Bifurcation and chaos in power systems

A detailed example of a power system model with load dynamics is studied by investigating qualitative changes or bifurcations in its behaviour as a reactive power demand at one load bus is increased. In addition to the saddle-node bifurcation often associated with voltage collapse, we find other bifurcation phenomena which include Hopf bifurcation, cyclic fold bifurcation, period doubling bifurcation, and the emergence of chaos. The presence of these dynamic bifurcations motivates a re-examination of the role of saddle-node bifurcations in the voltage collapse phenomenon. In fact, simulation results suggest that voltage collapse may take place before the reactive power demand is increased to the system steady-state operating limit where a saddle-node bifurcation is detected. We also consider the role that the algebraic constraints imposed by some load models may play in the global analysis of the attractors of the system. Implications for power system operations are drawn.     

On generic dynamics related to a 3-fold zero eigenvalue

In this paper, we consider the bifurcation and stability behaviour of a nonlinear autonomous system in the vicinity of a generic codimension-3 critical point characterized by a triple-zero eigenvalue. The analysis is based on simplified differential equations in normal form and the unification technique coupled with the intrinsic harmonic balancing procedure. The equilibrium solutions, Hopf bifurcations and bifurcations into two-dimensional tori are studied in detail, and the associated stability conditions are presented as well. All results are expressed in terms of system coefficients. An example drawn from nonlinear control systems is analysed to demonstrate the direct applicability of the theory     

Examples of non-degenerate and degenerate cuspidal loops in planar systems

In this paper we study some examples of non-degenerate and degenerate cuspidal loops in planar systems. A cuspidal loop is a codimension-three homoclinic orbit given by the intersection of the separatrices of an equilibrium of cusp type. Using a Dulac map analysis and asymptotic expansions we study the stability in the neighbourhood of a cusp point. As a first example we consider an enzyme-catalysed reaction model exhibiting a non-degenerate cuspidal loop. All the codimension-one and-two homoclinic bifurcations present in the unfolding of the corresponding cuspidal loop are found in such a realistic model. Finally, the unfolding of a codimension-five Bogdanov-Takens bifurcation is analysed. A degenerate codimension-four cuspidal loop appearing in this system is located on a non-degenerate cuspidal loop curve, and part of the unfolding of such singularity is shown.     

Global dynamics of a family of 3-D Lotka–Volterra systems

In this article, we analyse the flow of a family of three-dimensional Lotka–Volterra systems restricted to an invariant and bounded region. The behaviour of the flow in the interior of this region is simple: either every orbit is a periodic orbit or orbits move from one boundary to another. Nevertheless, the complete study of the limit sets in the boundary allows one to understand the bifurcations which take place in the region as a global bifurcation that we denote by focus-centre-focus bifurcation.     

Periodic solutions and their bifurcations in a non-smooth second-order delay differential equation

We consider a non-smooth second order delay differential equation (DDE) that was previously studied as a model of the pupil light reflex. It can also be viewed as a prototype model for a system operated under delayed relay control. We use the explicit construction of solutions of the non-smooth DDE hand-in-hand with a numerical continuation study of a related smoothed system. This allows us to produce a comprehensive global picture of the dynamics and bifurcations, which extends and completes previous results. Specifically, we find a rich combinatorial structure consisting of solution branches connected at resonance points. All new solutions of the smoothed system were subsequently constructed as solutions of the non-smooth system. Furthermore, we show an example of the unfolding in the smoothed system of a non-smooth bifurcation point, from which infinitely many solution branches emanate. This shows that smoothing of the DDE may provide insight even into bifurcations that can only occur in non-smooth systems.     

直线运动柔性梁非线性动力学--组合参数共振与内共振联合激励

针对基础直线运动柔性梁，基于Kane方程建立了相应的非线性动力学方程。采用多尺度法并结合笛卡尔坐标变换，导出了系统受前两阶模态间3：1内共振及其组合参激共振时的非线性调制方程组，数值求解了该方程组的定常解及相应的稳定性问题。研究表明，系统的平凡响应与双模态非平凡响应共存，由内共振所产生的非平凡响应皆为不稳定的鞍点，平凡及非平凡解分支都存在Hopf分岔现象，一些稳定的极限环随参数变化最终经倍周期分岔后产生混沌运动。     

Generic multiparameter bifurcation from a manifold

The geometry of generic k -parameter bifurcation from an n -manifold is discussed for all values of k , n with particular emphasis on the case n = 2 (the case n = 1 being dealt with in earlier work). Such bifurcations typically arise in the study of equilibrium states of dynamical systems with continuous (for example, spherical or toroidal) symmetry which undergo small symmetry-breaking perturbations, and in the use of Melnikov maps for detecting bifurcations of periodic orbits from resonance. Detailed analysis is given in the interesting case n = 2, k = 3 where the local geometry partly resembles unfolding of a degenerate wavefront or Legendrian collapse.     

