Nonhyperbolic Case Research Articles

Stability and bifurcations of a host–parasitoid model with general host escape function and general stocking upon parasitoids

This paper analyzes the generalization of a model presented in J. Bektešević, V. Hadžiabdić, S. Kalabušić, M. Mehuljić and E. Pilav [Dynamics of a class of host–parasitoid models with external stocking upon parasitoids, Adv. Differ. Equ. 2021(31) (2021)]. The study explores the behavior of the solution near equilibrium points when the system has different outcomes, such as extinction, infinitely many exclusion points or unique exclusion and coexistence. We prove global stability for the extinction and host-exclusion equilibrium. We also investigate the non-hyperbolic case of parasitoid-exclusion equilibrium and delve deeper into the 1:1 resonance. The transcritical bifurcation occurs at the host-exclusion equilibrium, indicating a threshold for host population invasion through transcritical bifurcation. Moreover, the local dynamics around the coexisting equilibrium can be highly complex due to the appearance of the Neimark–Sacker and period-doubling bifurcations. We provide the explicit form of the first Lyapunov exponent for the Neimark–Sacker bifurcation. Through numerical examples, we illustrate the theoretical findings.

Qualitative properties, bifurcations and chaos of a discrete predator–prey system with weak Allee effect on the predator

In this work, the qualitative properties of the fixed points in the non-hyperbolic and degenerate cases, codimension-one bifurcations and Marotto’s chaos of a discrete predator–prey system with weak Allee effect on the predator are investigated. Utilizing the center manifold theorem and the reduction principle, the qualitative property of each fixed point in the non-hyperbolic case is explored. Based on the approximate flow theory, the qualitative property of the boundary fixed point in the degenerate case is investigated. Making use of the center manifold theorem and the bifurcation theory, all the potential codimension-one bifurcation types of the co-existence fixed point are explored, including flip bifurcation and Neimark–Sacker bifurcation. For each type of these bifurcations, not only the concise non-degenerate condition expressed by the system parameters is given, but also the analytical expression of the system orbit caused by the bifurcation is derived. Based on the non-negativity of the bifurcation critical values, the non-resonant and non-degenerate conditions for the occurrence of these bifurcations, the system parameter plane is divided into several regions using tools such as the polynomial complete discriminant system theory, real root separation theory, dichotomy method, implicit function theorem and Cardan’s formula. In each region, the direction and stability of these bifurcations are studied, which helps to theoretically explain the impact of Allee effect on the system. By proving that the co-existence fixed point is a snap-back repeller under appropriate conditions, it is obtained that the system is chaotic. Numerical simulations are completely consistent with all the theoretical analyses. The research results show that the impact of Allee effect on the population system depends not only on the assumption and construction of the system, but also on the system parameter and initial value of the system.

Directed harmonic currents near non-hyperbolic linearizable singularities

AbstractLet$(\mathbb {D}^2,\mathscr {F},\{0\})$be a singular holomorphic foliation on the unit bidisc$\mathbb {D}^2$defined by the linear vector field$$ \begin{align*} z \frac{\partial}{\partial z}+ \unicode{x3bb} w \frac{\partial}{\partial w}, \end{align*} $$where$\unicode{x3bb} \in \mathbb {C}^*$. Such a foliation has a non-degenerate singularity at the origin${0:=(0,0) \in \mathbb {C}^2}$. LetTbe a harmonic current directed by$\mathscr {F}$which does not give mass to any of the two separatrices$(z=0)$and$(w=0)$. Assume$T\neq 0$. The Lelong number ofTat$0$describes the mass distribution on the foliated space. In 2014 Nguyên (see [16]) proved that when$\unicode{x3bb} \notin \mathbb {R}$, that is, when$0$is a hyperbolic singularity, the Lelong number at$0$vanishes. Suppose the trivial extension$\tilde {T}$across$0$is$dd^c$-closed. For the non-hyperbolic case$\unicode{x3bb} \in \mathbb {R}^*$, we prove that the Lelong number at$0$:(1)is strictly positive if$\unicode{x3bb}>0$;(2)vanishes if$\unicode{x3bb} \in \mathbb {Q}_{<0}$;(3)vanishes if$\unicode{x3bb} <0$andTis invariant under the action of some cofinite subgroup of the monodromy group.

Cross ratios and cubulations of hyperbolic groups

Many geometric structures associated to surface groups can be encoded in terms of invariant cross ratios on their circle at infinity; examples include points of Teichmüller space, Hitchin representations and geodesic currents. We add to this picture by studying cocompact cubulations of arbitrary Gromov hyperbolic groups G. Under weak assumptions, we show that the space of cubulations of G naturally injects into the space of G-invariant cross ratios on the Gromov boundary partial _{infty }G. A consequence of our results is that essential, hyperplane-essential, cocompact cubulations of hyperbolic groups are length-spectrum rigid, i.e. they are fully determined by their length function. This is the optimal length-spectrum rigidity result for cubulations of hyperbolic groups, as we demonstrate with some examples. In the hyperbolic setting, this constitutes a strong improvement on our previous work [4]. Along the way, we describe the relationship between the Roller boundary of a mathrm{CAT(0)} cube complex, its Gromov boundary and—in the non-hyperbolic case—the contracting boundary of Charney and Sultan. All our results hold for cube complexes with variable edge lengths.

