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Journal of Siberian Federal University. Mathematics & Physics, 2019

One of the main problems in the qualitative theory of real planar differential equations is to de... more One of the main problems in the qualitative theory of real planar differential equations is to determinate the number of limit cycles for a given planar differential system. As we all know, this is a very difficult problem for a general polynomial system. Therefore, many mathematicians study some systems with special conditions. To obtain the number of limit cycles as many as possible for a planar differential system, we usually take in consideration of the bifurcation theory. In recent decades, many new results have been obtained (see [9, 10]). The number of medium amplitude limit cycles bifurcating from the linear center ẋ= y, ẏ = −x for the following three kind of generalized polynomial Liénard differential systems { ẋ = y,

In this work, we determine conditions for planar systems of the form dx dt = P 5 (x, y) + xR 6 (x... more In this work, we determine conditions for planar systems of the form dx dt = P 5 (x, y) + xR 6 (x, y) dy dt = Q 5 (x, y) + yR 6 (x, y) S (a, b, c, u, v, w) where P 5 (x, y) = ax 5 + bx 4 y + cx 3 y 2 + ux 2 y 3 + vxy 4 + wy 5 , Q 5 (x, y) = −6wx 5 + ax 4 y + (b − 11w) x 3 y 2 + cx 2 y 3 + (u − 6w) xy 4 + vy 5 , R 6 (x, y) = x 6 + 3x 4 y 2 + 3x 2 y 4 + y 6 , and where a, b, c, u, v and w are real constants, to possess nonalgebraic limit cycles. Moreover we proof that this non-algebraic limit cycle, when it exists it can be explicitly given. This is done as an application of former theorems gives description of the existence of the non-algebraic limit cycles of the family of systems: dx dt = P n (x, y) + xR n (x, y) , dy dt = Q n (x, y) + yR n (x, y) , where P n (x, y) , Q n (x, y) and R n (x, y) are homogenous polynomials of degree n, n and m respectively with n < m and n is odd, m is even. The tool for proving our result is based on methods developed in [1] and [2].

We determine conditions for planar systems of the form dx dt=-x 3 +2(1+3λ)yx 2 -2xy 2 +(1+4λ)y 3 ... more We determine conditions for planar systems of the form dx dt=-x 3 +2(1+3λ)yx 2 -2xy 2 +(1+4λ)y 3 +x(ax 4 +bx 2 y 2 +cy 4 ),dy dt=-2(1+4λ)x 3 -yx 2 -(6λ+1)xy 2 -2y 3 +y(ax 4 +bx 2 y 2 +cy 4 ), where a,b,c and λ are real constants, to possess non-algebraic limit cycles. This is done as an application of some theorems providing the existence of non-algebraic limit cycles of the family of systems: dx dt=P n (x,y)+xR n (x,y),dy dt=Q n (x,y)+yR n (x,y), where P n (x,y),Q n (x,y) and R n (x,y) are homogeneous polynomials of degree n. The tool for proving our result is based on methods developed by Khalil I. T. Al-Dosary [Int. J. Math. 18, No. 2, 179–189 (2007; Zbl 1121.34036)] and A. Gasull, H. Giacomini and J. Torregrosa [J. Comput. Appl. Math. 200, No. 1, 448–457 (2007; Zbl 1171.34021)].

Journal of Siberian Federal University. Mathematics & Physics, 2019

One of the main problems in the qualitative theory of real planar differential equations is to de... more One of the main problems in the qualitative theory of real planar differential equations is to determinate the number of limit cycles for a given planar differential system. As we all know, this is a very difficult problem for a general polynomial system. Therefore, many mathematicians study some systems with special conditions. To obtain the number of limit cycles as many as possible for a planar differential system, we usually take in consideration of the bifurcation theory. In recent decades, many new results have been obtained (see [9, 10]). The number of medium amplitude limit cycles bifurcating from the linear center ẋ= y, ẏ = −x for the following three kind of generalized polynomial Liénard differential systems { ẋ = y,

In this work, we determine conditions for planar systems of the form dx dt = P 5 (x, y) + xR 6 (x... more In this work, we determine conditions for planar systems of the form dx dt = P 5 (x, y) + xR 6 (x, y) dy dt = Q 5 (x, y) + yR 6 (x, y) S (a, b, c, u, v, w) where P 5 (x, y) = ax 5 + bx 4 y + cx 3 y 2 + ux 2 y 3 + vxy 4 + wy 5 , Q 5 (x, y) = −6wx 5 + ax 4 y + (b − 11w) x 3 y 2 + cx 2 y 3 + (u − 6w) xy 4 + vy 5 , R 6 (x, y) = x 6 + 3x 4 y 2 + 3x 2 y 4 + y 6 , and where a, b, c, u, v and w are real constants, to possess nonalgebraic limit cycles. Moreover we proof that this non-algebraic limit cycle, when it exists it can be explicitly given. This is done as an application of former theorems gives description of the existence of the non-algebraic limit cycles of the family of systems: dx dt = P n (x, y) + xR n (x, y) , dy dt = Q n (x, y) + yR n (x, y) , where P n (x, y) , Q n (x, y) and R n (x, y) are homogenous polynomials of degree n, n and m respectively with n < m and n is odd, m is even. The tool for proving our result is based on methods developed in [1] and [2].

We determine conditions for planar systems of the form dx dt=-x 3 +2(1+3λ)yx 2 -2xy 2 +(1+4λ)y 3 ... more We determine conditions for planar systems of the form dx dt=-x 3 +2(1+3λ)yx 2 -2xy 2 +(1+4λ)y 3 +x(ax 4 +bx 2 y 2 +cy 4 ),dy dt=-2(1+4λ)x 3 -yx 2 -(6λ+1)xy 2 -2y 3 +y(ax 4 +bx 2 y 2 +cy 4 ), where a,b,c and λ are real constants, to possess non-algebraic limit cycles. This is done as an application of some theorems providing the existence of non-algebraic limit cycles of the family of systems: dx dt=P n (x,y)+xR n (x,y),dy dt=Q n (x,y)+yR n (x,y), where P n (x,y),Q n (x,y) and R n (x,y) are homogeneous polynomials of degree n. The tool for proving our result is based on methods developed by Khalil I. T. Al-Dosary [Int. J. Math. 18, No. 2, 179–189 (2007; Zbl 1121.34036)] and A. Gasull, H. Giacomini and J. Torregrosa [J. Comput. Appl. Math. 200, No. 1, 448–457 (2007; Zbl 1171.34021)].