accurate predictions, i.e. fourth-order harmonic balance approximation, on frequencies, amplitudes, and approximation expressions for periodic solutions emerging from Hopf bifurcation are provided. We also detect the stability and location of these periodic solutions. Numerical simulations not only confirm the theoretical analysis results but also illustrate some complex oscillations such as a cascade of period-doubling bifurcation, quasi-periodic solution, and period-doubling route to chaos. All these results improve the understanding of the dynamics of the model.

Investigation of Chaotic and Strange Nonchaotic Phenomena in Nonautonomous Wien-Bridge Oscillator with Diode Nonlinearity

We have studied the chaotic and strange nonchaotic phenomena of a simple quasiperiodically forced Wien bridge oscillator circuit with diode as the only nonlinearity in this electronic oscillator system responsible for various nonlinear behaviors. Both the experimental results and the numerical simulation results for their confirmation are provided to show the bifurcation process. Various measures used for the numerical confirmation of SNA are power spectrum, maximal Lyapunov exponent, path of translational variables, mean square displacement, projection of poincaré section, log-log plot, and autocorrelation function. Based upon the numerical results, the birth of SNAs has been identified in the band merging route, intermittency route, and blowout bifurcation route. In addition, the birth of SNAs has been analyzed with peculiar mechanism, namely, “0-1 Test” employing the one state dynamical variable.

Numerical and Experimental Studies on Nonlinear Dynamics and Performance of a Bistable Piezoelectric Cantilever Generator

A piezo-magneto-elastically coupled distributed-parameter model of a bistable piezoelectric cantilever generator is developed by using the generalized Hamilton principle. The influence of the spacing between two adjacent magnets on the static bifurcation characteristics of the system is studied and the range of magnet spacing corresponding to the bistable states is obtained. Numerical and experimental studies are carried out to analyze the bifurcation, response characteristics, and their impact on the electrical output performance under varying external excitations. Results indicate that interwell limit cycle motion of the beam around the two centers corresponds to optimum power output; interwell chaotic motion and multiperiodic motion including intrawell oscillations are less effective. At a given frequency, the phenomena of symmetric-breaking and amplitude-phase modulation are observed with increase of base excitation. Both period-doubling bifurcation and intermittency routes to chaotic motion in the bistable system are found. It can be observed that the power output is not proportional to the excitation level because of the bifurcation behaviours.

Dynamical modeling and analysis of buck converter operating in pseudo-continuous conduction mode

Taking buck converter operating in pseudo-continuous conduction mode (PCCM) for example, through a detailed description of the switch state of the switching converter, its accurate discrete-time model is established in this paper. On the basis of the model, bifurcation diagrams of the PCCM buck converter with the variations of circuit parameters are obtained, including load resistance, equivalent series resistance (ESR) of inductor, inductance, capacitance, reference current, and input voltage. And the complex dynamical behaviors existing in PCCM buck converter, such as subharmonic oscillation, period-double bifurcation and chaos, are revealed. Under different load resistances, time-domain simulation waveforms and phase portraits of PCCM buck converter are obtained by Runge-Kutta algorithm based on the piecewise smooth switch model. The working states of PCCM buck converter, reflected by the time-domain waveforms and phase portraits, are consistent well with those described by the bifurcation diagrams. It is shown that the time-domain simulation results verify the validation of the discrete-time model.#br#From theoretical analysis and simulation results, some conclusions can be obtained below. 1) When the load resistance gradually decreases, PCCM buck converter has a unique bifurcation route, i. e. , from PCCM period-1 state, PCCM multi-period oscillation via period-double bifurcation, chaos, CCM-PCCM multi-period oscillation, to CCM period-1 state via inverse period-double bifurcation. What is more, the bifurcation analysis with the load resistance serving as parameter indicates that the PCCM buck converter is more suitable for light load conditions, and its stable state will be lost and operation mode can be shifted (from PCCM to CCM) with increasing the load. 2) The ESR of inductor is closely related to the power loss and will affect the stability of the PCCM converter. The larger the ESR, the more the power loss will be. However, the PCCM converter is more stable if the ESR is larger. 3) Period-double bifurcation or inverse period-double bifurcation exists in the PCCM buck converter with the other circuit parameters varied in a wide range except for the load resistance, and there are three working states of buck operating in PCCM, i.e., stable period-1 state, multi-period sub-harmonic oscillation, and chaos. The research results in this paper are useful for designing and controlling PCCM switching converter.

Analysis of Dynamical Behaviors in a Continuous Chaotic System Without Šil'nikov Orbits

This paper discusses a simple yet interesting three-dimensional autonomous quadratic chaotic system, which has two fixed points of saddle-focus type but does not belong to the class of Šil'nikov type since there are neither homoclinic nor heteroclinic orbits. Dynamical behaviors of this chaotic system are investigated in detail through analyzing bifurcation routes, Lyapunov exponents and Poincaré mappings. These investigations partially reveal the mechanism of how chaos in non-Šil'nikov type of systems is generated.

