Location Of Bifurcation Point Research Articles

Multi-parametric optimization for controlling bifurcation structures

Bifurcations organize the dynamics of many natural and engineered systems. They induce qualitative and quantitative changes to a system’s dynamics, which can have catastrophic consequences if ignored during design. In this paper, we propose a general computational method to control the local bifurcations of dynamical systems by optimizing design parameters. We define an objective functional that enforces the appearance of local bifurcation points at targeted locations or even encourages their disappearance. The methodology is an efficient alternative to bifurcation tracking techniques capable of handling many design parameters ( > 10 2 ). The method is demonstrated on a Duffing oscillator featuring a hardening cubic nonlinearity and an autonomous van der Pol-Duffing oscillator coupled to a nonlinear tuned vibration absorber. The finite-element model of a clamped-free Euler–Bernoulli beam, coupled with a reduced-order modelling technique, is also used to show the extension to the shape optimization of more complicated structures. Results demonstrate that several local bifurcations of various types can be handled simultaneously by the bifurcation control framework, with both parameter and state target values.

Detecting Stability and Bifurcation Points in Control-Based Continuation for a Physical Experiment of the Zeeman Catastrophe Machine

Feedback-based control of dynamical systems enables tracking of unstable branches that cannot be reached by the free evolution of a simulated or physical system. As the stationary points are stabilized with these control methods, the stability information on the original system gets lost in the data from the transient behavior reaching these equilibria. We introduce, by means of a physical experiment of the Zeeman catastrophe machine, a method to construct the whole bifurcation diagram including stability information about the underlying uncontrolled equilibria and the location of bifurcation points.

Mathematical modeling of the population dynamics of a distinct interactions type system with local dispersal

Distinct biotic interactions in multi-species communities are a ubiquitous force in the natural ecosystem, and this force is an essential determinant of community stability and species coexistence outcomes. We conduct numerical simulations and bifurcation analysis of partial differential equations to gain better understanding and ecological insights into how predation (a), predator handling time (h), and local dispersal affect multi-species community dynamics. This system consists of resource-mutualist-exploiter-competitor interactions and local dispersal. From the inspection of our numerical simulations and co-dimension one bifurcation analysis findings, we discover several critical values that correspond to transcritical bifurcation, subcritical and supercritical Hopf bifurcations. This occurs as we vary the bifurcation parameters a and h in this complex ecological system under symmetric and asymmetric dispersal scenarios. Furthermore, the interplay between these local bifurcation points results in an exciting co-dimension two bifurcations, i.e., Bogdanov-Takens and cusp bifurcation points, respectively, which act as the synchronization points in this complex ecological system. From an ecological viewpoint, we find that (i) the effect of the no-dispersal scenario supports the maintenance of species biodiversity when the predation strength is moderate; (ii) symmetric dispersal induces both subcritical and supercritical Hopf bifurcation and support species diversity for moderate predation strength; and (iii) asymmetric dispersal promotes species diversity as it simplifies the bifurcation changes in dynamics by eliminating the subcritical bifurcations that trigger uncertainty, and this dispersal mechanism mediates species coexistence outcomes. Fundamentally, stable limit cycles have been reported as predator handling time varies in some ecological models; however, we observed in our bifurcation analysis the emergence of the unstable limit cycle as predator handling time changes. We discover that intense predator handling time destabilizes this complex ecological community. In general, our results demonstrate the influential roles of predation, predator handling time, and local dispersal in determining this system’s coexistence dynamics. This knowledge provides a better understanding of species conservation and biological control management.

Non-premixed flame propagation inside and outside the different three-way tubes after the self-ignition of pressurized hydrogen

Accidental leakage of pressurized hydrogen into the pipeline is often accompanied by the non-premixed self-ignition flame, whilst the complex structure of pipeline affects the flame propagation. In this study, the propagation of self-ignition hydrogen flame in different three-way tubes is studied and the flame evolution near the two exits is compared. Effects of the tube length and the location of bifurcation points are investigated. Results show that the bifurcation structure has the attenuation effect on the intensity of flame and shock wave, and they attenuate more sharply in the branch pipeline. But the flame intensity can increase again downstream the bifurcation point in the trunk pipeline, and this increase is not continuous and decrease again as the tube length increases. In addition, the flame evolution can be different near two exits of the tube. As the bifurcation point moves downstream, the number of Mach disk and the time required to form the stable flame will increase near the exit of branch pipeline. Despite different flame evolution, the flame morphology and propagation are all controlled by the shock wave structures before the formation of the vortex, but the flame changes dramatically and grows rapidly to form the stable lift-off flame after the formation of the vortex.

