Narrow Results

This paper introduces a new practical method for distinguishing chaotic, periodic and quasi-periodic orbits based on a new criterion, and apply it to investigate the local bifurcations of the Chen system. Conditions for supercritical and subcritical bifurcations are obtained, with their parameter domains specified. The analytic results are also verified by numerical simulation studies.

A structure-induced bifurcation of nonsmooth nonlinear systems is studied and illustrated on electric power system models. The consequence of structure-induced bifurcation is an immediate instability induced by generator reactive power limits. It is numerically shown that structure-induced bifurcation can occur at both small power systems and large-scale power systems. Without taking the structure-induced bifurcation into account in defining power system operating limits, the resulting operating limits can be overly optimistic.

The local bifurcation boundary and steady-state security boundary of parameter-dependent electric power system models are computed and studied. A computational procedure for numerically constructing the local bifurcation boundary and the steady-state security boundary is proposed. Then the proposed computational procedure is applied to large power systems to compute the local bifurcation boundary and steady-state security boundary. Numerical studies reveal the characteristics and the convexity properties of these boundaries. The impact of the physical limitation of the generator reactive capability on the local bifurcation boundary and the steady-state security boundary are also investigated.

In this article we extend the theory of local bifurcation in one-dimensional autonomous maps to one-dimensional nonautonomous periodic maps. We give the necessary conditions for the main types of local bifurcation in one-dimensional periodic maps.

In continuous-time dynamical systems, a periodic orbit becomes a fixed point on a certain Poincaré section. The eigenvalues of the Jacobian matrix at this fixed point determine the local stability of the periodic orbit. Analogously, a quasi-periodic orbit (2-torus) becomes an invariant closed curve (ICC) on a Poincaré section. From the Lyapunov exponents of an ICC, we can determine the time average of the exponential divergence rate of the orbit, which corresponds to the eigenvalues of a fixed point. We denote the Lyapunov exponent with the smallest nonzero absolute value as the Dominant Lyapunov Exponent (DLE). A local bifurcation manifests as a crossing or touch of the DLE locus with zero. However, the type of bifurcation cannot be determined from the DLE. To overcome this problem, we define the Dominant Lyapunov Bundle (DLB), which corresponds to the dominant eigenvectors of a fixed point. We prove that the DLB of a 1-torus in a map can be classified into four types: A⁺ (annulus and orientation preserving), A^- (annulus and orientation reversing), M (Möbius band), and F (focus). The DLB of a 2-torus in a flow can be classified into three types: A⁺ × A⁺, A^- × M (equivalently M × A^- and M × M), and F × F. From the results, we conjecture the possible local bifurcations in both cases. For the 1-torus in a map, we conjecture that type A⁺ and A^- DLBs correspond to a saddle-node and period-doubling bifurcations, respectively, whereas a type M DLB denotes a double-covering bifurcation, and type F relates to a Neimark–Sacker bifurcation. Similarly, for the 2-torus in a flow, we conjecture that type A⁺ × A⁺ DLBs correspond to saddle-node bifurcations, type A^- × M DLBs to double-covering bifurcations, and type F × F DLBs to the Neimark–Sacker bifurcations. After introducing the mathematical concepts, we provide a DLB-calculating algorithm and illustrate all of the above bifurcations by examples.

Recent years have witnessed a growing trend towards the development and deployment of distributed generation (DG). It is shown that electric distribution networks with DGs can encounter two types of local bifurcations: saddle-node bifurcation and structure-induced bifurcation. The structure-induced bifurcation occurs when a transition between two structures of the distribution network takes place due to limited amount of reactive power supports from renewable energies. The saddle-node bifurcation occurs when the underlying distribution network reaches the limit of its delivery capability. The consequence of structure-induced bifurcation is an immediate instability induced by reactive power limits of renewable energy. It is numerically shown that both types of local bifurcations can occur at both small distribution networks and large-scale distribution networks with DGs. Physical explanations of these two local bifurcations are provided. Studies of local bifurcations in distribution networks provide insights regarding how to design controls to enhance distribution networks with DGs.

