Period-doubling Bifurcation Points Research Articles

Period-doubling cascade route to chaos in an initially curved microbeam resonator exposed to fringing-field electrostatic actuation

AbstractThis paper explores the chaotic dynamics of a piezoelectrically laminated initially curved microbeam resonator subjected to fringing-field electrostatic actuation, for the first time. The resonator is fully clamped at both ends and is coated with two piezoelectric layers, encompassing both the top and bottom surfaces. The nonlinear motion equation which is obtained by considering the nonlinear fringing-field electrostatic force, includes geometric nonlinearities due to the mid-plane stretching and initial curvature. The motion equation is discretized using Galerkin method and the reduced order system is numerically integrated over the time for the time response. The variation of the first three natural frequencies with respect to the applied electrostatic voltage is determined and the frequency response curve is obtained using the combination of shooting and continuation methods. The bifurcation points have been examined and their types have been clarified based on the loci of the Floquet exponents on the complex plane. The period-doubled branches of the frequency response curves originating from the period doubling (PD) bifurcation points are stablished. It's demonstrated that the succession PD cascades leads to chaotic behavior. The chaotic behavior is identified qualitatively by constructing the corresponding Poincaré section and analyzing the response's associated frequency components. The bifurcation diagram is obtained for a wide range of excitation frequency and thus the exact range in which chaotic behavior occurs for the system is determined. The chaotic response of the system is regularized and controlled by applying an appropriate piezoelectric voltage which shifts the frequency response curve along the frequency axis.

Nonlinear dynamics of a piezoelectrically laminated initially curved microbeam resonator exposed to fringing-field electrostatic actuation

In this paper, the nonlinear dynamics of a piezoelectrically sandwiched initially curved microbeam subjected to fringing-field electrostatic actuation is investigated. The governing motion equation is derived by minimizing the Hamiltonian over the time and discretized to a reduced-order model using the Galerkin technique. The modelling accounts for nonlinearities due to the fringing-field electrostatic force, initial curvature and mid-plane stretching. The electrostatic force is numerically computed using finite element simulation. The nonlinear dynamics of the microbeam in the vicinity of primary resonance is investigated, and the bifurcation types are determined by investigating the location of the Floquet exponents and their configuration with respect to the unit circle on the complex plane. The branches on the frequency–response curves, which originate from the period-doubling bifurcation points, are introduced, and the transition from period-1 to period-2 response is demonstrated by slight sweep of the excitation frequency over the time. The effect of DC and AC electrostatic excitation and the piezoelectric excitation on the response of the system are examined, and their effect on the bifurcation types is determined. The force response curves assuming the AC voltage as the bifurcation parameter are also introduced; it is illustrated that in contrast to in-plane electrostatic excitation, in fringing field-based resonators the resonator is not limited by pull-in instability, which is substantially confining the amplitude of the motion in in-plane resonators.

Bifurcation Analysis of Periodic Oscillation in a Hematopoietic Stem Cells Model with Time Delay Control

Underlying the state feedback control, the complex dynamical disease of the hematopoietic stem cells model based on Mackey’s mathematical description is analyzed. The bifurcating periodical oscillation solutions of the system are continued by applying numerical simulation method. The limit point cycle bifurcation and period doubling bifurcation are observed frequently in the continuation process. The attraction basins of the positive equilibrium solution shrink as the differentiate rate is ascending and the observed Mobiüs strain is simulated with boundary as the period-2 solution. The period doubling bifurcation leads to period-2, period-4, and period-8 solutions which are simulated. Starting from period doubling bifurcation point, the continuation of the bifurcating solution routes to homoclinic solution is finished. The simulation results improve the comprehension related to the spontaneous dynamical character manifested in the hematopoietic stem cells model.

Local bifurcation structure of a bouncing ball system with a piecewise polynomial function for table displacement.

The system in which a small rigid ball is bouncing repeatedly on a heavy flat table vibrating vertically, so-called the bouncing ball system, has been widely studied. Under the assumption that the table is vibrating with a piecewise polynomial function of time, the bifurcation diagram changes qualitatively depending on the order of the polynomial function. We elucidate the mechanism of the difference in the bifurcation diagrams by focusing on the two-period solution. In addition, we derive the approximate curve of the branch close to the period-doubling bifurcation point in the case of the piecewise cubic function of time for the table vibration. We also performed numerical calculation, and we demonstrate that the approximations well reproduce the numerical results.