Anatomy study and clinical significance about the spatial distribution of the recurrent branch of median nerve

Objective To provide anatomical data for avoiding the injury of the recurrent branch of median nerve (RBMN) through the proximal operative incision of the carpal thenar region. Methods Thirty adult upper limb specimens (16 formaldehyde fixed specimens and 14 fresh frozen specimens) were dissected. The origin of the RBMN, the angle between the RBMN and the longitudinal axis of the median nerve main trunk, the location of the first bifurcation point of the RBMN, the distribution of the superficial branch and the deep branch of the RBMN were measured, and the relationship between the RBMN and the proximal, middle and distal regions of the thenar was observed. Results The origin of the trunk of the RBMN located in the proximal and middle third of the thenar area account for 46.7% and 52.3%, respectively. After the RBMN originated from the main trunk of the median nerve, it moved towards the radial side at an average angle of 48.6° (range, 20° to 70°). The first bifurcation point of RBMN located in the proximal and middle third of the thenar area account for 56.7% and 43.3%, respectively. 96.7% deep branch and 56.7% superficial branch were located in the proximal third of the thenar area. Conclusion The RBMN and its terminal branches were mostly located in the proximal third of the thenar area, and any operation in this area should pay attention to avoiding injury. Key words: Median nerve; Anatomy; Recurrent branch; Clinical significance

Stability controlled crack initiation in nacre-like composite materials

Recent research has focused on the toughening mechanism of nacre in view of the cracking process. Crack initiation, often characterized as the sudden concentration of strain, has a strong relation with the emergence of the system’s multi-solution and the change in convexity of the strain energy function. In this study, we use a cohesive spring system representing the micro-structure of nacre, and the Monte-Carlo (MC) algorithm to trigger the transition of the solution mode when the system loses its global stability. The stability evolution of the material can be divided into three stages. In stage I, the global and local stabilities are maintained and the strain energy function remains strictly convex. In stage II, the system loses local stability but global stability is maintained. In stage III, the system loses global stability and the strain concentration occurs. Stage II originates from the combination of shear and debond interface mechanisms and enhances the system’s stability four-fold. Under certain conditions, stage II disappears with a simultaneous loss of both local and global stabilities. This phenomenon can be reproduced by both theoretical prediction and MC simulation. Several factors have shown relations with stability evolution, such as irreversible dissipation, micro-structure size, shear/debond cohesive strength, and the local cohesive bifurcation point. This study offers a view different from conventional fracture mechanics for exploring crack-toughening mechanisms of nacre.

A meeting point of entropy and bifurcations in cross-diffusion herding

A cross-diffusion system modelling the information herding of individuals is analysed in a bounded domain with no-flux boundary conditions. The variables are the species' density and an influence function which modifies the information state of the individuals. The cross-diffusion term may stabilize or destabilize the system. Furthermore, it allows for a formal gradient-flow or entropy structure. Exploiting this structure, the global-in-time existence of weak solutions and the exponential decay to the constant steady state is proved in certain parameter regimes. This approach does not extend to all parameters. We investigate local bifurcations from homogeneous steady states analytically to determine whether this defines the validity boundary. This analysis shows that generically there is a gap in the parameter regime between the entropy approach validity and the first local bifurcation. Next, we use numerical continuation methods to track the bifurcating non-homogeneous steady states globally and to determine non-trivial stationary solutions related to herding behaviour. In summary, we find that the main boundaries in the parameter regime are given by the first local bifurcation point, the degeneracy of the diffusion matrix and a certain entropy decay validity condition. We study several parameter limits analytically as well as numerically, with a focus on the role of changing a linear damping parameter as well as a parameter controlling the cross-diffusion. We suggest that our paradigm of comparing bifurcation-generated obstructions to the parameter validity of global-functional methods could also be of relevance for many other models beyond the one studied here.

