Some open problems in low dimensional dynamical systems

DOI10.1007/s40324-021-00244-3zbMath1487.37024arXiv2012.02524OpenAlexW3138940724MaRDI QIDQ2062256

Armengol Gasull

Publication date: 27 December 2021

Published in: S\(\vec{\text{e}}\)MA Journal (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/2012.02524

zbMATH Keywords

centerlimit cycleglobal asymptotic stabilityopen problemsAbel differential equationconjecturesperiod functionfewnomialspiecewise linear differential equation

Mathematics Subject Classification ID

Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations (34C07) Dynamics induced by flows and semiflows (37C10) Periodic orbits of vector fields and flows (37C27) Generic properties, structural stability of dynamical systems (37C20) Attractors of solutions to ordinary differential equations (34D45) Dynamical aspects of attractors and their bifurcations (37G35) Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.) (34C08) Solving polynomial systems; resultants (13P15) Lists of open problems (00A27)

Related Items (10)

Rational solutions of Abel differential equations ⋮ New criterions on stability and order of analytic nilpotent foci ⋮ Counting configurations of limit cycles and centers ⋮ Stability of singular limit cycles for Abel equations revisited ⋮ On the uniqueness of limit cycles in second-order oscillators ⋮ Hilbert number for a family of piecewise nonautonomous equations ⋮ Critical periods of the sum of two quasi-homogeneous Hamiltonian vector fields ⋮ Planar systems and Abel equations ⋮ On the Hilbert number for piecewise linear vector fields with algebraic discontinuity set ⋮ Limit cycles in piecewise polynomial Hamiltonian systems allowing nonlinear switching boundaries

