相关论文: The Pitchfork Bifurcation
We consider the Dirichlet problem for semilinear elliptic equations on a bounded domain which is diffeomorphic to a ball and investigate bifurcation from a given (trivial) branch of solutions, where the radius of the ball serves as…
Bifurcation equations, non-degeneracy and transversality conditions are obtained for the fold, transcritical, pitchfork and flip bifurcations for periodic points of one dimensional implicitly defined discrete dynamical systems. The backward…
Starting with a novel definition of divided differences, this essay derives and discusses the basic properties of, and facts about, (univariate) divided differences.
The notion of concept drift refers to the phenomenon that the distribution, which is underlying the observed data, changes over time; as a consequence machine learning models may become inaccurate and need adjustment. While there do exist…
A systematic theory is introduced for calculating the derivatives of quaternion matrix function with respect to quaternion matrix variables. The proposed methodology is equipped with the matrix product rule and chain rule and it is able to…
A q-difference analogue of the Painlev\'e III equation is considered. Its derivations, affine Weyl group symmetry, and two kinds of special function type solutions are discussed.
We propose a simple computational procedure in order to resolve the degeneracy, which invariably exists on the background of fluid motion associated with a channel of infinite extent. The procedure is applied to elucidate the bifurcation…
The bifurcation diagram of a model nonlinear Langevin equation with delayed feedback is obtained numerically. We observe both direct and oscillatory bifurcations in different ranges of model parameters. Below threshold, the stationary…
Recent developments in Seiberg-Witten theory and relations with Complex Geometry.
The notion of symmetry in polynomial rings with several indeterminates is generalized to polynomial rings over finite fields. Families of extensions of the projective line over a finite field of constants possessing this property are…
The paper investigates emergence of pitchfork bifurcations in the chemical systems. The systems may be split conditionally by two groups, the strong and the weak, depending upon system complexity and its potential response to the external…
Symmetry breaking is reported for continuous families of solitons in the nonlinear Schr\"odinger equation with a two-dimensional complex potential. This symmetry-breaking bifurcation is forbidden in generic complex potentials. However, for…
This paper develops a new divergence that generalizes relative entropy and can be used to compare probability measures without a requirement of absolute continuity. We establish properties of the divergence, and in particular derive and…
Standard bifurcation theory is concerned with families of vector fields $dx/dt = f(x,\lambda)$, $x \in \R^n$, involving one or several constant real parameters $\lambda$. Viewed as a differential equation for the pair $(x,\lambda)$, we…
A new approach to disintegration of measures is presented, allowing one to drop the usually taken separability assumption. The main tool is a result on fibers in the spectrum of algebra of essentially bounded functions established recently…
We introduce a new non-degeneracy condition at infinity for a real or a mixed polynomial mapping $F$ which allows us to approximate its bifurcation locus in terms of certain Newton polyhedra. We derive a sufficiency result for the Jacobian…
Arclength continuation and branch switching are enormously successful algorithms for the computation of bifurcation diagrams. Nevertheless, their combination suffers from three significant disadvantages. The first is that they attempt to…
We prove new results on the derivative of the Minkowski question mark function. Some of our theorems are non-improvable.
We numerically study bifurcations of attractors of the H\'enon map with additive bounded noise with spherical reach. The bifurcations are analysed using a finite-dimensional boundary map. We distinguish between two types of bifurcations:…
We consider the system -\Delta u_j + a(x)u_j = \mu_j u_j^3 + \be\sum_{k\ne j}u_k^2u_j, u_j>0, \qquad j=1,...,n, on a possibly unbounded domain $\Om\subset\R^N$, $N\le3$, with Dirichlet boundary conditions. The system appears in nonlinear…