Dynamic bifurcations in continuous process of bioethanol production under aerobic conditions using Saccharomyces cerevisiae

This article presents a numerical study of the oscillatory phenomena occurring in the bioethanol production process by aerobic continuous fermentation using Saccharomyces cerevisiae. The growth of biomass cells is described by a structured and nonsegregated kinetic model. In the first stage of the research, a sensitivity analysis and bifurcation analysis of the mathematical model describing the process was made. As bifurcation parameters, the concentration of glucose in the feed stream and the dilution rate were used. The analysis exposed the presence of subcritical and supercritical Hopf bifurcation points and period-doubling bifurcation points. Areas of stable and unstable limit cycles were also detected. In the second stage, the dynamics of the system in these areas was investigated. The prepared Feigenbaum diagrams have revealed opulent dynamics in the areas of oscillatory behaviours along with chaotic phenomena. A method of qualitative interpretation of limit cycles based on the analysis of the rate profiles of dominant metabolic pathways has also been proposed. With its help it is possible to precisely determine the metabolic state of biomass cells in the areas of oscillatory solutions and to specify the nature of the generated oscillations. The performed analysis provides important information about the oscillating phenomena occurring in the bioethanol production process, and its results can be used in the design, optimization and synthesis of modern control systems.

A double-side electrically-actuated arch microbeam for pressure sensing applications

In this study, we consider a pressure sensor whose main component is a clamped-clamped shallow arched microbeam. Two fixed electrodes are used to actuate the microbeam in the transverse direction. The lower electrode is powered by a combination of DC and AC voltages sources to excite the microbeam (drive mode). The upper electrode is polarized with a DC voltage and used to detect the change in the capacitance resulting from the transverse vibrations of the microbeam (sense mode). We formulate a fully-coupled multi-physics model of the electrically-actuated shallow arch microbeam combining the nonlinear Euler-Bernoulli beam theory with the nonlinear Reynolds equation governing the surrounding fluid domains (drive and sense zones). The model captures the inherent nonlinear physical aspects including the mid-plane stretching, the squeeze film damping, and fringing field effect. We validate the developed model by quantitatively comparing our nonlinear frequency-response against existing experimental results. We conduct static analysis to identify the occurrence of snap-through and pull-in instability for different initial midpoint rises. The coupled eigenvalue problem is also solved to compute the damped natural frequencies along with their corresponding beam and fluid mode shapes. The obtained natural frequencies are in good agreement with those reported in the literature. High sensitivity of the natural frequency to pressure variations is observed when decreasing the gap distance separating the microbeam from the fixed electrodes and reducing the initial midpoint rise of the curved microbeam.We also investigate the effect of breaking the symmetry of the microstructure on its dynamic response by introducing a slight perturbation in the mass distribution. The stability analysis reveals the occurrence of a new period doubling bifurcation point. Of interest, the location of this bifurcation point is observed to be sensitive to the pressure of the surrounding fluid medium. As such, we propose to exploit this nonlinear feature for design enhancement of the pressure sensor.

The bifurcation of periodic orbits and equilibrium points in the linked restricted three-body problem with parameter ω.

This paper is devoted to the bifurcation of periodic orbits and libration points in the linked restricted three-body problem (LR3BP). Inherited from the classic circular restricted three-body problem (CR3BP), it retains most of the dynamical structure of CR3BP, while its dynamical flow is dominated by angular velocity ω and Jacobi energy C. Thus, for the first time, the influence of the angular velocity in the three-body problem is discussed in this paper based on ω-motivated and C-motivated bifurcation. The existence and collision of equilibrium points in the LR3BP are investigated analytically. The dynamic bifurcation of the LR3BP under angular velocity variation is obtained based on three typical kinds of periodic orbits, i.e., planar and vertical Lyapunov orbits and Halo orbits. More bifurcation points are supplemented to Doedel's results in the CR3BP for a global sketch of bifurcation families. For the first time, a new bifurcation phenomenon is discovered that as ω approaches to 1.4, two period-doubling bifurcation points along the Halo family merge together. It suggests that the number and the topological type of bifurcation points themselves can be altered when the system parameter varies in LR3BP. Thus, it is named as "bifurcation of bifurcation" or "secondary bifurcation" in this paper. At selected values of ω, the phase space structures of equilibrium points L2 and L3 are revealed by Lie series method numerically, presenting the center manifolds on the Poincaré section and detecting three patterns of evolution for center manifolds in LR3BP.