A codimension-two resonant bifurcation from a heteroclinic cycle with complex eigenvalues

Robust heteroclinic cycles between equilibria lose stability either through local bifurcations of their equilibria or through global bifurcations. This paper considers a global loss of stability termed a 'resonant' bifurcation. This bifurcation is usually associated with the birth or death of a nearby periodic orbit, and generically occurs in either a supercritical or subcritical manner. For a specific robust heteroclinic cycle between equilibria with complex eigenvalues we examine the codimension-two point that separates the supercritical and subcritical. We investigate the bifurcation structure and show the existence of further bifurcations of periodic orbits.     

Bifurcation sequences in the interaction of resonances in a model deriving from nonlinear rotordynamics: the zipper

Using numerical continuation we show a new bifurcation scenario involving resonant periodic orbits in a parametrized four-dimensional autonomous system deriving from nonlinear rotordynamics. The scenario consists of a carefully orchestrated sequence of transcritical bifurcations in which branches of periodic solutions are exchanged. Collectively, the bifurcations resemble the action of a zipper. An underlying governing mechanism clearly exists but still has to be uncovered. For a range of parameter values the sequence of bifurcations forms a global connection between a Sil'nikov bifurcation and (partial) mode-locking. The homoclinic bifurcation is introduced into the system by a Takens-Bogdanov bifurcation. The system also features an interaction between two chaotic Sil'nikov bifurcations.     

Mode Interactions With Symmetry

We prove several results concerning problems invariant under the action of an arbitrary compact Lie group Γ. These include the existence of mixed-mode solutions and secondary Hopf bifurcations. We provide a definition of mode interaction which applies to a wide selection of problems and consider the unfolding of the equations characterizing such problems. Where appropriate, we distinguish the case when I acts trivially on one of the modes     

A tertiary hopf bifurcation with applications to problems with symmetries

We develop explicit criteria for the occurrence of a tertiary Hopf bifurcation, and stability of the bifurcating orbits, in a special class of two-parameter systems of ordinary differential equations. We use these results to discuss tertiary Hopf and torus bifurcations in some bifurcation problems with symmetries such as steady-state-Hopf and Hopf-Hopf interaction problems. To analyse (and even detect) these bifurcations we use invariant coordinates and rescaling techniques     

弦-梁耦合系统的动力学行为分析

本文研究了弦-梁耦合系统在初始平衡解处的稳定性与分岔情况,给出了特征值随阻尼参数的变化情况,并利用稳定性分析和特征值分析等解析方法,得到了初始平衡解、周期解和拟周期解的稳定边界以及导致Hopf分岔和2维胎面等分岔解的临界分岔曲线。最后,利用数值模拟方法研究了弦-梁耦合系统的稳定性与分岔情况。     

On Abbe's Theory of Image Formation in the Microscope

We consider a new class of bifurcations that may arise in multiphoton processes inside a coherently driven optical cavity involving more than one mode of the radiation field. For a non-saturable, non-linear medium, a single bifurcation point exists where the symmetric solution bifurcates into a number of non-symmetric solutions. The nature of the bifurcation ranges from a simple pitchfork bifurcation, in the case of four-wave mixing, to more complicated phenomena in higher-order processes. A saturable non-linear medium exhibits similar behaviour for low-input intensities; however, as the input intensity is increased the medium saturates, and at a second bifurcation point the symmetric branch regains its stability. The presence of fluctuations assures the accessibility of the symmetric branch. Thus, for example, for a two-photon absorbing medium we have the possibility of optical tristability involving one symmetric solution and two non-symmetric solutions.     

Bifurcations in dynamical systems with interior symmetry

We propose a definition of interior symmetry in the context of general dynamical systems. This concept appeared originally in the theory of coupled cell networks, as a generalization of the idea of symmetry of a network. The notion of interior symmetry introduced here can be seen as a special form of forced symmetry breaking of an equivariant system of differential equations. Indeed, we show that a dynamical system with interior symmetry can be written as the sum of an equivariant system and a ‘perturbation term’ which completely breaks the symmetry. Nonetheless, the resulting dynamical system still retains an important feature common to systems with symmetry, namely, the existence of flow-invariant subspaces. We define interior symmetry breaking bifurcations in analogy with the definition of symmetry breaking bifurcation from equivariant bifurcation theory and study the codimension one steady-state and Hopf bifurcations. Our main result is the full analogues of the well-known Equivariant Branching Lemma and the Equivariant Hopf Theorem from the bifurcation theory of equivariant dynamical systems in the context of interior symmetry breaking bifurcations.     

