Dynamic Bifurcation Analysis Research Articles

Dynamic Behavior and Bifurcation Analysis of a Difference Equation Including Exponential Terms

Dynamic Behavior and Bifurcation Analysis of a Difference Equation Including Exponential Terms

Dynamic Behavior and Bifurcation Analysis of a Modified Reduced Lorenz Model

This study introduces a newly modified Lorenz model capable of demonstrating bifurcation within a specified range of parameters. The model demonstrates various bifurcation behaviors, which are depicted as distinct structures in the diagram. The study aims to discover and analyze the existence and stability of fixed points in the model. To achieve this, the center manifold theorem and bifurcation theory are employed to identify the requirements for pitchfork bifurcation, period-doubling bifurcation, and Neimark–Sacker bifurcation. In addition to theoretical findings, numerical simulations, including bifurcation diagrams, phase pictures, and maximum Lyapunov exponents, showcase the nuanced, complex, and diverse dynamics. Finally, the study applies the Ott–Grebogi–Yorke (OGY) method to control the chaos observed in the reduced modified Lorenz model.

Dynamics of neural fields with exponential temporal kernel.

We consider the standard neural field equation with an exponential temporal kernel. We analyze the time-independent (static) and time-dependent (dynamic) bifurcations of the equilibrium solution and the emerging spatiotemporal wave patterns. We show that an exponential temporal kernel does not allow static bifurcations such as saddle-node, pitchfork, and in particular, static Turing bifurcations. However, the exponential temporal kernel possesses the important property that it takes into account the finite memory of past activities of neurons, which Green's function does not. Through a dynamic bifurcation analysis, we give explicit bifurcation conditions. Hopf bifurcations lead to temporally non-constant, but spatially constant solutions, but Turing-Hopf bifurcations generate spatially and temporally non-constant solutions, in particular, traveling waves. Bifurcation parameters are the coefficient of the exponential temporal kernel, the transmission speed of neural signals, the time delay rate of synapses, and the ratio of excitatory to inhibitory synaptic weights.

Setting Safe Operation Conditions for Acetyl Chloride Hydrolysis through Dynamic Modelling and Bifurcation Analysis

Acetyl chloride hydrolysis is a highly sensitive exothermic reaction that has presented several industrial safety issues. In the present study, a multiparameter mathematical model, previously developed and applied to simulate the oscillatory thermal behavior of an experimental continuous stirred tank reactor, was used to determine the static/dynamic bifurcation behavior of this reactive system. The values predicted by the model showed good agreement with the experimental data reported in the literature. Full topological classification of its fixed points and iterative maps was obtained: unique solutions (stable and unstable), multiple solutions, cyclic envelope, and bifurcation objects of codimension 1 (e.g., fold and Hopf’s points) and codimension 2 (e.g., cusp and generalized Hopf and Bogdanov-Takens points) have been uncovered. The emphasis of the analysis is to determine safe operating conditions through understanding these topological features and manipulating the reactor design and operating parameters.

Dynamic Behaviours and Bifurcation Analysis of a Filippov Ecosystem with Fear Effect

Evidences of the fear effect of the prey are well documented, which can greatly affect the dynamics of the predator-prey system. In this study, considering that the fear effect of the prey is triggered on as the density of the predator reaches and exceeds a threshold value, we develop a Filippov system of predator-prey model with the fear effect. In addition, we also include a modify factor of the growth rate of the prey when they adopt the antipredator behaviours due to the fear effect. We initially analyze the dynamics of the two subsystems, including the existence and stability of the equilibria. Utilizing the theory of the Filippov system, we discuss the sliding dynamics, i.e., the existence of sliding region and sliding equilibria. By choosing the threshold as the bifurcation parameter, we investigate the bifurcations near the regular equilibria. The solution curve has three cases: crossing the threshold curve, sliding on the threshold curve, and approaching the pseudoequilibrium. Finally, we numerically verified the existence of the global sliding bifurcation near the regular equilibrium and also the touching bifurcation.