Period Doubling Bifurcation Point Detection Strategy with Nested Layer Particle Swarm Optimization

This paper proposes a bifurcation point detection strategy based on nested layer particle swarm optimization (NLPSO). The NLPSO is performed by two particle swarm optimization (PSO) algorithms with a nesting structure. The proposed method is tested in detection experiments of period doubling bifurcation points in discrete-time dynamical systems. The proposed method directly detects the parameters of period doubling bifurcation regardless of the stability of the periodic point, but require no careful initialization, exact calculation or Lyapunov exponents. Moreover, the proposed method is an effective detection technique in terms of accuracy, robustness usability, and convergence speed.

The distinct stochastic and deterministic dynamics between period-adding and period-doubling bifurcations of neural bursting patterns

Period-adding bifurcations and period-doubling bifurcations of neural firing patterns, which were both observed in the biological experiment on a neural pacemaker and simulated in a theoretical model (Chay model), manifested different stochastic dynamics near the bifurcation points. For period-adding bifurcations, a noise-induced stochastic bursting, whose behavior is stochastic transition between period-k and period-( $$k+1$$ ) bursts, lying between period-k and period-( $$k+1$$ ) burstings ( $$k=1,2,3$$ ). For period-doubling bifurcations, period-1 bursting is changed to period-2 bursting firstly and then to period-4 bursting. No stochastic firing patterns similar to those lying in the period-adding bifurcation were detected. Using the method of the fast–slow variables dissection, the deterministic burstings in both period-adding bifurcation and period-doubling bifurcation are classified into “fold/homoclinic” bursting with a saddle-node point and a saddle-homoclinic point, which behaves as a critical phase sensitive to noisy disturbance. For the bursting pattern near the period-doubling bifurcation point, the trajectory of burstings is far from the saddle-homoclinic point. Near the period-adding bifurcation points from period-k to period-( $$k+1$$ ) burstings, the trajectories of the bursting patterns pass through the neighborhood of the saddle-homoclinic point and exhibit a platform. The platform appears after the k-th spike for the period-k bursting and between the k-th spike and the ( $$k+1$$ )-th spike for the period-( $$k+1$$ ) bursting. For some bursts of period-k bursting, noise can induce a novel spike near the platform to form a burst with $$k+1$$ spikes, and for some bursts of period-( $$k+1$$ ) bursting, the last spike can be terminated by noise to form a burst with k spikes. It is the cause that the stochastic bursting whose behavior is stochastic transition between period-k burst and period-( $$k+1$$ ) burst is induced by noise, and the stochastic transition happens within or near the platform, i.e., the neighborhood of the saddle-homoclinic point. More detailed transition dynamics can be explained by the stable/unstable manifold of the saddle point. The underlying deterministic dynamics between period-adding and period-doubling bifurcation points that can lead to distinct stochastic dynamics are identified, which are helpful for understanding the roles of noise and provide critical phase to apply control strategy to modulate the firing patterns.

Inverse period-doubling bifurcations determine complex structure of bursting in a one-dimensional non-autonomous map.

We propose a simple one-dimensional non-autonomous map, in which some novel bursting patterns (e.g., "fold/double inverse flip" bursting, "fold/multiple inverse flip" bursting, and "fold/a cascade of inverse flip" bursting) can be observed. Typically, these bursting patterns exhibit complex structures containing a chain of inverse period-doubling bifurcations. The active states related to these bursting can be period-2(n) (n = 1, 2, 3,…) attractors or chaotic attractors, which may evolve to quiescence by a chain of inverse period-doubling bifurcations when the slow excitation decreases through period-doubling bifurcation points of the map. This accounts for the complex inverse period-doubling bifurcation structures observed in bursting patterns. Our findings enrich the possible routes to bursting as well as the underlying mechanisms of bursting.