Calculation of Local Bifurcation Points in Piecewise Nonlinear Discrete‐Time Dynamical Systems

SUMMARYThis paper proposes a method for calculation of local bifurcation points in discrete‐time dynamical systems with piecewise nonlinear characteristics (PNDDS). First, an ‐dimensional PNDDS, which has two piecewise nonlinear maps, is shown and its variational equation is derived. Next, a calculation method for the local bifurcation points that utilizes the conditional equation for the periodic solution and the characteristic equation is proposed. It is essential to calculate the derivatives of the map with an initial value and with a bifurcation parameter to obtain the bifurcation points continuously in the parameter space. The above calculation process is a key component of the proposed method, and is explained in detail. Finally, we apply the proposed method to a two‐dimensional PNDDS and calculate the local bifurcation points in order to confirm the validity of the proposed method.

Calculation Method of Local Bifurcation Point in Piecewise Nonlinear Discrete-Time Dynamical Systems

This paper proposes a calculation method of local bifurcation points for discrete-time dynamical systems with piecewise nonlinear characteristic (PNDDS). First, an n-dimensional PNDDS, which has two piecewise nonlinear maps, is shown and its variational equation is derived. Next, the calculation method of the local bifurcation points which utilizes the conditional equation of the periodic solution and the characteristic equation is proposed. Then, we have to calculate derivatives of the map with an initial value and with a bifurcation parameter to obtain the bifurcation points continuously in the parameter space. The above calculation process is a key of the proposed method, and we explain it in detail. Finally, we apply the proposed method to a two-dimensional PNDDS and calculate the local bifurcation points for confirming its validity.

A Dynamic Analysis for an Anaerobic Digester: Stability and Bifurcation Branches

This work presents a dynamic analysis for an anaerobic digester, supported on the analytical application of the indirect Lyapunov method. The mass-balance model considered is based on two biological reaction pathways and involves both Monod and Haldane representations of the specific biomass growth rates. The dilution rate, the influent concentration of chemical oxygen demand (COD), and the influent concentration of volatile fatty acids (VFA) are considered as stability parameters. Several characteristics are determined analytically for the normal operation equilibrium point: (i) equilibrium coordinates, (ii) parameter conditions that lead to positive values of the equilibrium state variables, (iii) parameter conditions for locally stable nature of the equilibrium, (iv) coordinates for the local bifurcation points—fold and transcritical—, and (v) coordinates of the crossing between bifurcation points. These factors are computed analytically and explicitly as expressions of the dilution rate and the influent concentrations of COD and VFA.

Simulation and Analysis of Three-Dimensional Arterial Trees

For hemodynamic simulation and heat transfer analysis of blood, it is necessary to generate a three-dimensional model of arterial tree. Referring to the approach of Constrained Constructive Optimization (CCO), a new method based on bifurcation law, Poiseuilles law and mass conservation was proposed to model arterial trees directly in simulation objects. The formulas for calculating the bifurcation ratios and the way to determine the location of bifurcation point were presented in the paper. In this model, the radius of the root segment was constant, the terminal pressure was unknown and the bifurcation exponent was variable. In order to demonstrate the feasibility of this method, an arterial tree was constructed in a spherical tissue.

Bifurcation control of a cubic symmetry discrete chaotic system

A direct and effective linear-controller is employed to exactly control the locations of bifurcation points, both the symmetry-breaking bifurcation and the period-doubling bifurcation, in a cubic symmetry discrete system. Moreover, both the sensibility and the symmetry to the initial values of the system are analyzed. The lack of the solution branches due to the symmetry-breaking bifurcation can be reinstated temporarily by selecting the corresponding basins of attraction. The effectiveness of the controller is verified by numerical simulations.

Hopf bifurcation control of the Rössler system based on Washout filter controller

The Washout filter is adopted to control Hopf bifurcation in Rössler system. The effect of its parameters on the position of bifurcation point, the bifurcation type, and the amplitude of periodic solution is discussed in detail. Based on Routh criterion, the stability region in parameter space is found out firstly. Then the influence of the filter’s time constant and the linear gain on the location of bifurcation point is analyzed. By using the Direct Normal Form method, the normal form of the controlled Rössler system is deduced and the coefficient of resonant term is expressed as a function of the nonlinear gain. It is revealed that, by changing the sign and the absolute value of the real part of the coefficient, the nonlinear gain can modify the periodic solution’s amplitude and Hopf bifurcation type. All the results obtained are verified by numerical simulation.