Inertia‐based spectrum slicing for symmetric quadratic eigenvalue problems

SummaryIn the quadratic eigenvalue problem (QEP) with all coefficient matrices symmetric, there can be complex eigenvalues. However, some applications need to compute real eigenvalues only. We propose a Lanczos‐based method for computing all real eigenvalues contained in a given interval of large‐scale symmetric QEPs. The method uses matrix inertias of the quadratic polynomial evaluated at different shift values. In this way, for hyperbolic problems, it is possible to make sure that all eigenvalues in the interval have been computed. We also discuss the general nonhyperbolic case. Our implementation is memory‐efficient by representing the computed pseudo‐Lanczos basis in a compact tensor product representation. We show results of computational experiments with a parallel implementation in the SLEPc library.

Superfractal Approximation of Functions

The methods of superfractal approximation of sets introduced in 2005–2011 by M. Barnsley, et al. are modified for the approximation of functions. Nonlinear operators are introduced in the space of bounded functions. The limit behavior of this operator sequence is investigated in a function space (in a sense of pointwise and uniform convergence). We consider a nonhyperbolic case in which not all plane maps specifying the operator in the function space are contractive and propose sufficient conditions for the convergence of approximations and estimates of the errors for this kind of approximation (similar to the collage theorem for fractal approximation).

Convergence of rank-type equations

Convergence results are presented for rank-type difference equations, whose evolution rule is defined at each step as the kth largest of p univariate difference equations. If the univariate equations are individually contractive, then the equation converges to a fixed point equal to the kth largest of the individual fixed points of the univariate equations. Examples are max-type equations for k = 1, and the median of an odd number p of equations, for k = ( p + 1)/2. In the non-hyperbolic case, conjectures are stated about the eventual periodicity of the equations, generalizing long-standing conjectures of G. Ladas.

Leading coefficient problem for polynomial-like iterative equations

Because of difficulties in applying fixed point theorems, most of known results on the polynomial-like iterative equation ∑ j = 1 n λ j f j ( x ) = F ( x ) were given by assuming λ 1 > 0 and the existence of solutions under the most natural assumption λ n > 0 is an interesting problem, called “Leading Coefficient Problem”. For this problem locally expansive invertible C 1 solutions are obtained in the expansive case and the non-hyperbolic case in [W. Zhang, On existence for polynomial-like iterative equations, Results Math. 45 (2004) 185–194] and C 0 increasing solutions are constructed in [B. Xu, W. Zhang, Construction of continuous solutions and stability for the polynomial-like iterative equation, J. Math. Anal. Appl. 325 (2007) 1160–1170]. In this paper we discuss C 1 solutions for more combinations between expansive mappings and contractive ones and combinations between increasing mappings and decreasing ones.

Isolated periodic minima are unstable

A classical result, studied, among others, by Carathéodory [C. Carathéodory, Calculus of Variations and Partial Differential Equations of the First Order, Chelsea, New York, 1989], says that, at least generically, periodic minimizers are hyperbolic, and consequently, unstable as solutions of the associated Euler–Lagrange equation. A new version of this fact, also valid in the nonhyperbolic case, is given.

Trajectory-based optimal linearization for nonlinear autonomous vector fields

This paper deals with an optimal approximation in the least square sense of nonlinear vector fields. The optimal approximation consists of a linearization along a trajectory that approximates the nonlinear solution from the initial state to the equilibrium position. It is shown that the optimal linearization can be seen as a generalization of the classical linearization. Furthermore, the optimal linearization can approximate the derivative at the equilibrium point, and the order of the method is the same as the nonlinearity, since the approximation depends on the initial state. We also show that the method can be used to study the asymptotic stability of the equilibrium of a nonlinear vector fields, especially in the nonhyperbolic case. Simulation shows good agreement between the linearized and the nonlinear systems.

An L 2 Norm Trajectory-Based Local Linearization for Low Order Systems

This paper presents a linear transformation for low order nonlinear autonomous differential equations. The procedure consists of a trajectory-based local linearization, which approximates the nonlinear system in the neighborhood of its equilibria. The approximation is possible even in the non-hyperbolic case which is of a particular interest. The linear system is derived using an L 2 norm optimization and the method can be used to approximate the derivative at the equilibrium position. Unlike the classical linearization, the L 2 norm linearization depends on the initial state and has the same order as the nonlinearity. Simulation results show good agreement of the suggested method with the nonlinear system.

On the hadamard—perron lemma

A formal generalization of the Hadamard—Perron lemma to the nonhyperbolic case is considered.

A Family of Linearizations of Autonomous Ordinary Differential Equations with Scalar Nonlinearity

This paper deals with a method for the linearization of nonlinear autonomous differential equations with a scalar nonlinearity. The method consists of a family of approximations which are time independent, but depend on the initial state. The family of linearizations can be used to approximate the derivative of the nonlinear vector field, especially at equilibrium points, which are of particular interest, it can be used also to determine the asymptotic stability of equilibrium point, especially in the non-hyperbolic case. Using numerical experiments, we show that the method presents good agreement with the nonlinear system even in the case of highly nonlinear systems.