Size-dependent parametric dynamics of imperfect microbeams

The nonlinear parametric dynamics of a geometrically imperfect microbeam subject to a time-dependent axial load is investigated in this paper. Based on the Euler–Bernoulli beam theory and the modified couple stress theory, continuous models for kinetic and potential energies are developed and balanced via use of Hamilton's principle. A model reduction procedure is carried out by applying the Galerkin scheme coupled with an assumed-mode technique, yielding a high-dimensional second-order reduced-order model. A linear analysis is performed upon the linear part of the reduced-order model in order to obtain the linear size-dependent natural frequencies. A nonlinear analysis is performed on the reduced-order model using the pseudo-arclength continuation method and a direct time-integration technique, yielding generalised coordinates, and hence the system parametric response. It is shown that, the steady-state frequency-response curves possess a trivial solution, both stable and unstable, throughout the solution space, separated by period-doubling bifurcation points, from which non-trivial solution branches bifurcate.

Nonlinear resonances of a single-wall carbon nanotube cantilever

The dynamics of an electrostatically actuated carbon nanotube (CNT) cantilever are discussed by theoretical and numerical approaches. Electrostatic and intermolecular forces between the single-walled CNT and a graphene electrode are considered. The CNT cantilever is analyzed by the Euler–Bernoulli beam theory, including its geometric and inertial nonlinearities, and a one-mode projection based on the Galerkin approximation and numerical integration. Static pull-in and pull-out behaviors are adequately represented by an asymmetric two-well potential with the total potential energy consisting of the CNT elastic energy, electrostatic energy, and the Lennard-Jones potential energy. Nonlinear dynamics of the cantilever are simulated under DC and AC voltage excitations and examined in the frequency and time domains. Under AC-only excitation, a superharmonic resonance of order 2 occurs near half of the primary frequency. Under both DC and AC loads, the cantilever exhibits linear and nonlinear primary and secondary resonances depending on the strength of the excitation voltages. In addition, the cantilever has dynamic instabilities such as periodic or chaotic tapping motions, with a variation of excitation frequency at the resonance branches. High electrostatic excitation leads to complex nonlinear responses such as softening, multiple stability changes at saddle nodes, or period-doubling bifurcation points in the primary and secondary resonance branches.

Period doubling of the nonlinear dynamical system of an electrostatically actuated micro-cantilever

The paper presents an investigation of the nonlinear dynamical system of an electrostatically actuated micro-cantilever by the incremental harmonic balance (IHB) method. An efficient approach is proposed to tackle the difficulty in expanding the nonlinear terms into truncated Fourier series. With the help of this approach, periodic and multi-periodic solutions are obtained by the IHB method. Numerical examples show that the IHB solutions, provided as many as harmonics are taken into account, are in excellent agreement with numerical results. In addition, an iterative algorithm is suggested to accurately determine period doubling bifurcation points. The route to chaos via period doublings starting from the period-1 or period-3 solution are analyzed according to the Floquet and the Feigenbaum theories.

Numerical bifurcation analysis of the bipedal spring-mass model

The spring–mass model and its numerous extensions are currently one of the best candidates for templates of human and animal locomotion. However, with increasing complexity, their applications can become very time-consuming. In this paper, we present an approach that is based on the calculation of bifurcations in the bipedal spring–mass model for walking. Since the bifurcations limit the region of stable walking, locomotion can be studied by computing the corresponding boundaries. Originally, the model was implemented as a hybrid dynamical system. Our new approach consists of the transformation of the series of initial value problems on different intervals into a single boundary value problem. Using this technique, discontinuities can be avoided and sophisticated numerical methods for studying parametrized nonlinear boundary value problems can be applied. Thus, appropriate extended systems are used to compute transcritical and period-doubling bifurcation points as well as turning points. We show that the resulting boundary value problems can be solved by the simple shooting method with sufficient accuracy, making the application of the more extensive multiple shooting superfluous. The proposed approach is fast, robust to numerical perturbations and allows determining complete manifolds of periodic solutions of the original problem.

English

Here, we consider a one dimensionalnonlinear cubic map to find out a few inherent attributes i.e.fixed points, periodic points, bifurcation values of periods 2 � , � = 0,1,2,3,4 … … … … … . ...We use suitable numerical methods and have shown how the period doubling bifurcation points ultimately converge to the Feigenbaum constant. We have calculated Feigenbaumvalue also. We have further verified our findings with the help of bifurcation diagram, Lyapunavexponent, time series analysis of the map. Computer software package 'Mathematica' and 'C-program' are used prudentially to implement numerical algorithms for our purpose. Keywords: Fixed points, Periodic points, Bifurcation points, Feigenbaum constant, Lyapunov exponent, Time series.