Dynamic shimmy behavior and bifurcation analysis of driver–vehicle–road system

Dynamic shimmy behavior and bifurcation analysis of driver–vehicle–road system

Dynamic modeling and bifurcation analysis of blade-disk rotor system supported by rolling bearing

The dynamic modeling and bifurcation analysis are carried out for a blade-disk rotor system supported by rolling bearing in this paper. Considering the nonlinearity of rolling bearings and the flexible coupling between multi-stage blades and disk, the coupled dynamics model of the blade-disk rotor system is established by using the finite element method. Subsequently, with numerical simulations the steady-state responses of the nonlinear system are obtained. Using the bifurcation diagrams, the time histories, whirling orbits, Poincaré maps and power spectrums, the parametric studies are conducted in detail to investigate the system's bifurcation characteristics. The results indicate that considering the blade mass and stiffness will create a new resonance peak near the first-order resonance of the rotor system with linearized rolling bearing stiffness. For the rotor system with nonlinear rolling bearing forces, with the increase of the blade length, the system motion pattern becomes complex, slender blades will lead to chaotic motion at high speed, and blade stiffness and mass will obviously increase the rotational speed region of quasi-periodic motion. Moreover, the Young's modulus of the blade significantly affects the bifurcation characteristics of the system, i.e., small Young's modulus values make the motion form complex, which leads the range of quasi-periodic motion and chaotic motion increased obviously. The results of this research would be helpful to deepen comprehensive understanding of the nonlinear dynamic characteristics of blade-disk rotor system supported by rolling bearing.

Dynamic bifurcation for a three-species cooperating model

A dynamic bifurcation analysis on a three-species cooperating model was presented and it was proved that the problem bifurcated an attractor as the parameter λ crossed the critical value λ0. The analysis was based on the attractor bifurcation theory together with the central manifold reduction.

Dynamic behavior and bifurcation analysis of a deterministic and stochastic coupled logistic map system

In the present work, a system of two linear coupled logistic map is studied. Local stability analysis of the fixed points of the proposed system is investigated. The system occurs transcritical, flip, and Neimark-Sacker bifurcations, which are analyzed by both center manifold theory and bifurcation theory. For any non-linear system that represents a real-world model affected by noise, white noise is included in the system and its effect on fixed points is analyzed via the technique of stochastic sensitivity function. The phenomenon of noise-induced transitions between closed invariant curves is discussed. Finally, numerical simulations are performed with the aid of Matlab to assure the agreements with analytical results obtained.

Nonlinear dynamic characteristics and bifurcation analysis of Ti–Zr–Ni quasicrystal as hydrogen storage material

In this article, the nonlinear dynamic characteristics and bifurcation of a Ti–Zr–Ni quasicrystal impacted by hydrogen atoms are studied. New nonlinear damping terms are proposed to express the delay characteristics of Ti–Zr–Ni quasicrystal, and the accurate natural frequency is obtained by the harmonic balance method. A new method based on the developed largest Lyapunov exponent is proposed to analyze the local stability of any point in the system, and the system's global stability is determined. Finally, a new way to realize the switch between hydrogen storage and release based on stochastic Hopf bifurcation is proposed. The results of theoretical analysis and numerical simulation show that the system's motion can be switched between a periodic orbit and a balanced point near the bifurcation boundary with little energy consumption, which is helpful for hydrogen storage and release.

Nonlinear dynamic behavior and bifurcation analysis of a rotating viscoelastic size-dependent beam based on non-classical theories

In this paper, the free and harmonically forced vibrations of a viscoelastic rotating microbeam have been analyzed by exploiting non-classical theories, such as the Modified Couple Stress Theory and Modified Strain Gradient Theory (MSGT). A continuous model is derived and reduced to a well-known Duffing–Rayleigh system via a Galerkin projection; then, the Multiple Scales Method is used to obtain amplitude equations. Free and forced nonlinear oscillations are studied via classical and non-classical theories, and their corresponding results are compared with each other. An expression for natural frequency is obtained in which the effects of the internal damping and small-scale parameters are obvious. It is observed that the natural frequency obtained using the MSGT has the highest value, while the Classical Theory (CT) predicts the lowest natural frequency value. Also, it is seen that by increasing the hub radius, the influence of rotational speed on natural frequency is increased. According to obtained results, the responses corresponding to the CT become completely stable for the lowest value of the excitation force and the highest value of the external damping coefficient. For MSGT, responses are stable when the values of the excitation force and external damping coefficients are highest and lowest, respectively. It is shown that for the MSGT, the system is stable when the internal damping coefficient is small, while for the CT, the beam is stable for the highest value of the internal damping coefficient. The system parameters—such as the detuning parameter, damping coefficients, and excitation force—affect the solution, such a way that the system may possess one or two stable spirals depending on the values of those parameters, as shown in phase plane portraits.