Secondary instability and tertiary states in rotating plane Couette flow

AbstractRecent experimental studies have shown rich transition behaviour in rotating plane Couette flow (RPCF). In this paper we study the transition in supercritical RPCF theoretically by determination of equilibrium and periodic orbit tertiary states via Floquet analysis on secondary Taylor vortex solutions. Two new tertiary states are discovered which we name oscillatory wavy vortex flow (oWVF) and skewed vortex flow (SVF). We present the bifurcation routes and stability properties of these new tertiary states and, in addition, we describe a bifurcation procedure whereby a set of defected wavy twist vortices is approached. Further to this, transition scenarios at flow parameters relevant to experimental works are investigated by computation of the set of stable attractors which exist on a large domain. The physically observed flow states are shown to share features with states in our set of attractors.

Mechanistic investigation of the reactivity of disilene with nitrous oxide: A DFT study

We have reported the mechanistic investigation of the reaction of N2O addition to disilene, trans-[(TMS)2N(η1-Me5C5)SiSi(η1-Me5C5)N(TMS)2] (1t), employing density functional theory (BP86/TZVP//BP86/SVP) calculations. The potential energy surfaces of the title reaction are broadly classified under three pathways. Pathway I deals with the direct N2O additions to 1t affording the trans-dioxadisiletane ring compound Pt whereas in the same pathway we report a different bifurcation route from intermediate 2t. This route portrays the isomerization of trans-monooxadisiletane species 2t prior to the second N2O addition, finally leading to the cis-isomeric product Pc. Different possibilities for isomerization of disilene 1t to 1c were studied in pathway II. The cis-disilene (1c) formed can subsequently react with two N2O molecules affording the cis-product Pc. Pathway III details the formation of silanone type intermediate 6, which subsequently combine with another silanone to afford loosely bound intermediates 7 and 8 respectively. The two separated silanone fragments in the isomeric intermediates 7 and 8 can then dimerizes to furnish the desired products. Among all the calculated potential energy surfaces, pathway III remains the most preferred route for disilene oxygenation under normal experimental condition. The present investigation about disilene reactivity will provide a deeper understanding on silylene chemistry and will exhibit promising applicability in main group chemistry as a whole.

Extended chaotic domain in the long optical fibers based on SBS process

Chaotic stimulated Brillouin scattering (SBS) in optical fibers with weak external optical feedback has been investigated experimentally and theoretically. However, only the bifurcation route to chaos through period-one and quasi-periodic emission was discovered in the former work because the chaotic domain is very short in the general nonlinear system. In order to control the chaos process and observe finer periodic-orbit structures, a novel experiment was designed to extend the chaotic domain by using long optical fibers in the SBS system. In this experiment, the period-one, period-doubling, period-four and period-eight cycle routes to the chaos laser process as well as the more details of chaos, have been observed.

Precise modeling and dynamic characteristics of valley V2 controlled Boost converter

A discrete iterative map model of valley V2 controlled Boost converter is established, based on which the bifurcation diagrams are obtained with the variation of input voltage, output capacitance and its equivalent series resistance (ESR). Jacobi matrix at a fixed point is also derived, and according to it, the converter stability is analyzed using characteristic values and maximum Lyapunov exponent, thus the correctness of bifurcation analysis is validated. The effect of input voltage, output capacitance and its ESR on the dynamic characteristics of valley V2 controlled Boost converter is mainly investigated. It is found that as the input voltage increases continuously, the valley V2 controlled Boost converter changes from continuous conduction mode (CCM) period-1 to CCM period-2 due to period-doubling bifurcation, and comes into CCM chaos due to border collision bifurcation. The converter has the same bifurcation routes at output capacitance and its ESR: with gradual reduction of output capacitance or its ESR, the valley V2 controlled Boost converter behaves the evolutive dynamic behavior from CCM period-1 to CCM period-2, CCM period-4, CCM period-8, and CCM chaos. Finally, the simulation and experimental circuits are set up, and the correctness of theoretical analysis is verified by simulation and experimental results.

The Chaos Dynamic of Multiproduct Cournot Duopoly Game with Managerial Delegation

Although oligopoly theory is generally concerned with the single-product firm, what is true in the real word is that most of the firms offer multiproducts rather than single products in order to obtain cost-saving advantages, cater for the diversity of consumer tastes, and provide a barrier to entry. We develop a dynamical multiproduct Cournot duopoly model in discrete time, where each firm has an owner who delegates the output decision to a manager. The principle of decision-making is bounded rational. And each firm has a nonlinear total cost function due to the multiproduct framework. The Cournot Nash equilibrium and the local stability are investigated. The tangential bifurcation and intermittent chaos are reported by numerical simulations. The results show that high output adjustment speed can lead to output fluctuations which are characterized by phases of low volatility with small output changes and phases of high volatility with large output changes. The intermittent route to chaos of Flip bifurcation and another intermittent route of Flip bifurcation which contains Hopf bifurcation can exist in the system. The study can improve our understanding of intermittent chaos frequently observed in oligopoly economy.