Dynamic Behavior and Control in an Industrial Fluidized-Bed Polymerization Reactor

In an industrial fluidized-bed polymerization reactor, tight temperature control is of utmost importance to ensure that the temperature in the reaction zone is kept above the dew point of reactants, yet below the melt point of the polymer. Even within a narrow temperature range, temperature excursions must be avoided because they can result in low catalyst productivity and significant changes in product properties. Furthermore, if the reactor temperature is under open-loop conditions, these reactors are prone to instability and undesired nonlinear dynamic behaviors. In this work, a first principles model, including the recycle stream and the heat exchange system, is used to describe an industrial fluidized-bed polymerization reactor of the UNIPOL process. A detailed study of the dynamic behavior of this process is carried out, with the location of bifurcation points and system stabilization by PID controller designed via optimization in the frequency domain. Nonetheless, if the manipulated variable (cooling water valve position) saturates, the reactor operates without a feedback temperature controller, leading to oscillatory behavior and limit cycles. For this case, it has been also demonstrated the necessity of using auxiliary manipulated variables to decrease the polymerization heat generation and to help control the reactor temperature.

Non-linear vibration and stability analysis of a partially supported conveyor belt by a distributed viscoelastic foundation

The main source of transverse vibration of a conveyor belt is frictional contact between pulley and belt. Also, environmental characteristics such as natural dampers and springs affect natural frequencies, stability and bifurcation points of system. These phenomena can be modeled by a small velocity fluctuation about mean velocity. Also, viscoelastic foundation can be modeled as the dampers and springs with continuous characteristics. In this study, non-linear vibration of a conveyor belt supported partially by a distributed viscoelastic foundation is investigated. Perturbation method is applied to obtain a closed form analytic solutions. Finally, numerical simulations are presented to show stiffness, damping coefficient, foundation length, non-linearity and mean velocity effects on location of bifurcation points, natural frequencies and stability of solutions.

Stability estimation of SVCs in distribution systems

AbstractThis paper discusses the stability and an interactive phenomenon of one SVC, two or more sets of SVC, an SVC and an SVR in distribution systems. In particular, we focus on the saddle‐node bifurcation and Hopf bifurcation points. The two local bifurcation points show a boundary of a stable domain. © 2007 Wiley Periodicals, Inc. Electr Eng Jpn, 160(3): 1–11, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/eej.20333

Local and global bifurcation in periodic scalar ODEs

Following the lines of Yu. Ilyashenko and W. Li, we propose a definition of local and global bifurcations for the family of differential equations x ′ = f ( t , x , λ ) , where f ∈ C ( R 3 ) is locally Lipschitz continuous with respect to x , and T -periodic in t . Then, we prove that a value of the parameter λ is a global bifurcation value if and only if there exists a local bifurcation point (including bifurcation points at infinity) for this value of the parameter.

Comportamentos dinâmicos em um reator industrial de polimerização em fase gasosa

(artigo convidado): Nos reatores de polimerização em fase gasosa, o controle de temperatura em uma estreita faixa de temperatura é importantíssimo para assegurar que a temperatura na zona de reação seja mantida acima do ponto de orvalho dos reagentes, assim como abaixo do ponto de fusão do polímero. Outro ponto importante é que se a temperatura estiver em malha aberta, estes reatores estão propensos à instabilidade e ao surgimento de ciclos limites, entretanto podem ser facilmente estabilizados com um controlador de temperatura apropriado. Neste trabalho é realizado um estudo detalhado do comportamento dinâmico de um reator industrial de polimerização em leito fluidizado, com a localização de pontos de bifurcação e estabilização do sistema com controlador PID projetado via otimização no domínio da freqüência.

Shape design of pin-jointed multistable compliant mechanisms using snapthrough behavior

A general approach is presented for generating pin-jointed multistable compliant mechanisms using snapthrough behavior. An optimization problem is formulated for minimizing the total structural volume under constraints on the displacements at the specified nodes, stiffnesses at initial and final states, and load factors to lead to snapthrough behavior. The design variables are cross-sectional areas and the nodal coordinates. It is shown in the numerical examples that several mechanisms can be naturally found as a result of optimization starting from randomly selected initial solutions. It is also shown that no local bifurcation point exists along the equilibrium path, and the obtained mechanism is not sensitive to initial imperfections.

Stability Estimation of SVCs in Distribution Systems

This paper presents stability and an interactive phenomenon of one SVC, two or more sets of SVC, a SVC and a SVR in distribution systems. In particular, we focus on the saddle-node bifurcation and Hopf bifurcation points. The two local bifurcation points show a boundary of a stable domain.