Impulsive control of bifurcations

In this paper, an impulsive control of bifurcations method is developed. Sufficient conditions for the asymptotical stability of an impulsive control system are derived. Comparison has been made between it and a typical feedback control method. Numerical simulations are cited to illustrate the methodology and to verify the theoretical results.     

Feedback tuning of bifurcations

The present paper studies a feedback regulation problem that arises in at least two different biological applications. The feedback regulation problem under consideration may be interpreted as an adaptive control problem for tuning bifurcation parameters, and it has not been studied in the control literature. The goal of the paper is to formulate this problem and to present some preliminary results.     

分段线性连续系统中的同宿分岔

对于平面上分段线性的连续系统研究了同宿轨的存在性及同宿分岔问题.该系统同宿轨的存在性可以归结为两种情况：一种是由一个可见鞍点和一个可见焦点（或中心）组成的系统;另一种是由两个稳定性相反的结点重合于原点组成的系统.本文对第一种情况给出了同宿轨存在的充要条件,并研究了相应的同宿分岔问题.     

Feedback stabilization of bifurcations in multivariable nonlinear systems—Part I: equilibrium bifurcations

Under certain non‐degeneracy conditions, necessary and sufficient conditions for stabilizability are obtained for multi‐input nonlinear systems possessing a simple equilibrium bifurcation with the critical mode being linearly uncontrollable. Stabilizability is defined as the existence of a sufficiently smooth state feedback such that the bifurcation of the closed‐loop system is a supercritical pitchfork bifurcation, which is equivalent to local asymptotic stability of the system at the bifurcation point. Stabilizability is reduced to the existence of solutions to a third order algebraic inequality, coupled with a quadratic algebraic equation, with the unknowns being the feedback gains. Explicit conditions for the existence of solutions to the algebraic equation and the inequality are derived. Part of the sufficient conditions are equivalent to the rank conditions of augmented matrices. Conditions under which there exists a stabilizing linear feedback law are also derived. The theoretical results are applied to active control of rotating stall in axial compressors using bleed valve as actuator. Copyright © 2006 John Wiley & Sons, Ltd.     

Feedback stabilization of bifurcations in multivariable nonlinear systems—Part II: Hopf bifurcations

In this paper we derive necessary and sufficient conditions of stabilizability for multi‐input nonlinear systems possessing a Hopf bifurcation with the critical mode being linearly uncontrollable, under the non‐degeneracy assumption that stability can be determined by the third order term in the normal form of the dynamics on the centre manifold. Stabilizability is defined as the existence of a sufficiently smooth state feedback such that the Hopf bifurcation of the closed‐loop system is supercritical, which is equivalent to local asymptotic stability of the system at the bifurcation point. We prove that under the non‐degeneracy conditions, stabilizability is equivalent to the existence of solutions to a third order algebraic inequality of the feedback gains. Explicit conditions for the existence of solutions to the algebraic inequality are derived, and the stabilizing feedback laws are constructed. Part of the sufficient conditions are equivalent to the rank conditions of an augmented matrix which is a generalization of the Popov–Belevitch–Hautus (PBH) rank test of controllability for linear time invariant (LTI) systems. We also apply our theory to feedback control of rotating stall in axial compression systems using bleed valve as actuators. Copyright © 2006 John Wiley & Sons, Ltd.     