A nonlinear dynamical systems framework for structural diagnosis and prognosis

This work aims to establish a nonlinear dynamics framework for diagnosis and prognosis in structural dynamic systems. The objective is to develop an analytically sound means for extracting features, which can be used to characterize damage, from modal-based input-output data in complex hybrid structures with heterogeneous materials and many components. Although systems like this are complex in nature, the premise of the work here is that damage initiates and evolves in the same phenomenological way regardless of the physical system according to nonlinear dynamic processes. That is, bifurcations occur in healthy systems as a result of damage. By projecting a priori the equations of motion of high-dimensional structural dynamic systems onto lower dimensional center, or so-called ‘damage’, manifolds, it is demonstrated that model reduction near bifurcations might be a useful way to identify certain features in the input-output data that are helpful in identifying damage. Normal forms describing local co-dimension one and two bifurcations (e.g. transcritical, subcritical pitchfork, and asymmetric pitchfork bifurcations) are assumed to govern the initiation and evolution of damage in a low-order model. Real-world complications in damage prognosis involving spatial bifurcations, global bifurcation phenomena, and the sensitivity of damage to small changes in initial conditions are also briefly discussed.     

3-mode Interactions with O(2) Symmetry and a Model for Taylor-Couette flow

We analyse the bifurcations of a general ordinary dififerential equation where is equivariant under an action of the group O(2) on. The equation represents the most general nonlinear local interaction of three O(2)-symmetric modes:a steady-state mode with mode-number k, and two periodic (Hopf) modes with mode-numbers l and m. The parameter λ is a bifurcation parameter, and α₁, α₂are unfolding parameters that split the individual modes apart. The system is assumed to be in Birkhoff normal form, so that f also commutes with an action of the 2-torus T². We discuss the existence and stability of bifurcating branches and how these break the O(2) × T² symmetry.Depending on the precise mode-numbers k l m we find up to 31 symmetry classes of possible solutions including six that combine all three modes, and thus cannot be found in any 2-mode interaction. We also discuss the possible occurrence of Sacher-Naimark torus bifurcations, providing a further 10 solution types, and 'slow drift'bifurcations.

This 10-dimensional system can occur generically in O(2)-symmetric bifurcation problems having two extra parameters, and in principle is applicable to a wide range of physical systems. The discussion here is motivated by the observed pattern formation in the Taylor–Couette system, the flow of a fluid contained between coaxial rotating cylinders. It arises by seeking a 'hidden organizing centre' that combines two previous mode-interaction models of this system: a 6-dimensional Hopf–steady-state model due to Chossat and looss (1985) and Golubitsky and Stewart (1986), and an 8-dimensional Hopf–Hopf model due to Chossat, Demay and looss (1987). We interpret the general results on the 10-dimensional system in the context of Taylor–Couette flow, giving schematic pictures of the associated flow patterns. The model incorporates almost all of the observed non-chaotic flows in the Taylor–Couette experiment into a single finite-dimensional dynamical system. It predicts the possible occur- rence of four new flow patterns (corresponding to four of the six possible solutions that combine all three modes). Theseform invariant 3-tori, and may be described as superimposed twisted vortices, superimposed wavy vortices, and two types of twisted wavy vortices. Possible torus bifurcations from states in the 10-dimensional model include various modulated spirals, three types of modulated twisted vortices, three types of modulated wavy vortices, modulated superimposed spirals, modulated interpenetrating spirals, modulated superimposed ribbons, and modulated interpenetrating ribbons. However, whether any of these new states and torus bifurcations can actually occur in Taylor–Couette flow at suitable parameter values, and if so whether they can occur stably, depend upon more detailed numerical calculations than we have performed     

Degenerate bifurcations and catastrophe sets via frequency analysis

We present some bifurcation conditions using the well-known stability analysis of feedback systems. A general ordinary differential equation system is formulated in two parts: one that considers the linear part and the other that includes the memoryless nonlinear part, in a similar way as the describing function. The bifurcation conditions are obtained using the results of the generalized Nyquist stability criterion (GNSC) with some explicit formulae derived from some properties of the complex variable

We analyse simultaneously both static and dynamic (Hopf) bifurcations and their degeneracies in a rich example, a continuous stirred-tank reactor (CSTR), in which two consecutive, irreversible, first-order reactions A→B→C occur     

On the k-jet degeneracy of singular points

This paper proves that the stable set of the origin for a gradient vector field whose k-jet vanishes at the origin is the union of immersed invariant C^k submanifolds of the ambient space. Furthermore, we prove a generalization of the Hopf bifurcation theorem for planar vector fields with vanishing k-jet at the origin. The asymptotic properties of the period of oscillations is determined and a criterion to determine the stability of the bifurcation closed orbits is given. Finally we give bounds for the dimensions of the stable and centre manifolds for a semi-Riemiannian gradient vector field whose first jet is non-vanishing at a singular point