Introduction The center and cyclicity problems Time-reversibility and a polynomial subalgebra Limit cycles, centers and time-reversibility in systems of polynomial differential equations Valery Romanovski CAMTP – Center for Applied Mathematics and Theoretical Physics University of Maribor, Krekova 2, SI-2000 Maribor, Slovenia October 21, 2010 Valery Romanovski Limit cycles, centers and time-reversibility in systems of polynomial
Introduction The center and cyclicity problems Time-reversibility and a polynomial subalgebra Table of contents 1 Introduction Predator-prey equations 16th Hilbert’s problem and related problems 2 The center and cyclicity problems The center variety of the quadratic system The cyclicity of the quadratic system 3 Time-reversibility and a polynomial subalgebra Valery Romanovski Limit cycles, centers and time-reversibility in systems of polynomial
Introduction Predator-prey equations The center and cyclicity problems 16th Hilbert’s problem and related problems Time-reversibility and a polynomial subalgebra Lotka-Volterra equations Consider a biological system in which two species interact, one a predator and one its prey. They evolve in time according to the pair of the equations: dx dt = x ( α − β y ) , dy dt = − y ( γ − δ x ) where, y is the number of some predator; x is the number of its prey; x and dy dx dt = ˙ dt = ˙ y represent the growth of the two populations against time t ; Valery Romanovski Limit cycles, centers and time-reversibility in systems of polynomial
Introduction Predator-prey equations The center and cyclicity problems 16th Hilbert’s problem and related problems Time-reversibility and a polynomial subalgebra The prey equation: dx dt = α x − β xy . The prey are assumed to have an unlimited food supply, and to reproduce exponentially unless subject to predation; this exponential growth is represented by the term α x . The rate of predation upon the prey is assumed to be proportional to the rate at which the predators and the prey meet; this is represented by β xy . The predator equation: dy dt = δ xy − γ y . δ xy - the growth of the predator population. γ y represents the loss rate of the predators due to either natural death or emigration; it leads to an exponential decay in the absence of prey. The equation expresses the change in the predator population as growth fueled by the food supply, minus natural death. Valery Romanovski Limit cycles, centers and time-reversibility in systems of polynomial
Introduction Predator-prey equations The center and cyclicity problems 16th Hilbert’s problem and related problems Time-reversibility and a polynomial subalgebra 16th Hilbert’s problem and related problems x = P n ( x , y ) , ˙ y = Q n ( x , y ) , ˙ ( A ) P n ( x , y ) , Q n ( x , y ) , are polynomials of degree n . Let h ( P n , Q n ) be the number of limit cycles of system (A) and let H ( n ) = sup h ( P n , Q n ) . The question of the second part of the 16th Hilbert’s problem: find a bound for H ( n ) as a function of n . (The problem is still unresolved even for n = 2.) A simpler problem: is H ( n ) finite? Unresolved. Valery Romanovski Limit cycles, centers and time-reversibility in systems of polynomial
Introduction Predator-prey equations The center and cyclicity problems 16th Hilbert’s problem and related problems Time-reversibility and a polynomial subalgebra 16th Hilbert’s problem and related problems An even simpler problem: is h ( P n , Q n ) finite? Chicone and Shafer (1983) proved that for n = 2 a fixed system (A) has only finite number of limit cycles in any bounded region of the phase plane. Bam` on (1986) and V. R (1986) proved that h ( P 2 , Q 2 ) is finite. Il’yashenko (1991) and Ecalle (1992): h ( P n , Q n ) is finite for any n . Valery Romanovski Limit cycles, centers and time-reversibility in systems of polynomial
Introduction Predator-prey equations The center and cyclicity problems 16th Hilbert’s problem and related problems Time-reversibility and a polynomial subalgebra Local Hilbert’s 16th problem Find an upper bound for the number of limit cycles in a neighborhood of elementary singular point. This problem is called the cyclicity problem or the local Hilbert’s 16th problem. Valery Romanovski Limit cycles, centers and time-reversibility in systems of polynomial