Multiscale analysis of defective multiple-Hopf bifurcations

An adapted version of the Multiple Scale Method is formulated to analyze 1:1 resonant multiple Hopf bifurcations of discrete autonomous dynamical systems, in which, for quasi-static variations of the parameters, an arbitrary number m of critical eigenvalues simultaneously crosses the imaginary axis. The algorithm therefore requires discretizing continuous systems in advance. The method employs fractional power expansion of a perturbation parameter, both in the state variables and in time, as suggested by a formal analogy with the eigenvalue sensitivity analysis of nilpotent (defective) matrices, also illustrated in detail. The procedure leads to an order-m differential bifurcation equation in the complex amplitude of the unique critical eigenvector, which is able to capture the dynamics of the system around the bifurcation point. The procedure is then adapted to the specific case of a double Hopf bifurcation (m = 2), for which a step-by-step, computationally-oriented version of the method is furnished that is directly applicable to solve practical problems. To illustrate the algorithm, a family of mechanical systems, subjected to aerodynamic forces triggering 1:1 resonant double Hopf bifurcations is considered. By analyzing the relevant bifurcation equation, the whole scenario is described in a three-dimensional parameter space, displaying rich dynamics.     

复合材料层合板的双Hopf分叉分析

利用高维非线性系统的Hopf分叉定理,研究复合材料层合板的双Hopf分叉.研究了一类受面内激励和横向外激励联合作用下的复合材料层合板在主参数共振—1∶1内共振情况下的双Hopf分叉.首先利用多尺度法得到系统的平均方程,经过简化得到了系统的分叉响应方程.根据对分叉响应方程的分析,得到了系统平衡解的稳定性临界曲线,并给出了系统产生双Hopf分叉的条件.利用数值方法得到系统在参数平面上的分叉集,通过对不同分叉区域的分析,我们发现随着参数的变化复合材料层合板存在不同的周期运动现象.     

Stability region bifurcations of nonlinear autonomous dynamical systems: Type‐zero saddle‐node bifurcations

The behavior of stability regions of nonlinear autonomous dynamical systems subjected to parameter variation is studied in this paper. In particular, the behavior of stability regions and stability boundaries when the system undergoes a type‐zero sadle‐node bifurcation on the stability boundary is investigated in this paper. It is shown that the stability regions suffer drastic changes with parameter variation if type‐zero saddle‐node bifurcations occur on the stability boundary. A complete characterization of these changes in the neighborhood of a type‐zero saddle‐node bifurcation value is presented in this paper. Copyright © 2010 John Wiley & Sons, Ltd.     

Local bifurcations and feasibility regions indifferential-algebraic systems

The dynamics of a large class of physical systems such as the general power system can be represented by parameter-dependent differential-algebraic models of the form x˙=f and 0=g. Typically, such constrained models have singularities. This paper analyzes the generic local bifurcations including those which are directly related to the singularity. The notion of a feasibility region is introduced and analyzed. It consists of all equilibrium states that can be reached quasistatically from the current operating point without loss of local stability. It is shown that generically loss of stability at the feasibility boundary is caused by one of three different local bifurcations, namely the saddle-node and Hopf bifurcations and a new bifurcation called the singularity induced bifurcation which is analyzed precisely here for the first time. The latter results when an equilibrium point is at the singular surface. Under certain transversality conditions, the change in the eigenstructure of the system Jacobian at the equilibrium is established and the local dynamical structure of the trajectories near this bifurcation point is analyzed     

一个干摩擦Duffing振子的滑动分岔分析

分析了一个干摩擦Duffing振子在不同参数下混沌和周期轨及不同周期轨之间的共存与转换.干摩擦振子属于Filippov系统,会发生特有的粘滞现象.分析发现,从穿越轨线转换到粘滞--滑移轨线不仅可经穿越滑动分岔和切换滑动分岔实现,也可经两个相邻的穿越滑动分岔和多滑动分岔实现,而从粘滞-滑移轨线转换到非穿越轨线须经擦边滑动...     

A fifth-order multiple-scale solution for Hopf bifurcations

The subject of this paper is the multiple-time-scale analysis of Hopf bifurcations up to fifth-order nonlinearities. It is shown how an asymptotic fifth-order expansion captures the change in the nature of the limit cycle from stable to unstable and viceversa. The formulation is validated by applying it to a simple mechanical system for which there exists an analytical limit-cycle solution. Applications include the pre- and post-flutter behavior of a typical section with nonlinear spring having a stable limit cycle (supercritical Hopf bifurcation) that turns into an unstable one, because of fifth-order nonlinearities.     

