Recently, dynamic systems with hidden attractors have aroused increasing attentions in the community of nonlinear dynamics because the singularity of hidden attractors can lead to unexpected phenomena. Hidden attractors have been observed in many engineering problems such as the absolute stability in automatic control and phase-locked loops (PLL) in electric circuits.Although a hidden attractor is usually defined as an attractor containing a basin of attraction which has no intersection with the adjacency of equilibria, several newly discovered infinite-equilibrium systems has been classified as hidden attractors from the computational point of view. Messias&nbsp;et al. [1] presented two memristor models characterized by three-dimensional piecewise-linear and cubic ODEs. At a suitable parameter range, the systems possess infinitely many stable equilibria as well as infinitely many stable periodic orbits. Jafari and Sprott [2] performed a computer search for three-dimensional, quadratic nonlinear systems with a line equilibrium. Kengne&nbsp;et al. [3] investigated the symmetric break of an infinite-equilibrium system with a curve of equilibria.&nbsp;&nbsp;Chen [4]&nbsp;et al. proposed a discrete system with one-line equilibria and used it to design a pseudorandom number generator for image encryption.&nbsp;&nbsp;&nbsp;&nbsp;In fact, infinite-equilibrium systems also exist in planar infinite-equilibrium systems (Andronov [5]). More recently, Luo systematically presented infinite-equilibrium dynamic systems.&nbsp;&nbsp;The infinite equilibria can be nonhyperbolic. To determine the stability of such equilibria, center manifold and normal form theories can be applied. In 2019, Luo [6] presented a new method for local analysis of nonlinear dynamic systems through the transformation of the original system into a dynamical system in each specific eigenvector direction. The more generalized local stability and bifurcations analysis in nonlinear polynomial systems can be seen from Luo [7, 8] as well. Inspired by Luo’s work, Xing and Luo&nbsp;in 2021 proposed an infinite-line-equilibrium system based on a quadratic oscillator. It was numerically demonstrated that the infinite-equilibrium boundary can significantly impact the dynamics of the original dynamic system. This allows one to control the dynamics of a system by constructing infinite-equilibrium boundary through a control law. Therefore, this paper proposes a novel control law to confine the dynamics of a quadratic oscillator through constructing the infinite-equilibrium boundary on the separatrix of the oscillator.This paper begins with the construction of infinite equilibria for the quadratic oscillator using a control law, followed by the local analysis of the controlled quadratic oscillator. The global analysis of such a system is performed through the phase portrait generated from first integrals of the system. Numerical simulations of the perturbed system with/without the implementation of the control law are presented. The feasibility of the control scheme is numerically demonstrated from this paper.&nbsp;References[1]&nbsp;Messias, M., Nespoli, C., and Botta, V. A., 2010, “Hopf Bifurcation from Lines of Equilibria Without Parameters in Memristor Oscillators,”&nbsp;International Journal of Bifurcation and Chaos,&nbsp;20(02),&nbsp;437–450.[2] Jafari, S., and Sprott, J. C., 2013, “Simple Chaotic Flows with a Line Equilibrium,”&nbsp;Chaos, Solitons &amp; Fractals,&nbsp;57,&nbsp;79–84.[3] Korneev, I. A., Slepnev, A. V., Semenov, V. V., and Vadivasova, T. E., 2020, “Forced Synchronization of an Oscillator with a Line of Equilibria,”&nbsp;The European Physical Journal Special Topics,&nbsp;229(12–13),&nbsp;2215–2224.[4] Chen, E., Min, L., and Chen, G., 2017, “Discrete Chaotic Systems with One-line Equilibria and Their Application to Image Encryption,”&nbsp;International Journal of Bifurcation and Chaos,&nbsp;27(03), 1750046.[5]&nbsp;Andronov, A.A., Leontovich, E.A., Gordon, I.I., and Maier, A.G., (1971),&nbsp;Qualitative Theory of Second-order Dynamic Systems, John Wiley &amp; Sons, LTD., Chichester.[6]&nbsp;Luo, A.C.J., 2019, “On Stability and Bifurcation of Equilibriums in Nonlinear Systems,”&nbsp;Journal of Vibration Testing and System Dynamics,&nbsp;3(2),&nbsp;147-232.[7]&nbsp;Luo, A.C.J., 2020, “The Stability and Bifurcation of Equilibriums in (2m)th-degree Polynomial Systems,”&nbsp;Journal of Vibration Testing and System Dynamics,&nbsp;4(1), 1-42[8]&nbsp;Luo, A.C.J. 2020, “The Stability and Bifurcation of Equilibriums in (2m+1)th-degree Polynomial Systems,”&nbsp;Journal of Vibration Testing and System Dynamics,&nbsp;4(2), 93-144.&nbsp;

Controlling the Dynamics of a Quadratic Oscillator Using Infinite-Equilibriam

Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2021/volume/85543

ASME 2021 International Mechanical Engineering Congress and Exposition (IMECE2021)

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