Cites Work

• Nonsmooth mechanics. Models, dynamics and control
• An upper bound of the index of an equilibrium point in the plane
• From Abel equations to Jacobian conjecture
• Global behaviour of the period function of the sum of two quasi-homogeneous vector fields
• Periodic solutions of singular equations
• On a Dulac function for the Kukles system
• On Poncelet's maps
• Classical Liénard equations of degree \(n\geqslant 6\) can have \([\frac{n-1}{2}+2\) limit cycles]
• On the Casas-Alvero conjecture
• Uniqueness of limit cycles for Liénard differential equations of degree four
• Limit cycles of quadratic systems in the plane
• A new uniqueness criterion for the number of periodic orbits of Abel equations
• Limit cycles of differential equations
• Configurations of fans and nests of limit cycles for polynomial vector fields in the plane
• Nonsmooth modeling and simulation for switched circuits.
• On the number of critical periods for planar polynomial systems of arbitrary degree
• Center configurations of Hamiltonian cubic systems
• Upper bounds for the number of limit cycles through linear differential equations
• A cubic system with thirteen limit cycles
• Particle trajectories in linear water waves
• The period in the Lotka-Volterra system is monotonic
• On the number of solutions of the equation \(\sum^n_{j=0}a_j(t)x^j,0\leq t\leq 1\), for which \(x(0)=x(1)\)
• A proof of a conjecture of Loewner and of the conjecture of Caratheodory on umbilic points
• An algebraic formula for the degree of a \(C^\infty\) map germ. Sur une inegalite a la Minkowski pour les multiplicites
• The period function of a Hamiltonian quadratic system
• On polynomial Hamiltonian planar vector fields
• Planar vector field versions of Carathéodory's and Loewner's conjectures
• The period function for Hamiltonian systems with homogeneous nonlinearities
• A polynomial counterexample to the Markus-Yamabe conjecture
• An umbilical point on a non-real-analytic surface
• Counting real connected components of trinomial curve intersections and \(m\)-nomial hypersurfaces
• Non-smooth dynamical systems
• Polynomial automorphisms and the Jacobian conjecture
• Mathematical problems for the next century
• A simple counterexample to Kouchnirenko's conjecture
• Limit cycles for rigid cubic systems
• Invariant algebraic curves of large degree for quadratic system
• A new criterion for controlling the number of limit cycles of some generalized Liénard equations
• A solution to the bidimensional global asymptotic stability conjecture
• The period function of some polynomial systems of arbitrary degree
• Asymptotic stability of linearizations of a planar vector field with a singular point implies global stability
• Phase portraits of planar semi-homogeneous vector fields. II
• The number of periodic solutions of polynomial differential equations
• Limit cycles for some families of smooth and non-smooth planar systems
• Higher order Melnikov analysis for planar piecewise linear vector fields with nonlinear switching curve
• A survey on algebraic and explicit non-algebraic limit cycles in planar differential systems
• New lower bound for the number of critical periods for planar polynomial systems
• Periodic orbits of discrete and continuous dynamical systems via Poincaré-Miranda theorem
• Asymptotic lower bounds on Hilbert numbers using canard cycles
• Transversal conics and the existence of limit cycles
• A counterexample for singular equations with indefinite weight
• Piecewise-smooth dynamical systems. Theory and applications
• Limit cycles of the classical Liénard differential systems: a survey on the Lins Neto, de Melo and Pugh's conjecture
• Melnikov analysis in nonsmooth differential systems with nonlinear switching manifold
• The Casas-Alvero conjecture for infinitely many degrees
• Lower bounds for the number of limit cycles of trigonometric Abel equations
• On some open problems in planar differential systems and Hilbert's 16th problem
• The period function of reversible quadratic centers
• Qualitative theory of planar differential systems
• Sur les points singuliers multiples de systèmes dynamiques dans \(R^ 2\)
• The Gaussian moments conjecture and the Jacobian conjecture
• An explicit bound of the number of vanishing double moments forcing composition
• On Periodic Billiard Trajectories in Obtuse Triangles
• A note on LaSalle's problems
• Some Applications of the Extended Bendixson-Dulac Theorem
• Different Approaches to the Global Periodicity Problem
• Periodic Solutions of a Singular Equation With Indefinite Weight
• The periods of the Volterra-Lotka system.
• Limit Cycles for a Class of Abel Equations
• Bifurcation of critical periods from Pleshkan's isochrones
• More limit cycles than expected in Liénard equations
• DYNAMICS OF SOME RATIONAL DISCRETE DYNAMICAL SYSTEMS VIA INVARIANTS
• Obtuse Triangular Billiards II: One Hundred Degrees Worth of Periodic Trajectories
• Periodic billiard orbits are dense in rational polygons
• The discrete Markus–Yamabe problem
• Some Applications of the Euler-Jacobi Formula to Differential Equations
• Limit cycles for vector fields with homogeneous components
• Centennial History of Hilbert's 16th Problem
• On the integrability of Birkhoff billiards
• Simple Proofs of the Hadamard and Poincaré–Miranda Theorems Using the Brouwer Fixed Point Theorem
• Algebraic Limit Cycles in Piecewise Linear Differential Systems
• New lower bounds for the Hilbert numbers using reversible centers
• HILBERT'S 16TH PROBLEM AND BIFURCATIONS OF PLANAR POLYNOMIAL VECTOR FIELDS
• On Periodic Rational Difference Equations of Orderk
• A proof of the two-dimensional Markus-Yamabe Stability Conjecture and a generalization
• Polynomial systems: a lower bound for the Hilbert numbers
• On the Number of Limit Cycles in Generalized Abel Equations
• Stability index of linear random dynamical systems
• Around some extensions of Casas-Alvero conjecture for non-polynomial functions
• LIMIT CYCLES FOR GENERALIZED ABEL EQUATIONS
• LIMIT CYCLES FOR SOME ABEL EQUATIONS HAVING COEFFICIENTS WITHOUT FIXED SIGNS
• On rational functions orthogonal to all powers of a given rational function on a curve
• On Sloane’s Persistence Problem
• Palindromes by Addition
• Palindromes by Addition in Base Two
• Higher order polar germs
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