Discretized Fast–Slow Systems with Canards in Two Dimensions

AbstractWe study the problem of preservation of maximal canards for time discretized fast–slow systems with canard fold points. In order to ensure such preservation, certain favorable structure-preserving properties of the discretization scheme are required. Conventional schemes do not possess such properties. We perform a detailed analysis for an unconventional discretization scheme due to Kahan. The analysis uses the blow-up method to deal with the loss of normal hyperbolicity at the canard point. We show that the structure-preserving properties of the Kahan discretization for quadratic vector fields imply a similar result as in continuous time, guaranteeing the occurrence of maximal canards between attracting and repelling slow manifolds upon variation of a bifurcation parameter. The proof is based on a Melnikov computation along an invariant separating curve, which organizes the dynamics of the map similarly to the ODE problem.

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On the singularity structure of Kahan discretizations of a class of quadratic vector fields

European Journal of Mathematics ◽

10.1007/s40879-021-00479-4 ◽

2021 ◽

Author(s):

René Zander

Keyword(s):

Elliptic Curves ◽

Vector Fields ◽

Singularity Structure ◽

Quadratic Vector Fields ◽

Parameter Values

AbstractWe discuss the singularity structure of Kahan discretizations of a class of quadratic vector fields and provide a classification of the parameter values such that the corresponding Kahan map is integrable, in particular, admits an invariant pencil of elliptic curves.

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Contracting on Time

The American Economic Review ◽

10.1257/000282805775014452 ◽

2005 ◽

Vol 95 (5) ◽

pp. 1369-1385 ◽

Cited By ~ 47

Author(s):

Sergei Guriev ◽

Dmitriy Kvasov

Keyword(s):

Detailed Analysis ◽

Continuous Time ◽

Contract Theory ◽

Bilateral Trade ◽

Incomplete Contract ◽

Optimal Contracts ◽

Contract Duration ◽

Unilateral Termination ◽

Advance Notice ◽

The Relationship

The paper shows how time considerations, especially those concerning contract duration, affect incomplete contract theory. Time is not only a dimension along which the relationship unfolds, but also a continuous verifiable variable that can be included in contracts. We consider a bilateral trade setting where contracting, investment, trade, and renegotiation take place in continuous time. We show that efficient investment can be induced either through a sequence of constantly renegotiated fixed-term contracts; or through a renegotiation-proof “evergreen” contract—a perpetual contract that allows unilateral termination with advance notice. We provide a detailed analysis of properties of optimal contracts.

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Marking Vertices to Find Graph Isomorphism Mapping Based on Continuous-Time Quantum Walk

Entropy ◽

10.3390/e20080586 ◽

2018 ◽

Vol 20 (8) ◽

pp. 586 ◽

Cited By ~ 1

Author(s):

Xin Wang ◽

Yi Zhang ◽

Kai Lu ◽

Xiaoping Wang ◽

Kai Liu

Keyword(s):

Polynomial Time ◽

Continuous Time ◽

Graph Isomorphism ◽

Quantum Walk ◽

Isomorphism Problem ◽

Quantum Walks ◽

Regular Graphs ◽

Structure Preserving ◽

Topological Structures ◽

Time Quantum

The isomorphism problem involves judging whether two graphs are topologically the same and producing structure-preserving isomorphism mapping. It is widely used in various areas. Diverse algorithms have been proposed to solve this problem in polynomial time, with the help of quantum walks. Some of these algorithms, however, fail to find the isomorphism mapping. Moreover, most algorithms have very limited performance on regular graphs which are generally difficult to deal with due to their symmetry. We propose IsoMarking to discover an isomorphism mapping effectively, based on the quantum walk which is sensitive to topological structures. Firstly, IsoMarking marks vertices so that it can reduce the harmful influence of symmetry. Secondly, IsoMarking can ascertain whether the current candidate bijection is consistent with existing bijections and eventually obtains qualified mapping. Thirdly, our experiments on 1585 pairs of graphs demonstrate that our algorithm performs significantly better on both ordinary graphs and regular graphs.

