We present dissipative systems with unstable dynamics called the unstable dissipative systems which are capable of generating a multi-stable behavior, i.e., depending on its initial condition, the trajectory of the system converges to a specific attractor. Piecewise linear (PWL) systems are generated based on unstable dissipative systems, whose main attribute when they are switched is the generation of chaotic trajectories with multiple wings or scrolls. For this PWL system, a structure is proposed where both the linear part and the switching function depend on two parameters. We show the range of values of such parameters where the PWL system presents a multistable behavior and trajectories with multiscrolls.
References
1.
E.
Campos-Cantón
, J. G.
Barajas-Ramírez
, G.
Solís-Perales
, and R.
Femat
, “Multiscroll attractors by switching systems
,” Chaos
20
, 013116
(2010
).2.
D.
Angeli
, “Multistability in systems with counter-clockwise input-output dynamics
,” IEEE Trans. Autom. Control
52
, 596
–609
(2007
).3.
A.
Ghaffarizadeh
, N. S.
Flann
, and G. J.
Podgorski
, “Multistable switches and their role in cellular differentiation networks
,” BMC Bioinf.
15
(7
), S7
(2014
).4.
F.
Sagués
and I. R.
Epstein
, “Nonlinear chemical dynamics
,” Dalton Trans.
2003
, 1201
–1217
.5.
W. M.
Haddad
, Q.
Hui
, and J. M.
Bailey
, “Multistability, bifurcations, and biological neural networks: A synaptic drive firing model for cerebral cortex transition in the induction of general anesthesia
,” in Proceedings of the IEEE Conference on Decision and Control
(2011
), pp. 3901
–3908
.6.
M. S.
Patel
, U.
Patel
, A.
Sen
, G. C.
Sethia
, C.
Hens
, S. K.
Dana
, U.
Feudel
, K.
Showalter
, C. N.
Ngonghala
, and R. E.
Amritkar
, “Experimental observation of extreme multistability in an electronic system of two coupled Rössler oscillators
,” Phys. Rev. E
89
, 022918
(2014
).7.
S. J.
Gershman
, E.
Vul
, and J. B.
Tenenbaum
, “Multistability and perceptual inference
,” Neural Comput.
24
, 1
–24
(2012
).8.
P.
Jung
, S.
Butz
, M.
Marthaler
, M. V.
Fistul
, J.
Leppäkangas
, V. P.
Koshelets
, and A. V.
Ustinov
, “Multistability and switching in a superconducting metamaterial
,” Nat. Commun.
5
, 3730
(2014
).9.
R. V.
Mendes
, “Multistability in dynamical systems
,” in
Dynamical Systems: From Crystal to Chaos. Proceedings of the Conference in Honor of Gerar Rauzy on His 60th Birthday
, 1st ed., edited by J.
Gambaudo
, P.
Hubert
, P.
Tisseur
, and S.
Vaienti
(World Scientific
, Singapore
, 2000
), pp. 105
–113
.10.
J.
Kengne
, “On the dynamics of Chua's oscillator with a smooth cubic nonlinearity: Occurrence of multiple attractors
,” Nonlinear Dyn.
87
, 363
–375
(2017
).11.
P.
Giesl
, “On the determination of the basin of attraction of discrete dynamical systems
,” J. Differ. Equations Appl.
13
, 523
–546
(2007
).12.
Q.
Hui
, “Multistability analysis of discontinuous dynamical systems via finite trajectory length
,” in
World Automation Congress (WAC), 2014
(IEEE
, 2014
).13.
E.
Jiménez-López
, J. S.
González Salas
, L. J.
Ontañón-García
, E.
Campos-Cantón
, and A. N.
Pisarchik
, “Generalized multistable structure via chaotic synchronization and preservation of scrolls
,” J. Franklin Inst.
350
, 2853
–2866
(2013
).14.
C. R.
Hens
, R.
Banerjee
, U.
Feudel
, and S. K.
Dana
, “How to obtain extreme multistability in coupled dynamical systems
,” Phys. Rev. E
85
, 035202
(2012
).15.
J.
Kengne
, Z. N.
Tabekoueng
, and H. B.
Fotsin
, “Coexistence of multiple attractors and crisis route to chaos in autonomous third order Duffing-Holmes type chaotic oscillators
,” Commun. Nonlinear Sci. Numer. Simul.
36
, 29
–44
(2016
).16.
E.
Gilardi-Velázquez
, H. E.
Ontañón-García
, L. J.
Hurtado-Rodriguiez
, and D. G.
Campos-Cantón
, “Multistability in piecewise linear systems versus eigenspectra variation and round function
,” Int. J. Bifurcation Chaos
27
, 1730031
(2017
).17.
C.
Li
and J. C.
Sprott
, “Multistability in a butterfly flow
,” Int. J. Bifurcation Chaos
23
, 1350199
(2013
).18.
D. T. Z. T.
Njitacke
, J.
Kengne
, H. B.
Fotsin
, and A. N.
Negou
, “Coexistence of multiple attractors and crisis route to chaos in a novel memristive diode bridge-based Jerk circuit
,” Chaos, Solitons Fractals
91
, 180
–197
(2016
).19.
R.
Carvalho
, B.
Fernandez
, and R. V.
Mendes
, “From synchronization to multistability in two coupled quadratic maps
,” Phys. Lett. A
285
, 327
–338
(2001
).20.
V.
Astakhov
, A.
Shabunin
, W.
Uhm
, and W.
Kim
, “Multistability formation and synchronization loss in coupled Hénon maps: Two sides of the single bifurcational mechanism
,” Phys. Rev. E
63
, 056212
(2001
).21.
M.
Guzzo
, “Chaos and diffusion in dynamical systems through stable-unstable manifolds
,” in Space Manifold Dynamics
(Springer
, New York, NY
, 2010
), Chap. 2, pp. 97
–112
.22.
E.
Campos-Cantón
, R.
Femat
, and G.
Chen
, “Attractors generated from switching unstable dissipative systems
,” Chaos
22
, 033121
(2012
).23.
L. J.
Ontañón-García
, E.
Jiménez-López
, E.
Campos-Cantón
, and M.
Basin
, “A family of hyperchaotic multi-scroll attractors in R̂n
,” Appl. Math. Comput.
233
, 522
–533
(2014
).© 2018 Author(s).
2018
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