An analytical method for analyzing symmetry-breaking bifurcation and period-doubling bifurcation

A new modification of homotopy analysis method (HAM) is proposed in this paper. The auxiliary differential operator is specifically chosen so that more than one secular term must be eliminated. The proposed method can capture asymmetric and period-2 solutions with satisfactory accuracy and hence can be used to predict symmetry-breaking and period-doubling bifurcation points. The variation of accuracy is investigated when different number of frequencies are considered.

Nonlinear processes upon doubling the period of self-modulation oscillations in a solid-state ring laser

Nonlinear phenomena appearing in a solid-state ring laser upon approaching the period-doubling bifurcation point of self-modulation oscillations and inside the doubling region are studied theoretically and experimentally. The bifurcation appears due to the parametric interaction of self-modulation oscillations of the first kind with relaxation oscillations. It is found that the bifurcation diagrams, time dependences of the intensities and power spectrum can significantly differ for counterpropagating waves because of the amplitude nonreciprocity of the ring resonator and the inequality of the moduli of the feedback coefficients. It is shown that when the self-modulation period is doubled, the widths of spectral peaks corresponding the self-modulation frequency and the fundamental relaxation frequency decrease. Noise precursors of doubling bifurcation are studied. It is found that the distance between the peaks of noise precursors increases with increasing the noise intensity. It is demonstrated experimentally that the noise modulation leads to the bifurcation point displacement, which increases with increasing the noise.

Periodic solutions of a singularly perturbed delay differential equation

A singularly perturbed differential delay equation of the form (1) ϵ x ̇ ( t ) = − x ( t ) + f ( x ( t − 1 ) , λ ) exhibits slowly oscillating periodic solutions (SOPS) near the first period-doubling bifurcation point of the underlying map (obtained by setting ϵ = 0 ). For extremely small values of ϵ , these periodic solutions resemble square waves, which consist of sharp, O ( ϵ ) transition layers connecting intervals of approximately unit length. In this article, we obtain analytic expressions for these square-wave periodic solutions, by solving the corresponding transition layer equations, and show that they are in excellent agreement with numerical solutions for a range of values of ϵ and λ . We also derive analytic expressions for other periodic solutions which are odd harmonics of the SOPS, and numerically exhibit their instability near the first period doubling bifurcation point of the map. The numerical computations were performed using a high accuracy Chebyshev spectral scheme. We give a brief description together with a study of its accuracy and efficiency.

Bifurcation and route-to-chaos analyses for Mathieu–Duffing oscillator by the incremental harmonic balance method

Bifurcations and route to chaos of the Mathieu–Duffing oscillator are investigated by the incremental harmonic balance (IHB) procedure. A new scheme for selecting the initial value conditions is presented for predicting the higher order periodic solutions. A series of period-doubling bifurcation points and the threshold value of the control parameter at the onset of chaos can be calculated by the present procedure. A sequence of period-doubling bifurcation points of the oscillator are identified and found to obey the universal scale law approximately. The bifurcation diagram and phase portraits obtained by the IHB method are presented to confirm the period-doubling route-to-chaos qualitatively. It can also be noted that the phase portraits and bifurcation points agree well with those obtained by numerical time-integration.

Stochastic period-doubling bifurcation analysis of stochastic Bonhoeffer–van der Pol system

In this paper, the Chebyshev polynomial approximation is applied to the problem of stochastic period-doubling bifurcation of a stochastic Bonhoeffer?van der Pol (BVP for short) system with a bounded random parameter. In the analysis, the stochastic BVP system is transformed by the Chebyshev polynomial approximation into an equivalent deterministic system, whose response can be readily obtained by conventional numerical methods. In this way we have explored plenty of stochastic period-doubling bifurcation phenomena of the stochastic BVP system. The numerical simulations show that the behaviour of the stochastic period-doubling bifurcation in the stochastic BVP system is by and large similar to that in the deterministic mean-parameter BVP system, but there are still some featured differences between them. For example, in the stochastic dynamic system the period-doubling bifurcation point diffuses into a critical interval and the location of the critical interval shifts with the variation of intensity of the random parameter. The obtained results show that Chebyshev polynomial approximation is an effective approach to dynamical problems in some typical nonlinear systems with a bounded random parameter of an arch-like probability density function.