From Abel's differential equations to Hilbert's 16th problem

Periodic solutions to ordinary differential equations (34C25) Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations (34C07) Periodic orbits of vector fields and flows (37C27)

Cites Work

Non-autonomous equations related to polynomial two-dimensional systems
Polynomial systems: a lower bound for the Hilbert numbers
On the Number of Limit Cycles in Generalized Abel Equations
The Number of Periodic Solutions of the Equation Ż=z ^N +p ₁ (t )z ^{N −1} +…+p _N (t )
LIMIT CYCLES FOR GENERALIZED ABEL EQUATIONS
LIMIT CYCLES FOR SOME ABEL EQUATIONS HAVING COEFFICIENTS WITHOUT FIXED SIGNS
Resonance and nonlinearity: A survey
Lines of principal curvature on canal surfaces in R³
Hilbert's sixteenth problem
Bifurcation of unimodal maps
An introduction to the study of stellar structure.
Limit Cycles of Differential Equations
Nests of limit cycles in quadratic systems
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Application of the Abel equation of the 1st kind to inflation analysis of non-exactly solvable cosmological models
Periods of solutions of periodic differential equations.
Non-algebraic oscillations for predator-prey models
Further considerations on the number of limit cycles of vector fields of the form \(X(v)=Av+f(v)Bv\)
Quadratic vector fields in the plane have a finite number of limit cycles
Relativistic dissipative cosmological models and Abel differential equation
Classical Liénard equations of degree \(n\geqslant 6\) can have \([\frac{n-1}{2}+2\) limit cycles]
A new Chebyshev family with applications to Abel equations
Uniqueness of limit cycles for Liénard differential equations of degree four
Limit cycles for nonsmooth differential equations via Schwarzian derivative
Canard cycles for predator-prey systems with Holling types of functional response
A simple solution of some composition conjectures for Abel equations
Periodic orbits for perturbed non-autonomous differential equations
Limit cycles for two families of cubic systems
Periodic orbits in complex Abel equations
Explicit non-algebraic limit cycles for polynomial systems
A new uniqueness criterion for the number of periodic orbits of Abel equations
On the determination of the number of periodic (or closed) solutions of a scalar differential equation with convexity
Iterative approximation of limit cycles for a class of Abel equations
A cubic system with thirteen limit cycles
Quaternionic-valued ordinary differential equations. The Riccati equation
From the sixteenth Hilbert problem to tropical geometry
Limit cycles of vector fields of the form \(X(v)=Av+f(v)Bv\)
On the number of solutions of the equation \(\sum^n_{j=0}a_j(t)x^j,0\leq t\leq 1\), for which \(x(0)=x(1)\)
The Riccati equation
Cubic systems and Abel equations
Hilbert's 16th problem for quadratic vector fields
The limit cycle of the van der Pol equation is not algebraic
Limit cycles in the Holling-Tanner model
Bifurcation of planar vector fields and Hilbert's sixteenth problem
A relation between small amplitude and big limit cycles.
On the distribution and number of limit cycles for quadratic systems with two foci
On ``relaxation-oscillations.
Abel ODEs: Equivalence and integrable classes
Mathematical problems for the next century
Mathematical biology. Vol. 1: An introduction.
Periodic solutions of linear, Riccati, and Abel dynamic equations
Uniqueness of limit cycles for quadratic vector fields
Limit cycles for rigid cubic systems
Some theorems on the existence, uniqueness, and nonexistence of limit cycles for quadratic systems
Nonexistence of periodic solutions of a complex Riccati equation
Poincaré replies to Hilbert: on the future of mathematics ca. 1908
The number of periodic solutions of polynomial differential equations
Some open problems in low dimensional dynamical systems
Planar systems and Abel equations
Effectiveness of the Bendixson-Dulac theorem
Asymptotic lower bounds on Hilbert numbers using canard cycles
Bifurcations of limit cycles in complex networks of Hamiltonian systems and Lloyd's conjecture
Slow divergence integrals in classical Liénard equations near centers
Periodic solutions of a periodic scalar piecewise ODE
Planar nonautonomous polynomial equations: the Riccati equation
Lower bounds for the number of limit cycles of trigonometric Abel equations
Qualitative theory of planar differential systems
A survey of quadratic systems
Transcendental limit cycles via the structure of arbitrary degree invariant algebraic curves of polynomial planar vector fields
On the differential equations of the simplest boundary-layer problems
Existence of at most two limit cycles for some non-autonomous differential equations
Planar nonautonomous polynomial equations V. The Abel equation
Some Applications of the Extended Bendixson-Dulac Theorem
Extracting the time-dependent transmission rate from infection data via solution of an inverse ODE problem
The number of small-amplitude limit cycles of Liénard equations
Finiteness theorems for limit cycles
Limit Cycles for a Class of Abel Equations
Coexistence of algebraic and non-algebraic limit cycles, explicitly given, using Riccati equations
More limit cycles than expected in Liénard equations
DYNAMICS OF SOME RATIONAL DISCRETE DYNAMICAL SYSTEMS VIA INVARIANTS
Qualitative Tools for Studying Periodic Solutions and Bifurcations as Applied to the Periodically Harvested Logistic Equation
Separatrix and Limit Cycles of Quadratic Systems and Dulac's Theorem
Order of cyclicity of the singular point of lineard's polynomial vector fields
Limit cycles of a class of polynomial systems
On a Class of Differential Equations of Riccati Type
A Note on the Number of Limit Cycles in Certain Two-Dimensional Systems
Möbius Transformations and Periodic Solutions of Complex Riccati Equations
Differential Equations Defined by the Sum of two Quasi-Homogeneous Vector Fields
DEGENERATE EQUATIONS OF PENDULUM-TYPE
Centennial History of Hilbert's 16th Problem
Periodic oscillators, isochronous centers and resonance
New lower bounds for the Hilbert numbers using reversible centers
HILBERT'S 16TH PROBLEM AND BIFURCATIONS OF PLANAR POLYNOMIAL VECTOR FIELDS

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