Results 1 to 10 of about 3,088 (134)

Planar chemical reaction systems with algebraic and non-algebraic limit cycles. [PDF]

J Math Biol
Abstract The Hilbert number H(n) is defined as the maximum number of limit cycles of a planar autonomous system of ordinary differential equations (ODEs) with right-hand sides containing polynomials of degree at most $$n \in {{\mathbb {N}}}$$ n
Craciun G, Erban R.
europepmc   +6 more sources

Algebraic limit cycles for quadratic polynomial differential systems [PDF]

Discrete and Continuous Dynamical Systems - Series B, 2018
AbstractAlgebraic limit cycles in quadratic polynomial differential systems started to be studied in 1958, and a few years later the following conjecture appeared: quadratic polynomial differential systems have at most one algebraic limit cycle. We prove that a quadratic polynomial differential system having an invariant algebraic curve with at most ...
Claudia Valls, Jaume Llibre
exaly   +7 more sources

Number of Limit Cycles for Planar Systems with Invariant Algebraic Curves

Qualitative Theory of Dynamical Systems, 2023
Abstract For planar polynomials systems the existence of an invariant algebraic curve limits the number of limit cycles not contained in this curve. We present a general approach to prove non-existence of periodic orbits not contained in this given algebraic curve.
Armengol Gasull, Hector Giacomini, Gasull Armengol   +2 more
exaly   +4 more sources

Rational Lyapunov Functions and Stable Algebraic Limit Cycles [PDF]

IEEE Transactions on Automatic Control, 2014
The main goal of this technical note is to show that the class of systems described by a planar differential equation having a rational proper Lyapunov function has asymptotically stable sets which are either locally asymptotically stable equilibrium points, stable algebraic limit cycles or asymptotically stable algebraic graphics. The use of the Zubov
Emmanuel Moulay
exaly   +3 more sources

Uniqueness of Algebraic Limit Cycles for Quadratic Systems

Journal of Mathematical Analysis and Applications, 2001
All known quadratic systems (QS) having an algebraic limit cycle are contained in five families. The degree of the algebraic curve containing the limit cycle is 4 in four of these families and 2 for the other one. Furthermore, it is also known that if there is another QS having an algebraic limit cycle, it should have at least degree 5. The main result
Jaume Llibre, Hector Giacomini
exaly   +3 more sources

Explicit non-algebraic limit cycles for polynomial systems

Journal of Computational and Applied Mathematics, 2007
We consider a system of the form x'=P_n(x,y)+xR_m(x,y), y'=Q_n(x,y)+yR_m(x,y), where P_n(x,y), Q_n(x,y) and R_m(x,y) are homogeneous polynomials of degrees n, n and m, respectively, with n<=m. We prove that this system has at most one limit cycle and that when it exists it can be explicitly found.
Armengol Gasull, Joan Torregrosa, H Giacomini   +2 more
exaly   +4 more sources

Algebraic limit cycles of degree 4 for quadratic systems

Journal of Differential Equations, 2004
This interesting paper contains basic concepts on algebraic curves and some results on polynomial differential systems having invariant algebraic curves. The main result is the proof that quadratic systems have exactly four different families of algebraic limit cycles of degree four.
Jaume Llibre
exaly   +2 more sources

The Limit Cycle of the van der Pol Equation Is Not Algebraic

Journal of Differential Equations, 1995
It is shown that no solution curve of the so-called Liénard equation \((y'' + f(y)y' + g(y) = 0)\) lies in an algebraic curve if \(f\) and \(g\) are nonzero polynomials, if \(\deg f \geq \deg g\) and if \(f/g\) is not constant. The proof is based on a curious but elementary lemma that gives a necessary and sufficient condition for a solution curve of a
exaly   +3 more sources

Coexistence of Algebraic Limit Cycles and Small Limit Cycles of Two Classes of Near-Hamiltonian Systems with a Nilpotent Singular Point

Axioms
In this paper, two classes of near-Hamiltonian systems with a nilpotent center are considered: the coexistence of algebraic limit cycles and small limit cycles.
Huimei Liu, Meilan Cai, Feng Li
doaj   +2 more sources

Coexistence of algebraic and non-algebraic limit cycles for quintic polynomial differential systems

Electronic Journal of Differential Equations, 2017
In the work by Gine and Grau [11], a planar differential system of degree nine admitting a nested configuration formed by an algebraic and a non-algebraic limit cycles explicitly given was presented.
Ahmed Bendjeddou, Rachid Cheurfa
doaj   +2 more sources