We introduce a new method for investigation of dynamical systems which allows us to extract as much information as possible about potential system dynamics, based only on the form of equations describing it. The discussed tool of critical surfaces, defined by the zero velocity (and/or) acceleration field for particular variables of the system is related to the geometry of the attractors. Particularly, the developed method provides a new and simple procedure allowing to localize hidden oscillations. Our approach is based on the dimension reduction of the searched area in the phase space and has an advantage (in terms of complexity) over standard procedures for investigating full–dimensional space. The two approaches have been compared using typical examples of oscillators with hidden states. Our topological tool allows us not only to develop alternate ways of extracting information from the equations of motion of the dynamical system, but also provides a better understanding of attractors geometry and their capturing in complex cases, especially including multistable and hidden attractors. We believe that the introduced method can be widely used in the studies of dynamical systems and their applications in science and engineering.
REFERENCES
1.
C.
Li
, W.
Hu
, J. C.
Sprott
, and X.
Wang
, “Multistability in symmetric chaotic systems
,” Eur. Phys. J. Spec. Top.
224
, 1493
(2015
). 2.
B. C.
Bao
, H.
Bao
, N.
Wang
, M.
Chen
, and Q.
Xu
, “Hidden extreme multistability in memristive hyperchaotic system
,” Chaos Soliton Fractals
94
, 102
(2017
). 3.
J.
Kengne
, Z. T.
Njitacke
, and H. B.
Fotsin
, “Dynamical analysis of a simple autonomous jerk system with multiple attractors
,” Nonlinear Dyn.
83
, 751
(2016
). 4.
C. N.
Ngonghala
, U.
Feudel
, and K.
Showalter
, “Extreme multistability in a chemical model system
,” Phys. Rev. E
83
, 056206
(2011
). 5.
F.
Freyer
, J. A.
Roberts
, P.
Ritter
, and M.
Breakspear
, “A canonical model of multistability and scale-invariance in biological systems
,” PLOS Comput. Biol.
8
, 1
(2012
). 6.
E.
Kaslik
and S.
Sivasundaram
, “Impulsive hybrid discrete-time Hopfield neural networks with delays and multistability analysis
,” Neural Netw.
24
, 370
(2011
). 7.
A.
Chudzik
, P.
Perlikowski
, A.
Stefanski
, and T.
Kapitaniak
, “Multistability and rare attractors in van der Pol-Duffing oscillator
,” Int. J. Bifurc. Chaos
21
, 1907
(2011
). 8.
D.
Dudkowski
, A.
Prasad
, and T.
Kapitaniak
, “Perpetual points and hidden attractors in dynamical systems
,” Phys. Lett. A
379
, 2591
(2015
). 9.
Y.
Kuramoto
and D.
Battogtokh
, “Coexistence of coherence and incoherence in nonlocally coupled phase oscillators
,” Nonlinear Phenom. Complex Syst.
5
, 380
(2002
).10.
D. M.
Abrams
and S. H.
Strogatz
, “Chimera states for coupled oscillators
,” Phys. Rev. Lett.
93
, 174102
(2004
). 11.
P. J.
Menck
, J.
Heitzig
, N.
Marwan
, and J.
Kurths
, “How basin stability complements the linear-stability paradigm
,” Nat. Phys.
9
, 89
(2013
). 12.
A. N.
Pisarchik
and U.
Feudel
, “Control of multistability
,” Phys. Rep.
540
, 167
(2014
). 13.
G. A.
Leonov
, N. V.
Kuznetsov
, and V. I.
Vagaitsev
, “Localization of hidden Chuas attractors
,” Phys. Lett. A
375
, 2230
(2011
). 14.
G. A.
Leonov
, N. V.
Kuznetsov
, and V. I.
Vagaitsev
, “Hidden attractor in smooth Chua systems
,” Physica D
241
, 1482
(2012
). 15.
G. A.
Leonov
, N. V.
Kuznetsov
, and T. N.
Mokaev
, “Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion
,” Eur. Phys. J. Spec. Top.
224
, 1421
(2015
). 16.
G. A.
Leonov
, N. V.
Kuznetsov
, and T. N.
Mokaev
, “Hidden attractor and homoclinic orbit in Lorenz-like system describing convective fluid motion in rotating cavity
,” Commun. Nonlinear Sci. Numer. Simul.
28
, 166
(2015
). 17.
C.
Li
and J. C.
Sprott
, “Coexisting hidden attractors in a 4-D simplified Lorenz system
,” Int. J. Bifurc. Chaos
24
, 1450034
(2014
). 18.
Z.
Wei
, P.
Yu
, W.
Zhang
, and M.
Yao
, “Study of hidden attractors, multiple limit cycles from hopf bifurcation and boundedness of motion in the generalized hyperchaotic Rabinovich system
,” Nonlinear Dyn.
82
, 131
(2015
). 19.
V.-T.
Pham
, C.
Volos
, S.
Jafari
, S.
Vaidyanathan
, T.
Kapitaniak
, and X.
Wang
, “A chaotic system with different families of hidden attractors
,” Int. J. Bifurc. Chaos
26
, 1650139
(2016
). 20.
S.-K.
Lao
, Y.
