Abstract
We studied the existence of limit cycles for the quartic polynomial differential systems depending on parameters. To prove that, first, we used the formal series method based on Poincare’ ideas to determine the center-focus. Then, by the Hopf bifurcation theory, we obtained the sufficient condition for the existence of the limit cycles. Finally, we provided some numerical examples for illustration.
Highlights
In the twentieth century, progress in applied electronics has been rapid
One of the main issues in analysing the qualitative theory of differential equations is the limit cycle. e limit cycle was discovered by Poincare’ (1881–1886)
The problem of limit cycles has become increasingly critical and has attracted the attention of many pure and applied mathematicians, see for instance [1,2,3,4,5,6,7,8,9,10]. e necessary problems in the qualitative theory of differential equation include the existence of the limit cycle
Summary
Progress in applied electronics has been rapid. Physicists invented the triode vacuum tube, making stable self-excited oscillations of constant amplitude. E necessary problems in the qualitative theory of differential equation include the existence of the limit cycle. Many researchers have studied the existence of limit cycles for the quadratic and cubic systems, and numerous research achievements have been obtained, see for instance [3, 5, 7, 10,11,12,13,14,15,16,17,18]. We will study the existence of limit cycles for the following quartic polynomial differential systems depending on parameters: dx −y + kx2 + lxy, dt (6) dy x + mx2y + ny, dt where k, l, m, and n are parameters, in which the linear system is the center. By using the Hopf bifurcation theory, we obtained the sufficient condition for the existence of limit cycle for quartic system (6), which bifurcated from the equilibrium point (singular point).
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