Discrete dynamic equilibrium model for a complex problem of flutter interactions

Abstract The dynamic bifurcation analysis of the nonlinear oscillation of a simple fluidelastic structure is presented. This structure is a cantilever beam in a flow, and it behaves as a nonlinear system without potential energy. The structure demonstrates complex flutter behaviour that varies with the controlled flow velocity. We observe the flutter behaviour in a flow experiment, and the motion is characterised with this present model based on chaos theory of discrete dynamics. We can readily find the solution of the simple system, with which it is possible to create a map of the complex flutter behaviour.

Dynamic stability and bifurcation analysis of sandwich plate with considering FG core and FG-CNTRC face sheets

In this article, the dynamic stability and bifurcation analysis of sandwich plate made of functionally graded material (FGM) as a core and functionally graded carbon nanotubes-reinforced composite (FG-CNTRC) as face sheets at top and bottom of the core under lateral stochastic loads are investigated. The classical plate theory is employed as a suitable model in obtaining equilibrium equations using the variational principles. The material properties of each layer are considered as functions of the temperature-dependent environment. By applying the Fokker–Planck–Kolmogorov (FPK) equation on equilibrium equations of motion, the probability density function is obtained. Drawing probability density function and bifurcation diagrams indicates the influence of the aspect ratio of thickness ( h/hp), temperature changes, various distributions and volume fraction of carbon nanotube and power law index of functionally graded core on stability and bifurcation of sandwich plate. By enhancing volume fraction of carbon nanotube, length of the stable region (one equilibrium point) increases and this method can be employed for improving the stability of sandwich structures. The outcomes of this research could be employed to manufacture many composite structures in aircraft and automobiles.

Dynamic bifurcation analysis and mitigation of SSR in SMIB system

This paper presents a detailed study on Sub-synchronous resonance in a Single machine connected to an infinite bus system by employing Bifurcation Theory. The synchronous machine model considered is a two-axis model, the turbine system is a six mass system. A comparative study of Sub-synchronous resonance for the two-axis synchronous machine models 1.1, 2.1 and 2.2 under constant excitation is presented in this paper. The effect of adding an exciter, power system stabilizer and Static Synchronous Series Compensator to the SMIB system, incorporated with a 2.2 synchronous machine model on the bifurcations of SSR is also investigated. The results obtained on replacing the fixed series compensation of the line by Static Synchronous Series Compensator resulted in the mitigation of Sub-synchronous Resonance. The results obtained for all the above-considered cases are compared with eigenvalue analysis and validated using transient analysis<strong>.</strong>

A numerical framework for the stability and cardinality analysis of concentric tube robots: Introduction and application to the follow-the-leader deployment

Concentric tube robot (CTR) is a promising class of continuum robots for medical interventions given their compactness and dexterity. Their dexterity is in particular being used to achieve so called Follow-the-Leader (FTL) deployments, where the tip path draws the shape of the robot. During this kind of deployment they can however be subject to elastic instabilities, and the number of reachable configurations may vary for a given state of actuators. These cardinality and stability changes need therefore to be predicted during CTR design. Available methods and results are limited, with restrictive assumptions on number and properties of tubes. We therefore propose in this paper a numerical framework for the cardinality and stability assessment of CTR. It is based on the association of dynamic relaxation, continuation method and bifurcation analysis. The numerical framework is validated by reproducing reference results on the stability and cardinality of two-tube robots. Then, new results on three-tube CTR deploying in a FTL manner are presented. The framework genericity allows in particular to provide new insights on the behaviour of CTR with helical-shaped tubes.

Dynamic stability and bifurcation analysis in fractional thermodynamics

In mechanics, viscoelasticity was the first field of applications in studying geomaterials. Further possibilities arise in spatial non-locality. Non-local materials were already studied in the 1960s by several authors as a part of continuum mechanics and are still in focus of interest because of the rising importance of materials with internal micro- and nano-structure. When material instability gained more interest, non-local behavior appeared in a different aspect. The problem was concerned to numerical analysis, because then instability zones exhibited singular properties for local constitutive equations. In dynamic stability analysis, mathematical aspects of non-locality were studied by using the theory of dynamic systems. There the basic set of equations describing the behavior of continua was transformed to an abstract dynamic system consisting of differential operators acting on the perturbation field variables. Such functions should satisfy homogeneous boundary conditions and act as indicators of stability of a selected state of the body under consideration. Dynamic systems approach results in conditions for cases, when the differential operators have critical eigenvalues of zero real parts (dynamic stability or instability conditions). When the critical eigenvalues have non-trivial eigenspace, the way of loss of stability is classified as a typical (or generic) bifurcation. Our experiences show that material non-locality and the generic nature of bifurcation at instability are connected, and the basic functions of the non-trivial eigenspace can be used to determine internal length quantities of non-local mechanics. Fractional calculus is already successfully used in thermo-elasticity. In the paper, non-locality is introduced via fractional strain into the constitutive relations of various conventional types. Then, by defining dynamic systems, stability and bifurcation are studied for states of thermo-mechanical solids. Stability conditions and genericity conditions are presented for constitutive relations under consideration.