Shekofteh
, S.
Jafari
, and J. C.
Sprott
, “Cost function based on gaussian mixture model for parameter estimation of a chaotic circuit with a hidden attractor
,” Int. J. Bifurc. Chaos
24
, 1450010
(2014
). 21.
V.-T.
Pham
, S.
Vaidyanathan
, C. K.
Volos
, and S.
Jafari
, “Hidden attractors in a chaotic system with an exponential nonlinear term
,” Eur. Phys. J. Spec. Top.
224
, 1507
(2015
). 22.
S.
Brezetskyi
, D.
Dudkowski
, and T.
Kapitaniak
, “Rare and hidden attractors in van der Pol-Duffing oscillators
,” Eur. Phys. J. Spec. Top.
224
, 1459
(2015
). 23.
M.
Zakrzhevsky
, A.
Klokov
, V.
Yevstignejev
, E.
Shilvan
, and A.
Kragis
, “Nonlinear dynamics and rare attractors in driven damped pendulum systems
,” in 7th International DAAAM Baltic Conference Industrial Engineering
(Tallinn, Estonia
, 2010
).24.
A. V.
Klokov
and M. V.
Zakrzhevsky
, “Parametrically excited pendulum systems with several equilibrium positions: Bifurcation analysis and rare attractors
,” Int. J. Bifurc. Chaos
21
, 2825
(2011
). 25.
V.
Yevstignejev
, A.
Klokov
, R.
Smirnova
, and I.
Schukin
, “Rare attractors in typical nonlinear discrete dynamical models
,” in 4th IEEE International Conference on Nonlinear Science and Complexity
(Budapest, Hungary
, 2012
).26.
D.
Dudkowski
, S.
Jafari
, T.
Kapitaniak
, N. V.
Kuznetsov
, G. A.
Leonov
, and A.
Prasad
, “Hidden attractors in dynamical systems
,” Phys. Rep.
637
, 1
(2016
). 27.
G. A.
Leonov
and N. V.
Kuznetsov
, “Hidden attractors in dynamical systems: From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits
,” Int. J. Bifurc. Chaos
23
, 1330002
(2013
). 28.
N. V.
Kuznetsov
, G. A.
Leonov
, T. N.
Mokaev
, A.
Prasad
, and M. D.
Shrimali
, “Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system
,” Nonlinear Dyn.
92
, 267
(2018
). 29.
A.
Hastings
, K. C.
Abbott
, K.
Cuddington
, T.
Francis
, G.
Gellner
, Y.-C.
Lai
, A.
Morozov
, S.
Petrovskii
, K.
Scranton
, and M. L.
Zeeman
, “Transient phenomena in ecology
,” Science
361
, 6406
(2018
). 30.
S.
Jafari
, J. C.
Sprott
, and S. M. R. H.
Golpayegani
, “Elementary quadratic chaotic flows with no equilibria
,” Phys. Lett. A
377
, 699
(2013
). 31.
C.
Letellier
and J.-M.
Malasoma
, “Architecture of chaotic attractors for flows in the absence of any singular point
,” Chaos
26
, 063115
(2016
). 32.
D.
Dudkowski
, A.
Prasad
, and T.
Kapitaniak
, “Describing chaotic attractors: Regular and perpetual points
,” Chaos
28
, 033604
(2018
). 33.
A.
Prasad
, “Existence of perpetual points in nonlinear dynamical systems and its applications
,” Int. J. Bifurc. Chaos
25
, 1530005
(2015
). 34.
D.
Dudkowski
, A.
Prasad
, and T.
Kapitaniak
, “Perpetual points and periodic perpetual loci in maps
,” Chaos
26
, 103103
(2016
). 35.
D.
Dudkowski
, A.
Prasad
, and T.
Kapitaniak
, “Perpetual points: New tool for localization of coexisting attractors in dynamical systems
,” Int. J. Bifurc. Chaos
27
, 1750063
(2017
). 36.
P.
Brzeski
and L. N.
Virgin
, “Experimental investigation of perpetual points in mechanical systems
,” Nonlinear Dyn.
90
, 2917
(2017
). 37.
S.
Jafari
, F.
Nazarimehr
, J. C.
Sprott
, and S. M. R. H.
Golpayegani
, “Limitation of perpetual points for confirming conservation in dynamical systems
,” Int. J. Bifurc. Chaos
25
, 1550182
(2015
). 38.
F.
Nazarimehr
, B.
Saedi
, S.
Jafari
, and J. C.
Sprott
, “Are perpetual points sufficient for locating hidden attractors?
,” Int. J. Bifurc. Chaos
27
, 1750037
(2017
). 39.
S.
Sabarathinam
, C. K.
Volos
, and K.
Thamilmaran
, “Implementation and study of the nonlinear dynamics of a memristor-based Duffing oscillator
,” Nonlinear Dyn.
87
, 37
(2017
). 40.
V.
Varshney
, S.
Sabarathinam
, A.
Prasad
, and K.
Thamilmaran
, “Infinite number of hidden attractors in memristor-based autonomous Duffing oscillator
,” Int. J. Bifurc. Chaos
28
, 1850013
(2018
). © 2018 Author(s).
2018
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