Experimental study on chaos generation in an all-fiber erbium-doped fiber ring laser with a Mach-Zehnder interferometer

We experimentally study the chaotic behaviors in a compact all-fiber erbium-doped fiber ring laser (ED-FRL) with an added Mach-Zehnder interferometer (MZI) by using a phase-modulation method. A piezo-electric ceramic transducer (PZT) is incorporated in the MZI to introduce single-frequency phase modulation. The coexistence of intermittency and period-doubling bifurcation routes to chaos in the EDFRL system is observed by adjusting the modulation frequency or the phase modulation depth in the experiment. In addition, the EDFRL presents irregular multi-longitudinal-mode oscillation with a definite linear polarization when operating at intensity chaos state.

Numerical simulation of torus breakdown to chaos in an atmospheric-pressure dielectric barrier discharge

Understanding the routes to chaos occurring in atmospheric-pressure dielectric barrier discharge systems by changing controlling parameters is very important to predict and control the dynamical behaviors. In this paper, a route of a quasiperiodic torus to chaos via the strange nonchaotic attractor is observed in an atmospheric-pressure dielectric barrier discharge driven by triangle-wave voltage. By increasing the driving frequency, the discharge system first bifurcates to a quasiperiodic torus from a stable single periodic state, and then torus and phase-locking periodic state appear and disappear alternately. In the meantime, the torus becomes increasingly wrinkling and stretching, and gradually approaches a fractal structure with the nonpositive largest Lyapunov exponent, i.e., a strange nonchaotic attractor. After that, the discharge system enters into chaotic state. If the driving frequency is further increased, another well known route of period-doubling bifurcation to chaos is also observed.

Equivalent modeling and bifurcation analysis of V2 controlled buck converter

After dimension reduction, two boundary voltages of V2 controlled buck converter are deduced under different operation mode, based on which, its equivalent one-dimensional discrete-time model is established and complex nonlinear bifurcation behaviors are emphatically studied. Two boundary conditions under which shift between stable period-one state and subharmonic oscillation state and shift between continuous conduction mode (CCM) and discontinuous conduction mode (DCM) take place are derived by analyzing stability and operation mode. The research results show that in V2 controlled buck converter period-doubling bifurcation and border-collision bifurcation can occur with varying circuit parameters and the converter has different bifurcation routes at different circuit parameters. Simulation and experiment platforms are implemented and the corresponding results verify the validity of equivalent discrete-time model and theoretical analysis.

Dynamics of current controlled quadratic boost converters

The two inductor current boundaries are derived, and segmented smooth iteration mapping model of current controlled quadratic boost converter is established based on the detailed description of switching states. The comparative analyses of the nonlinear bifurcation behaviors are performed with input inductor current and the storage inductor current as current feedback respectively. Two boundary conditions for the transition from stable period-one state to sub-harmonic oscillation state and for the transition from discontinuous conduction mode to continuous conduction mode are derived by analyzing stability and operation mode. The operation-state regions corresponding to circuit parameter regions are estimated by utilizing the parameter-space maps. An experimental circuit is built and the results show that both different bifurcation routes and operation mode transition phenomenon are changed with parameter variation, the current controlled quadratic boost converter exhibits complicated dynamical behaviors. The experimental results verify the correctness of the theoretical analysis.

Complexity Analysis of a Cournot-Bertrand Duopoly Game Model with Limited Information

A Cournot-Bertrand mixed duopoly game model with limited information about the market and opponent is considered, where the market has linear demand and two firms have the same fixed marginal cost. The principles of decision-making are bounded rational. One firm chooses output and the other chooses price as decision variable, with the assumption that there is a certain degree of differentiation between the products offered by firms to avoid the whole market being occupied by the one that applies a lower price. The existence of Nash equilibrium point and its local stability of the game are investigated. The complex dynamics, such as bifurcation scenarios and route to chaos, are displayed using parameter basin plots by numerical experiment. The influences of the parameters on the system performance are discussed from the perspective of economics.

A Study on Stability of Limit Cycle Walking Model with Feet: Parameter Study

In this paper, two kinds of feet, namely, curved and flat feet, are added to limit cycle walking model to investigate its stability properties. Although both models are already proposed and are investigated, most previous works are focused on efficiency and gait behaviors. Only the stability properties of the simplest walking model conceived Garcia et al. are well defined. Therefore, there is still a need for a precise research on the effect of feet, especially in the view of local stability, bifurcation route to chaos, global stability, falling boundary and energy balance line. Therefore, this article revisits the stability analysis of limit cycle walking model with various foot shape. To analyze the effects of feet, we re-derive the equation of motion of modified models by adding one more parameter of foot shape than the simplest walking model. Also, the falling boundary and energy balance line of modified models are derived to get proper initial conditions for stable walking and to explain global stability. Simulation results show us that the curved feet can enlarge both stable walking range and area of basin of attraction while the case of flat feet depends on foot shape parameter.