Nonlinear dynamic characteristics and bifurcation analysis of Al-doped graphene impacted by hydrogen atoms

In this paper, the nonlinear dynamic characteristics and bifurcation of Al-doped graphene are studied. The nonlinear dynamic model of Al-doped graphene impacted by hydrogen atoms is developed where the nonlocal effect of a graphene is considered. The natural frequency of the system is obtained, the system's drift coefficient and diffusion coefficient are verified, and the stationary probability density function of the system's dynamic response is given. The condition of stochastic bifurcation is determined, and the fractal boundary of the safe basin is provided. Finally, the reliability function of the system is solved, and the probability density of the first-passage time is determined. Theoretical analysis and numerical simulation show that stochastic bifurcation occurs in the variation of parameters; the area of safe basin decreases when the intensity of the atom's impact increases. The results of this paper are helpful for the application of Al-doped graphene in hydrogen storage.

Dynamic modeling and bifurcation analysis for the methyl isocyanate hydrolysis reaction

The aim of this work was to characterize the methyl isocyanate hydrolysis reaction and to identify its operational criteria. The parametric sensitivity and dynamic stability methodologies were performed at the Bhopal disaster circumstances, over the relevant operating parameter space. Stable and unstable conditions, bifurcations points, turning points and oscillatory behavior were determined. The combined methodology give useful guidance on the operative conditions selection and the appropriate strategy to overcome hazardous situations. The obtained results demonstrated high sensitivity to small perturbations (thermal runaway) and prevalent oscillatory behavior. Moreover, the following critical parameters for the studied dynamic system were defined: the inverse residence time of 1.5700103 and the heat transfer coefficient of 752.394.

A boundary modulation formulation for cable’s non-planar coupled dynamics under out-of-plane support motion

Suspended cable’s non-planar resonant coupled dynamics under out-of-plane support motion is investigated by the multiple- scale method, with a boundary modulation formulation established and nonlinear dynamic responses analyzed. Explicitly, to cope with the difficulty due to moving boundary, the small resonant support motion is properly rescaled and incorporated into cable’s modulation equations as a boundary resonant modulation term, through constructing solvability conditions of the multi-scale expansions. And the boundary resonance dynamic coefficient, characterizing the boundary modulation effect, is derived analytically for cable’s two-to-one resonant coupled dynamics. Numerical results for cable’s non-planar coupled dynamic responses, including stability and bifurcation analysis for the equilibrium solutions of modulation equations, are obtained and presented in the end, with both saddle-node bifurcations and Hopf bifurcations detected.

A multi-stable energy harvester: Dynamic modeling and bifurcation analysis

A theoretical investigation is conducted on the dynamic and energetic characteristics of a multi-stable bimorph cantilever energy harvester that uses magnetic attraction effect. The multi-stable energy harvester under study is composed of a bimorph cantilever beam with soft magnetic tip and two externally fixed permanent magnets that are arranged in series. With this configuration, the magnetic force and the moment that are exerted on the cantilever tip tend to be highly dependent on the magnetic field induced by the external magnets. Such an energy harvester can possess multi-stable potential functions, ranging from mono-stable to penta-stable. The mechanism that governs the formation of this multi-stability is thoroughly identified and examined thorough a bifurcation analysis performed on the system׳s equilibrium solutions. From this analysis, it is found that the transitions between these multi-stable states occur through very complicated bifurcation scenarios that include degenerate pitchfork bifurcations and mergers of pitchfork bifurcations or saddle-node bifurcations. Bifurcation set diagram is obtained, which is composed of five separate parametric regions, from mono- to penta-stability. The resulting stability map satisfactorily describes the multi-stable characteristics of the present energy harvester. In addition, the dynamic and energetic characteristics of the present multi-stable energy harvester are more thoroughly examined using its potential energy diagrams and a series of numerical simulations, and the obtained results are compared with those for the equivalent bi-stable cases.