Admissible perturbations refer to the perturbations that do not change the Mironenko reflecting function of the system, which are obtained for an autonomous three-dimensional quadratic generalized Langford system with five parameters. The obtained nonautonomous perturbed systems retain many of the qualitative properties of the original system solutions. In particular, the instability (in the sense of Lyapunov) of the equilibrium point, the presence of a periodic solution and its asymptotic stability (instability) are proved for the perturbed systems. The presence of similar chaotic attractors in the original and perturbed systems is shown by numerical simulation.

We consider an autonomous three-dimensional quadratic ODE system with nine parameters, which is a generalization of the Langford system. We derive conditions under which this system has infinitely many limit cycles. First, we study the equilibrium points of such systems and their eigenvalues. Next, we prove the non-local existence of an infinite set of limit cycles emerging by means of Andronov – Hopf bifurcation.

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In this work, we will consider an autonomous three-dimensional quadratic system of first-order ordinary differential equations, with five parameters and with symmetry relative to the z-axis, which generalize the Hopf–Langford system. By reformulating the system as a system of two second-order ordinary differential equations and using the Kosambi–Cartan–Chern (KCC) geometric theory, we will investigate this system from the perspective of Jacobi stability. We will compute the five invariants of KCC theory which determine the own geometrical properties of this system, especially the deviation curvature tensor. Additionally, we will search for necessary and sufficient conditions on the five parameters of the system in order to reach the Jacobi stability around each equilibrium point.

In this paper, we consider the problem of construction of real autonomous quadratic systems of three differential equations with the nonlocal existence of an infinite number of limit cycles. This means that an infinite number of limit cycles, emerging from the focus due to the Andronov – Hopf bifurcation, can exist in the phase space not only in the vicinity of the focus and not only for parameter values close to the bifurcation value. To solve this problem we use the method of determination of limit cycles as the curves of intersection of an invariant plane with a family of invariant elliptic paraboloids. Then the study of the limit cycles of the constructed system of the third order is reduced to the study of the corresponding system of the second order on each of the invariant elliptic paraboloids. The proof of the nonlocal existence of the limit cycle and the investigation of its stability for such a second-order system is carried out by constructing a topographic system of Poincaré functions or by transforming to polar coordinates.