Introduction The center variety of the quadratic system The center and cyclicity problems The cyclicity of the quadratic system Time-reversibility and a polynomial subalgebra Poincare (return) map � � α jl u j v l , β jl u j v l u = α u − β v + ˙ v = β u + α v + ˙ j + l =2 j + l =2 Poincare map P ( ρ ) = e 2 π α β ρ + η 2 ( α, β, α ij , β ij ) ρ 2 + η 3 ( α, β, α ij , β ij ) ρ 3 + . . . . Limit cycles ← → isolated fixed points of P ( ρ ). α changes the sign − > Hopf bifurcation W.l.o.g. we assume that α = 0 , β = 1. Then η k ( α ij , β ij ) are polynomials. Valery Romanovski Limit cycles, centers and time-reversibility in systems of polynomial
Introduction The center variety of the quadratic system The center and cyclicity problems The cyclicity of the quadratic system Time-reversibility and a polynomial subalgebra The Bautin ideal and Bautin’s theorem To study limit cycles in a system � α jl u j v l , � β jl u j v l u = − v + ˙ v = u + ˙ (1) j + l =2 j + l =2 we compute the Poincare map: P ( ρ ) = ρ + η 2 ( α ij , β ij ) ρ 2 + η 3 ( α ij , β ij ) ρ 3 + · · · + η k ( α ij , β ij ) ρ k . Let B = � η 3 , η 4 , . . . � ⊂ R [ α ij , β ij ] be the ideal generated by all focus quantities η i . There is k such that B = � η u 1 , η u 2 , . . . , η u k � . Valery Romanovski Limit cycles, centers and time-reversibility in systems of polynomial
Introduction The center variety of the quadratic system The center and cyclicity problems The cyclicity of the quadratic system Time-reversibility and a polynomial subalgebra The Bautin ideal and Bautin’s theorem Then for any s η s = η u 1 θ ( s ) + η u 2 θ ( s ) + · · · + η u k θ ( k ) k , 1 2 P ( ρ ) − ρ = η u 1 (1 + µ 1 ρ + . . . ) ρ u 1 + · · · + η u k (1 + µ k ρ + . . . ) ρ u k . Bautin’s Theorem If B = � η u 1 , η u 2 , . . . , η u k � then the cyclicity of system (1) (i.e. the maximal number of limit cycles which appear from the origin after small perturbations) is less or equal to k . Proof. Bautin N.N. Mat. Sb. (1952) v.30, 181-196 (Russian); Trans. Amer. Math. Soc. (1954) v.100 Roussarie R. Bifurcations of planar vector fields and Hilbert’s 16th problem (1998), Birkhauser. Valery Romanovski Limit cycles, centers and time-reversibility in systems of polynomial
Introduction The center variety of the quadratic system The center and cyclicity problems The cyclicity of the quadratic system Time-reversibility and a polynomial subalgebra P ( ρ ) = ρ + η 3 ( α ij , β ij ) ρ 3 + η 4 ( α ij , β ij ) ρ 4 + . . . . Center: η 3 = η 4 = η 5 = · · · = 0. Poincar´ e center problem Find all systems with a center at the origin within a given polynomial family Algebraic counterpart Find the variety of the Bautin ideal B = � η 3 , η 4 , η 5 . . . � . (This variety is called the center variety.) Valery Romanovski Limit cycles, centers and time-reversibility in systems of polynomial
Introduction The center variety of the quadratic system The center and cyclicity problems The cyclicity of the quadratic system Time-reversibility and a polynomial subalgebra An algebraic point of view The cyclicity problem Find an upper bound for the maximal number of limit cycles in a neighborhood of a center or a focus By Bautin’s theorem: Algebraic counterpart Find a basis for the Bautin ideal � η 3 , η 4 , η 5 , . . . � generated by all coefficients of the Poincar´ e map Valery Romanovski Limit cycles, centers and time-reversibility in systems of polynomial
Introduction The center variety of the quadratic system The center and cyclicity problems The cyclicity of the quadratic system Time-reversibility and a polynomial subalgebra Complexification n − 1 n − 1 � � a pq x p +1 y q ) , b qp x q y p +1 ) (2) x = i ( x − ˙ y = − i ( y − ˙ p + q =1 p + q =1 The change of time d τ = idt transforms (2) to the system n − 1 n − 1 � � a pq x p +1 y q ) , ˙ b qp x q y p +1 ) . x = ( x − ˙ y = − ( y − (3) p + q =1 p + q =1 Valery Romanovski Limit cycles, centers and time-reversibility in systems of polynomial
Introduction The center variety of the quadratic system The center and cyclicity problems The cyclicity of the quadratic system Time-reversibility and a polynomial subalgebra Poincar´ e-Lyapunov Theorem The system n n du dv � � α ij u i v j , β ij u i v j dt = − v + dt = u + (4) i + j =2 i + j =2 has a center at the origin (equivalently, all coefficients of the Poincar´ e map are equal to zero) if and only if it admits a first integral of the form Φ = u 2 + v 2 + � φ kl u k v l . k + l ≥ 2 Valery Romanovski Limit cycles, centers and time-reversibility in systems of polynomial
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