This note presents closed-form formulas for determining the critical points of general ndimensional differential equations. The formulas do not require commutating the eigenvalues of the Jacobian of a system. Based on the Hurwitz criterion, explicit necessary and sufficient conditions are obtained. Particular attention is focused on Hopf and double Hopf bifurcations. A model of induction machin...

where K0 = ⋃ t>0 tK is a closed convex cone. The typical problem one has to face is twofold: (1) find eigenvalues (which is nontrivial, unless K0 is a linear space): (2) ensure that some eigenvalues are bifurcation points (which is not always true, as counterexamples show). Much work has been done in this context; see [11], [15]–[18], [21]–[30], [32]–[34] and the references therein for a more c...

We consider a two-parametric family of one-dimensional piecewise smooth maps with one discontinuity point. The bifurcation structures in a parameter plane of the map are investigated, related to codimension-2 bifurcation points defined by the intersections of two border collision bifurcation curves. We describe the case of the collision of two stable cycles of any period and any symbolic sequen...

A number of parametric effects, such as suppression of period doubling, shift of the bifurcation point, scaling law relating the shift and the perturbation amplitude, influence of the detuning on the suppression, reaching of the maximum gain between the original and shifted bifurcation points, and scaling law for idler power are experimentally observed near period doubling bifurcation in a loss...

For differential-algebraic power systems, saddle-noddle bifurcation and Hopf bifurcation are both of universally existent phenomena in power systems. Usually Newton iteration method could be applied to the Moore-Spence system to compute saddle-noddle and Hopf bifurcation points directly. But the Moore-Spence system has very high dimension and causes much complexity in Jacobian matrix factorizat...

Resonance tongues are mode-locking regions of parameter space in which stable periodic solutions occur; they commonly occur, for example, near Neimark-Sacker bifurcations. For piecewise-smooth, continuous maps these tongues typically have a distinctive lens-chain (or sausage) shape in two-parameter bifurcation diagrams. We give a symbolic description of a class of “rotational” periodic solution...

This article investigates the indifference-attractor bifurcation of infinite horizon discrete-time optimal control problems with a single state variable. We show that these bifurcations are linked to a heteroclinic bifurcation scenario of the statecostate equations, and we analyse the consequences for the optimal solutions. In particular, we can characterise the bifurcation value at which indif...

This paper is devoted to the study of spatiotemporal dynamics of a diffusive Leslie–Gower predator–prey system with ratio-dependent Holling type III functional response under homogeneous Neumann boundary conditions. It is shown that the model exhibits spatial patterns via Turing (diffusion-driven) instability and temporal patterns via Hopf bifurcation. Moreover, the existence of spatiotemporal ...

| Bifurcations of equilibrium points and limit cycles in unidirectionally coupled three oscillators are studied. According to their symmetrical properties, we classify equilibrium points and limit cycles into three and eight di erent types, respectively. Possible oscillations in unidirectional coupled three oscillators are presented by calculating Hopf bifurcation sets of equilibrium points. We...

In this paper, we present an efficient method to compute singular points and singularity induced bifurcation points of differential-algebraic equations (DAEs) for a multimachine power system model. The algebraic part of the DAEs brings singularity issues into dynamic stability assessment of power systems. Roughly speaking, the singular points are points that satisfy the algebraic equations, but...