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We obtain 16 center conditions for a polynomial differential system with 27 parameters. DOI: 10.1134/S001226611302002X 1. Consider a system of the form x′ = yP0(x), y′ = −x+ P2(x)y + P3(x)y, (1) where P0(x) = 1+ ∑8 k=1 ckx , P2(x) = ∑7 i=0 aix , P3(x) = ∑10 j=0 bjx , and ai, bj, ck ∈ C, i = 0, . . . , 7, j = 0, . . . , 10, k = 1, . . . , 8. Along with (1), consider the cubic system of differential equations x′ = y(1+Dx+Px)+Hx+Qx, y′ = −x+Ax+3Bxy+Cy+Kx+3Lxy+Mxy+Ny, (2) where A,B,C,D,H,K,L,M,N,P,Q ∈ C. For H = Q = 0, the solution of the center–focus problem can be found in [1]. Let ω1(x) = Nx(H +Qx) − x(C +Mx)(H +Qx)(1 +Dx+ Px) + 3x(B + Lx)(H +Qx)(1 +Dx+ Px) + (1−Ax−Kx)(1 +Dx+ Px), ω2(x) = Nx(H +Qx) − x(2C +D + 2(M + P )x)(H +Qx)(1 +Dx+ Px)/3 + ((3B + 2H)/3 + (L+Q)x)(1 +Dx+ Px). Two successive changes of variables y = Y −Hx −Qx 1 +Dx+ Px2 and Y = yω1(x) 1 +Dx+ Px2 + ω2(x)y , and a change of the time scale reduce system (2) to system (1) [2]. The first focal quantity of system (1) has the form g1 = b0; next, g2 = b2 + 3a0b1 and g3 = 3(5a0a1 + a2)b1 + 3b4 + b3(13a0 + 2c1) + b2(15a0 + 3a1 + 5a0c1 + c2); the quantity g4 contains 38 terms; g5, 138 terms; g6, 438 terms; g7, 1239 terms; g8, 3201 terms; g9, 7670 terms; g10, 17266 terms; g11, 36865 terms; g12, 75231 terms; g13, 147600 terms; g14, 279729 terms; g15, 514052 terms; g16, 918955 terms; g17, 1602372 terms; g18, 2731567 terms; g19, 4561322 terms; and g20, 7473803 terms. Let us introduce the vector p = (a0, a1, . . . , a7, b0, b1, . . . , b10, c1, c2, . . . , c8) and form the ideal I = 〈g1, g2, . . .〉. Then V(I) = {p ∈ C : ∀g ∈ I, g(p) = 0} is the center manifold of system (1) [3, 4]. The singular point O(0, 0) of system (1) is a center if and only if p ∈ V(I). If the first three focal quantities are zero, then P3(x) = xQ(x), Q(x)P0(x) + 3Q(x)P2(x) = xR(x), 3R′(x)Q(x) − 5R(x)Q′(x) = xS(x)

We consider a system of differential equations that consists of two parts, a regularly perturbed and a singularly perturbed one. We assume that the matrix of the linear part of the regularly perturbed system has pure imaginary eigenvalues, while the matrix of the singularly perturbed part is hyperbolic; i.e., all of its eigenvalues have nonzero real parts.

We consider a mathematical model of a hybrid system in which the continuous dynamics induced at each instant of time by a certain system of differential equations from a given finite family is alternated with discrete operations sending commands either for an instantaneous switching from one system to another, or for an instantaneous reset from online coordinates to other coordinates, or for both actions together. We present a description of a hybrid system via systems with impulses. We suggest a new scheme, which permits one to reduce the problem of synthesis of controls for a hybrid system to an impulse control problem. The aim of the present paper is to find a control containing both ordinary components (that are not generalized functions) and higher-order generalized functions.

We generalize the localization method for invariant compact sets of an autonomous dynamical system to the case of a nonautonomous system of differential equations. By using this method, we solve the localization problem for the Vallis third-order dynamical system governing some processes in atmosphere dynamics over the Pacific Ocean. For this system, we construct a one-parameter family of localizing sets bounded by second-order surfaces and find the intersection of all sets of the family.

We construct the complete asymptotics of a periodic solution of a linear normal system of differential equations with high-frequency coefficients. We study the Lyapunov stability and instability of that solution. More specifically, we consider the critical case in which the matrix coefficient of the formally averaged stationary system has one eigenvector and one generalized (in the Vishik-Lyusternik sense) associated vector.