Journal of Applied Nonlinear Dynamics
Hidden Attractors: New Horizons in Exploring Dynamical Systems
Journal of Applied Nonlinear Dynamics 10(4) (2021) 671--673 | DOI:10.5890/JAND.2021.12.007
Maaita Jamal-Odysseas
Physics Department, Aristotle University of Thessaloniki, Thessaloniki 541 24, Greece Department of Mechanical Engineering, University of Western Macedonia, Bakola & Sialvera 50132, Kozani, Greece
Download Full Text PDF
Abstract
Nonlinear systems can be studied by their equivalent linearized systems in the region of equilibrium points. This fact led the study of dynamical systems to be focused near these points. On the contrary Hidden attractors can be found in systems with no equilibrium points, with one stable equilibrium or in systems with lines of equilibrium points. This forces us to ``sail in uncharted waters'' and opens new perspectives in the study of dynamical systems.
References
-
[1]  | Arrowsmith, D.K. and Place C.M. (1994), An Introduction to Dynamical
Systems, Cambridge University Press.
|
-
[2]  | Wiggins, S. (2003), Introduction to applied nonlinear dynamical
systems and chaos, (2), Springer Science and Business Media.
|
-
[3]  | Davis, H.T. (1962), Introduction to nonlinear differential and integral
equations, Dover.
|
-
[4]  | Guckenheimer, J. and Holmes, P. (1983), Nonlinear oscillations, dynamical
systems, and bifurcations of vector fields, Applied Mathematical Sciences, ({42}), Springer.
|
-
[5]  | Lorenz, E.N. (1963), Deterministic nonperiodic flow, Journal of the atmospheric sciences, 20(2), 130-141.
|
-
[6]  | Poincar\{e}, H. (2017), The three-body problem and the equations of
dynamics: Poincar\{e}s foundational work on dynamical systems theory.
(443), Springer.
|
-
[7]  | Mandelbrot, B.B. (1983). The fractal geometry of nature.
({173}). New York: WH freeman.
|
-
[8]  | Markus, L. and Yamabe, H. (1960), Global stability criteria for differential
systems, Osaka Math. J., ({12}), 305.
|
-
[9]  | Kuznetsov, N.V., Leonov, G.A., and Vagaitsev, V.I. (2010), Analytical-numerical method
for attractor localization of generalized Chuas system, IFAC Proc, (4), 29.
(IFAC-Papers Online).
|
-
[10]  | Dudkowski, D., Jafari, S., Kapitaniak, T., Kuznetsov, N.V., Leonov, G.A., and Prasad, A. (2016), Hidden attractors in dynamical systems, Physics Reports, (637),
1-50.
|
-
[11]  | Pham, V.T., Jafari, S., Volos, C., Wang, X., and Golpayegani, S.M.R.H. (2014), Is that really hidden? The presence of complex fixed-points in
chaotic flows with no equilibria, International Journal of Bifurcation and Chaos, 24(11).
|
-
[12]  | Xiong, W. and Chen, G.R. (2012), A chaotic system with only one
stable equilibrium, Communications in Nonlinear Science and Numerical Simulation.
|
-
[13]  | Molaie, M., Jafari, S., Sprott, J.C., and Golpayegani, S.M.R.H.
(2013), Simple chaotic flows with one stable equilibrium. International Journal of Bifurcation and Chaos, 23(11).
|
-
[14]  | Pham, V.-T., Volos, C., Vaidyanathan, S., Le, T., and Vu, V. (2015), A
memristor-based hyperchaotic system with hidden attractors: Dynamics,
synchronization and circuital emulating, J. Eng. Sci. Technol. Rev.
|
-
[15]  | Vaidyanathan, S., Volos, C.K., and Pham, V. (2015), Analysis, control,
synchronization and SPICE implementation of a novel 4-D hyperchaotic
Rikitake dynamo system without equilibrium, J. Eng. Sci. Technol. Rev.
|
-
[16]  | Leonov, G., Kuznetsov, N., Yuldashev, M., and Yuldashev, R. (2015), Hold-in,
pull-in, and lock-in ranges of PLL circuits: rigorous mathematical
definitions and limitations of classical theory, IEEE Trans. Circuits Syst. I. Regul. Pap.
|