Numerical analysis of bifurcations by continuation methods

A numerical calculation technique for computing bifurcation values of interconnected dynamical systems is presented. The technique is based on continuation methods in which the bifurcation value of interconnected dynamical systems can be calculated from the bifurcation value of a subsystem using a set of coupled differential equations. As an example, the value of the Hopf bifurcation of an interconnected dynamical system is calculated.     

On torus Burth bifurcations in the pulse-width system

The dynamics of the pulse-width system described by nonautonomous piecewise smooth differential equations is studied. In the parameter plane domains of different dynamic behaviour are determined both analytically and numerically. A special type of C-bifurcation resulting in the invariant torus birth from a stable equilibrium is discovered. It is shown that the periodic motion due to this bifurcation arises as an unstable focal cycle surrounded by a resonance or ergodic torus.     

Neural mass activity, bifurcations, and epilepsy

In this letter, we propose a general framework for studying neural mass models defined by ordinary differential equations. By studying the bifurcations of the solutions to these equations and their sensitivity to noise, we establish an important relation, similar to a dictionary, between their behaviors and normal and pathological, especially epileptic, cortical patterns of activity. We then apply this framework to the analysis of two models that feature most phenomena of interest, the Jansen and Rit model, and the slightly more complex model recently proposed by Wendling and Chauvel. This model-based approach allows us to test various neurophysiological hypotheses on the origin of pathological cortical behaviors and investigate the effect of medication. We also study the effects of the stochastic nature of the inputs, which gives us clues about the origins of such important phenomena as interictal spikes, interictal bursts, and fast onset activity that are of particular relevance in epilepsy.     

Control of near‐grazing dynamics and discontinuity‐induced bifurcations in piecewise‐smooth dynamical systems

This paper develops a rigorous control paradigm for regulating the near‐grazing bifurcation behavior of limit cycles in piecewise‐smooth dynamical systems. In particular, it is shown that a discrete‐in‐time linear feedback correction to a parameter governing a state‐space discontinuity surface can suppress discontinuity‐induced fold bifurcations of limit cycles that achieve near‐tangential intersections with the discontinuity surface. The methodology ensures a persistent branch of limit cycles over an interval of parameter values near the critical condition of tangential contact that is an order of magnitude larger than that in the absence of control. The theoretical treatment is illustrated with a harmonically excited damped harmonic oscillator with a piecewise‐linear spring stiffness as well as with a piecewise‐nonlinear model of a capacitively excited mechanical oscillator. Copyright © 2009 John Wiley & Sons, Ltd.     

Border-collision bifurcations in a model of Braess paradox

In Braess paradox adding an extra resource, and therefore an extra available choice, enriches the complexity of the game from a dynamic perspective. The analysis of the cycles and the bifurcations helps to visualize how this complexity changes, in a quite new way with respect to what is provided by the so far literature. We derive the conditions for the creation and the destruction of periodic cycles, as well as the analytical expressions of the bifurcation conditions, by studying the occurrence of border-collision bifurcations. We are also able to give a proof of the relation between the cost of the new resource and the existence of cycles of any given period, and also of the coexistence of equilibria, adding the path dependence to the problem.     

The controlled center dynamics of discrete time control bifurcations

In this paper, we introduce the Controlled center dynamics for nonlinear discrete time systems with uncontrollable linearization. This is a reduced order control system whose dimension is the number of uncontrollable modes and whose stabilizability properties determine the stabilizability properties of the full order system. After reducing the order of the system, the synthesis of a stabilizing controller is performed based on the reduced order control system. By changing the feedback, the stability properties of the controlled center dynamics will change, and thus the stability properties of the full order system will change too. Thus, choosing a feedback that stabilizes the controlled center dynamics allows stabilizing the full order system. This approach is a reduction technique for some classes of controlled differential equations.