Hidden attractors in electromechanical systems with and without equilibria

Abstract This paper studies hidden oscillations appearing in electromechanical systems with and without equilibria. Three different systems with such effects are considered: translational oscillator-rotational actuator, drilling system actuated by a DC-motor and drilling system actuated by induction motor. We demonstrate that three systems experience hidden oscillations in sense of mathematical definition. While some of these hidden oscillations can be easily seen in natural physical experiments, the localization of others requires special efforts.

[1]  T. E. Vadivasova,et al.  Numerical and experimental studies of attractors in memristor-based Chua’s oscillator with a line of equilibria. Noise-induced effects , 2015 .

[2]  G. Leonov,et al.  Hidden attractor in smooth Chua systems , 2012 .

[3]  Julien Clinton Sprott,et al.  Strange attractors with various equilibrium types , 2015 .

[4]  Michael Eckert,et al.  Arnold Sommerfeld: Science, Life and Turbulent Times 1868-1951 , 2013 .

[5]  Zhouchao Wei,et al.  Delayed feedback control and bifurcation analysis of the generalized Sprott B system with hidden attractors , 2015 .

[6]  Henk Nijmeijer,et al.  Analysis of Friction-Induced Limit Cycling in an Experimental Drill-String System , 2004 .

[7]  Sundarapandian Vaidyanathan,et al.  Hidden attractors in a chaotic system with an exponential nonlinear term , 2015 .

[8]  Yuri A. Kuznetsov,et al.  An event-driven method to simulate Filippov systems with accurate computing of sliding motions , 2008, TOMS.

[9]  Yu Feng,et al.  Switched generalized function projective synchronization of two hyperchaotic systems with hidden attractors , 2015 .

[10]  Alexander Medvedev,et al.  Multistability and hidden attractors in an impulsive Goodwin oscillator with time delay , 2015 .

[11]  I. VagaitsevV.,et al.  Localization of hidden Chua ’ s attractors , 2022 .

[12]  Nikolay V. Kuznetsov,et al.  Hidden attractors in Dynamical Systems. From Hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman Problems to Hidden Chaotic Attractor in Chua Circuits , 2013, Int. J. Bifurc. Chaos.

[13]  Julien Clinton Sprott,et al.  Recent new examples of hidden attractors , 2015 .

[14]  Sundarapandian Vaidyanathan,et al.  A 5-D hyperchaotic Rikitake dynamo system with hidden attractors , 2015 .

[15]  Nikolay V. Kuznetsov,et al.  Coincidence of the Gelig–Leonov–Yakubovich, Filippov, and Aizerman–Pyatnitskiy definitions , 2015 .

[16]  Viet-Thanh Pham,et al.  Synchronization and circuit design of a chaotic system with coexisting hidden attractors , 2015 .

[17]  Shouchuan Hu.  Differential equations with discontinuous right-hand sides☆ , 1991 .

[18]  Nathan van de Wouw,et al.  Control of mechanical motion systems with non-collocation of actuation and friction: A Popov criterion approach for input-to-state stability and set-valued nonlinearities , 2009, Autom..

[19]  J. Dugundji,et al.  Resonance oscillations in mechanical systems , 1976 .

[20]  G. Leonov,et al.  Localization of hidden Chuaʼs attractors , 2011 .

[21]  Tomasz Kapitaniak,et al.  Rare and hidden attractors in Van der Pol-Duffing oscillators , 2015 .

[22]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[23]  Nikolay V. Kuznetsov,et al.  Discontinuous differential equations: comparison of solution definitions and localization of hidden Chua attractors , 2015 .

[24]  Alexander L. Fradkov,et al.  Controlled passage through resonance in mechanical systems , 2011 .

[25]  Nikolay V. Kuznetsov,et al.  Analytical-numerical method for attractor localization of generalized Chua's system , 2010, PSYCO.

[26]  Werner Leonhard,et al.  Control of Electrical Drives , 1990 .

[27]  N. V. Kuznetsov,et al.  Hidden attractors in fundamental problems and engineering models. A short survey , 2015, 1510.04803.

[28]  Nikolay V. Kuznetsov,et al.  Hidden Periodic Oscillations in Drilling System Driven by Induction Motor , 2014 .

[29]  Nikolay V. Kuznetsov,et al.  Control of multistability in hidden attractors , 2015 .

[30]  M. A. Kiseleva,et al.  Oscillations of dynamical systems applied in drilling : analytical and numerical methods , 2013 .

[31]  Anirban Ray,et al.  Memristive non-linear system and hidden attractor , 2015, The European Physical Journal Special Topics.

[32]  T. N. Mokaev,et al.  Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion Homoclinic orbits, and self-excited and hidden attractors , 2015 .

[33]  Nikolay V. Kuznetsov,et al.  Hidden oscillations in mathematical model of drilling system actuated by induction motor with a wound rotor , 2014 .

[34]  Julien Clinton Sprott,et al.  Multistability in symmetric chaotic systems , 2015 .