Results 11 to 20 of about 76,964 (244)
Planar chemical reaction systems with algebraic and non-algebraic limit cycles. [PDF]
J Math BiolAbstract 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.
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The $16$th Hilbert problem on algebraic limit cycles [PDF]
Journal of Differential Equations, 2014 For real planar polynomial differential systems there appeared a simple version of the $16$th Hilbert problem on algebraic limit cycles: {\it Is there an upper bound on the number of algebraic limit cycles of all polynomial vector fields of degree $m ...
Xiang, Zhang
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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
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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
Chavarriga, Javier +2 more
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On the uniqueness of algebraic limit cycles for quadratic polynomial differential systems with two pairs of equilibrium points at infinity [PDF]
Geometriae Dedicata, 2017 Agraïments: The second author is partially supported by FCT/Portugal through UID/MAT/04459/2013.Algebraic limit cycles in quadratic polynomial differential systems started to be studied in 1958, and few years later the following conjecture appeared ...
Llibre, Jaume, Valls, Clàudia
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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.
Gasull, A. +2 more
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AxiomsIn 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
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Number of Limit Cycles for Planar Systems with Invariant Algebraic Curves
Qualitative Theory of Dynamical Systems, 2023 AbstractFor 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.
Gasull, Armengol, Giacomini, Hector
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On the 16th Hilbert problem for algebraic limit cycles
Journal of Differential Equations, 2010 For a polynomial planar vector field of degree n ≥ 2 with generic invariant algebraic curves we show that the maximum number of algebraic limit cycles is 1 + (n − 1)(n − 2)/2 when n is even, and (n − 1)(n − 2)/2 when n is odd. Furthermore, these upper bounds are reached.
Llibre Saló, Jaume +2 more
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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.
Chavarriga, Javier +2 more
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