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Planar quadratic vector fields with finite saddle connection on a straight line (convex case)

Qualitative Theory of Dynamical Systems ◽

10.1007/bf02972672 ◽

2005 ◽

Vol 6 (2) ◽

pp. 187-204 ◽

Cited By ~ 1

Author(s):

Paulo César Carrião ◽

Maria Elasir Seabra Gomes ◽

Antonio Augusto Gaspar Ruas

Keyword(s):

Vector Fields ◽

Saddle Connection ◽

Quadratic Vector Fields ◽

Straight Line ◽

Convex Case

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Twelve Limit Cycles in 3D Quadratic Vector Fields with Z3 Symmetry

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127418501390 ◽

2018 ◽

Vol 28 (11) ◽

pp. 1850139 ◽

Cited By ~ 3

Author(s):

Laigang Guo ◽

Pei Yu ◽

Yufu Chen

Keyword(s):

Limit Cycles ◽

Vector Fields ◽

Center Manifold ◽

Three Dimensional ◽

Center Manifold Theory ◽

Normal Form Theory ◽

Quadratic Vector Fields ◽

Fine Focus ◽

Form Theory ◽

Manifold Theory

This paper is concerned with the number of limit cycles bifurcating in three-dimensional quadratic vector fields with [Formula: see text] symmetry. The system under consideration has three fine focus points which are symmetric about the [Formula: see text]-axis. Center manifold theory and normal form theory are applied to prove the existence of 12 limit cycles with [Formula: see text]–[Formula: see text]–[Formula: see text] distribution in the neighborhood of three singular points. This is a new lower bound on the number of limit cycles in three-dimensional quadratic systems.

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Symmetries in central-force problems

Serbian Astronomical Journal ◽

10.2298/saj0367047m ◽

2003 ◽

pp. 47-52 ◽

Cited By ~ 2

Author(s):

V. Mioc ◽

M. Barbosu

Keyword(s):

Vector Fields ◽

Body Problem ◽

Blow Up ◽

Central Force ◽

Polar Coordinates ◽

Global Flow ◽

Two Body Problem ◽

Concrete Problem ◽

Central Force Problems ◽

Force Problem

The two-body problem in central fields (reducible to a central-force problem) models a lot of concrete astronomical situations. The corresponding vector fields (in Cartesian and polar coordinates, extended via collision-blow-up and infinity-blow-up transformations) exhibit nice symmetries that form eight-element Abelian groups endowed with an idempotent structure. All these groups are isomorphic, which is not a trivial result, given the different structures of the corresponding phase spaces. Each of these groups contains seven four-element subgroups isomorphic to Klein?s group. These symmetries are of much help in understanding various characteristics of the global flow of the general problem or of a concrete problem at hand, and are essential in searching for periodic orbits.

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Dynamics of a Single Species in a Fluctuating Environment under Periodic Yield Harvesting

Journal of Applied Mathematics ◽

10.1155/2013/167671 ◽

2013 ◽

Vol 2013 ◽

pp. 1-12 ◽

Cited By ~ 2

Author(s):

Mustafa Hasanbulli ◽

Svitlana P. Rogovchenko ◽

Yuriy V. Rogovchenko

Keyword(s):

Differential Equation ◽

Periodic Solutions ◽

Blow Up ◽

Single Species ◽

Bifurcation Parameter ◽

Positive Periodic Solutions ◽

Large Initial Data ◽

Fluctuating Environment ◽

Species Population ◽

Single Species Population

We discuss the effect of a periodic yield harvesting on a single species population whose dynamics in a fluctuating environment is described by the logistic differential equation with periodic coefficients. This problem was studied by Brauer and Sánchez (2003) who attempted the proof of the existence of two positive periodic solutions; the flaw in their argument is corrected. We obtain estimates for positive attracting and repelling periodic solutions and describe behavior of other solutions. Extinction and blow-up times are evaluated for solutions with small and large initial data; dependence of the number of periodic solutions on the parameterσassociated with the intensity of harvesting is explored. Asσgrows, the number of periodic solutions drops from two to zero. We provide bounds for the bifurcation parameter whose value in practice can be efficiently approximated numerically.

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