We classify the local bifurcations of quasi-periodic $d$-dimensional tori in maps (abbr. MT$d$) and in flows (abbr. FT$d$) for $d\ge 1$. It is convenient to classify these bifurcations into normal bifurcations and resonance bifurcations. Normal bifurcations of MT$d$ can be classified into four classes: namely, saddle-node, period doubling, double covering, and Neimark–Sacker bifurcations. Furthermore, normal bifurcations of FT$d$ can be classified into three classes: saddle-node, double covering, and Neimark–Sacker bifurcations. These bifurcations are determined by the type of the dominant Lyapunov bundle. Resonance bifurcations are well known as phase locking of quasi-periodic solutions. These bifurcations are classified into two classes for both MT$d$ and FT$d$: namely, saddle-node cycle and heteroclinic cycle bifurcations of the $(d-1)$-dimensional tori. The former is reversible, while the latter is irreversible. In addition, we propose a method for analyzing higher-dimensional tori, which uses one-dimensional tori in sections (abbr. ST$1$) and zero-dimensional tori in sections (abbr. ST$0$). The bifurcations of ST$1$ can be classified into five classes: saddle-node, period doubling, component doubling, double covering, and Neimark–Sacker bifurcations. The bifurcations of ST$0$ can be classified into four classes: saddle-node, period doubling, component doubling, and Neimark–Sacker bifurcations. Furthermore, we clarify the relationship between the bifurcations of ST$1$/ST$0$ and the bifurcations of MT$d$/FT$d$. We present examples of all of these bifurcations.

Local bifurcations occur in power systems, causing changes in power system dynamic behaviors. These local bifurcations include the saddle-node bifurcation, Hopf bifurcation, and structure-induced bifurcation. This paper presents a new type of bifurcation that can occur in the optimal power flow (OPF) problem. This new type of bifurcation, termed pseudo-pitchfork bifurcation, involves bifurcations of the feasible components of the OPF problem and the disappearance of local optimal power flow solutions. The main features of this special type of bifurcation are demonstrated on several power systems with different loading condition parameters and different constraint parameters. Then the computation consideration and physical meaning of the pseudo-pitchfork bifurcation are roughly discussed. It is also demonstrated that a pseudo-pitchfork bifurcation occurring between feasible components can help us interpret the loss or birth of optimal power flow solutions and can lead to powerful numerical methods for computing high-quality optimal power flow solutions.

Chaos is ubiquitous in nonlinear dynamical systems. It is a hidden state of deterministic disorder in certain regions of parameters and initial conditions of a nonlinear system. The path taken by the system to reach chaos is known as its route to chaos. The state of the system can drastically change from a stable fixed point or a periodic solution to chaos due to the effect of local or global bifurcation. The available tools, such as bifurcation diagrams, Lyapunov spectra, Poincaré maps, phase plots, and time series plots, explore the routes for such qualitative changes. These are tiresome procedures. For more efficient verifications, this work offers a guide or catalog for several routes to chaos (RTC) through which a continuous-time dynamical system enters or exits from chaos. Additionally, this review paper highlights various signatures associated with common RTC to make identification easier. The main emphasis is on enabling a reader to assess these RTC by merely observing the explained signatures in the bifurcation diagrams and the Poincaré maps.

In this paper, we study stability, bifurcation and spikes of positive stationary solutions of the following parabolic–elliptic chemotaxis system with singular sensitivity and logistic source:

We study local invariants of planar caustics, that is, invariants of Lagrangian maps from surfaces to ${ℝ}^2$ whose increments in generic homotopies are determined entirely by diffeomorphism types of local bifurcations of the caustics. Such invariants are dual to trivial codimension 1 cycles supported on the discriminant in the space $ℒ$ of the Lagrangian maps.

We obtain a description of the spaces of the discriminantal cycles (possibly non-trivial) for the Lagrangian maps of an arbitrary surface, both for the integer and mod 2 coefficients. It is shown that all integer local invariants of caustics of Lagrangian maps without corank 2 points are essentially exhausted by the numbers of various singular points of the caustics and the Ohmoto–Aicardi linking invariant of ordinary maps. As an application, we use the discriminantal cycles to establish non-contractibility of certain loops in $ℒ$.