ON THERMODYNAMICS OF RATIONAL MAPS. II: NON-RECURRENT MAPS

The pressure function p(t )o fa non-recurrent map is real analytic on some interval (0 ,t ∗ )w it ht∗ strictly greater than the dimension of the Julia set. The proof is an adaptation of the well known tower techniques to the complex dynamics situation. In general, p(t )n eed not be analytic on the whole positive axis. In this paper we study analyticity properties of the pressure function of non-recurrent maps. Our approach is based on the well known tower techniques adapted to the complex dynamics situation. The pressure function p(t), which is defined in terms of the Poincar´ es erie s( see (1.4)), carries essential information about ergodic and dimensional properties of the maximal measure. In particular, it characterizes the dimension spectrum of harmonic measure on the Julia set in the case of a polynomial dynamics. According to the classical theory of Sinai, Ruelle and Bowen, p(t )i s real analytic if the dynamics is hyperbolic ,t hat is, expanding on the Julia set. This fact is closely related to the so called ‘spectral gap’ phenomenon, which also implies other important features of hyperbolic dynamics such as the existence of equilibrium states, exponential decay of correlations, etc. The problem of extending (some parts of) the classical theory to the non-hyperbolic case has become one of the central themes in the ergodic theory of conformal dynamics. In the first part of this work [8], we provided a detailed analysis of the negative part t 0 is substantially more complicated (and more important). The main difficulty arises from the presence of singularities (critical points) on the Julia set. To circumvent this difficulty, we propose to use a tower construction which forces the dynamics to be expanding on some auxiliary space. The tower method has been widely used in the general theory of dynamical systems with some degree of hyperbolicity (see especially [12]), and in particular in 1-dimensional real dynamics, where the construction is known as Hof bauer’s tower .T oa pply this method in the complex case, it is natural to use some basic elements of the Yoccoz jigsaw puzzle structure (see [9]). We will discuss only the simplest type of non-hyperbolic behavior: the case of nonrecurrent dynamics (every critical point in the Julia set is non-recurrent, that is, is away from its iterates) without parabolic cycles (see [2 ]f orvarious characterizations

Holomorphic foliations desingularized by punctual blow-ups

Given F be a germ of codimension-one singular holomorphic foliation at the origin 0∈ C 3 . We assume that F can be desingularized by a certain sequence of punctual blow-ups producing only simple singularities (Definition 1). This case is studied in analogy with the case of Kleinian singularities of complex surfaces. It is proved that F is given by a simple poles closed meromorphic 1-form provided that, along the reduction process, the simple singularities exhibit a hyperbolic transverse type (Theorem 3). In the non-hyperbolic case, we prove the existence of a formal integrating factor if we interdict the existence of holomorphic first integrals for the transverse types (Theorem 4). The proof relies strongly on a result of Deligne regarding the fundamental group of the complement of algebraic curves in the complex projective plane.

Convergence of solutions in a nonhyperbolic case

Convergence of solutions in a nonhyperbolic case

Quantum manifestations of chaotic scattering.

We examine quantum chaotic scattering in the semiclassical regime for the two cases where the classical scattering is hyperbolic and nonhyperbolic. It is shown that in the nonhyperbolic case the energy-dependent S-matrix autocorrelation function C(\ensuremath{\varepsilon}) exhibits a cusp-shaped peak at \ensuremath{\varepsilon}=0 (where \ensuremath{\varepsilon} denotes the energy difference). This indicates that the fine scale fluctuations with energy of the S matrix are characteristically greatly enhanced in the nonhyperbolic case as compared with the hyperbolic case.

Fractal dimension in nonhyperbolic chaotic scattering.

In chaotic scattering there is a Cantor set of input-variable values of zero Lebesgue measure (i.e., zero total length) on which the scattering function is singular. For cases where the dynamics leading to chaotic scattering is nonhyperbolic (e.g., there are Kolmogorov-Arnol'd-Moser tori), the nature of this singular set is fundamentally different from that in the hyperbolic case. In particular, for the nonhyperbolic case, although the singular set has zero total length, we present strong evidence that its fractal dimension is 1.

On constructing Markov partitions by computer

Two methods to construct Markov partitions for two-dimensional systems are proposed. One is based on the existence of a known, or easily accessible by numerical analysis, hyperbolic fixed point; the other one, which is more general, is derived from Bowen's proof of the existence theorem of Markov partitions for hyperbolic systems. The methods are successfully implemented in two cases of hyperbolic systems: the linear automorphism $$\left( {\begin{array}{*{20}c} 1 & 1 \\ 1 & 2 \\ \end{array} } \right)$$ of the 2-torus and a nonlinear perturbation of it. The methods are applied also to the Henon mapping. In such nonhyperbolic case, however, they produce partitions of the Henon attractor which lack some essential properties.