We investigate the dynamics of a family of one-dimensional linear-power maps. This family has been studied by many authors mainly in the continuous case, associated with Nordmark systems. In the discontinuous case, which is much less studied, the map has vertical and horizontal asymptotes giving rise to new kinds of border collision bifurcations. We explain a mechanism of the interplay between smooth bifurcations and border collision bifurcations with singularity, leading to peculiar sequences of attracting cycles of periods $n,2n$, $4n-1$, $2(4n-1)$, ..., $n≥3$. We show also that the transition from invertible to noninvertible map may lead abruptly to chaos, and the role of organizing center in the parameter space is played by a particular bifurcation point related to this transition and to a flip bifurcation. Robust unbounded chaotic attractors characteristic for certain parameter ranges are also described. We provide proofs of some properties of the considered map. However, the complete description of its rich bifurcation structure is still an open problem.
Citation: |
Figure 1. 2D bifurcation diagrams of $f$ in the $(a, S(b))$-parameter plane, where $S(b) = \arctan (b)$, for $\gamma = 0.5 $ in (a) and $\gamma = 1.5$ in (b); striped regions are related to coexistence, colored regions to attracting cycles of different periods, uncolored regions to attracting cycles of higher periods or chaotic attractors, grey regions to divergence. In (c) examples of map $f$ are shown
Figure 7. In (a), repelling 4-cycle $ML^{3}$ of map $g$ corresponding to the repelling 5-cycle $R^{2}L^{3}$ of map $f.$ In (b), the enlargement of the small rectangle marked in (a), which shows also the graph of $g^{8}(x)$ with an attracting 8-cycle $L^{2}(LM)^{3}$ of map $g$ corresponding to the attracting 11-cycle $L^{2}(LR^{2})^{3}$ of map $f.$ Here $a = -2.16$, $b = 10$, $\gamma = 1.5.$
Figure 8. Qualitative representation of the S-fold and BCBs of map $g^{n}$ in a neighborhood of $x = 0, $ in all the possible cases, showing the shape of $g^{n}$ at the bifurcation and after: In (a) a fold-BCB; in (b) an S-fold, related to a local maximum of $g^{n}$ in $x = 0;$ in (c) a fold-BCB; in (d) an S-fold, related to a local minimum of $g^{n}$ in $x = 0.$
F. Angulo
and M. di Bernardo
, Feedback control of limit cycles: A switching control strategy based on nonsmooth bifurcation theory, IEEE Transactions on Circuits and Systems-I, 52 (2005)
, 366-378.
doi: 10.1109/TCSI.2004.841595.![]() ![]() ![]() |
|
V. Avrutin
, I. Sushko
and L. Gardini
, Cyclicity of chaotic attractors in one-dimensional discontinuous maps, Mathematics and Computers in Simulation, 95 (2014)
, 126-136.
doi: 10.1016/j.matcom.2012.07.019.![]() ![]() ![]() |
|
V. Avrutin
, P. S. Dutta
, M. Schanz
and S. Banerjee
, Influence of a square-root singularity on the behaviour of piecewise smooth maps, Nonlinearity, 23 (2010)
, 445-463.
doi: 10.1088/0951-7715/23/2/012.![]() ![]() ![]() |
|
M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems: Theory and Applications, Applied Mathematical Sciences 163, Springer-Verlag, London, 2008.
![]() ![]() |
|
M. di Bernardo
, C. Budd
and A. Champneys
, Grazing, skipping and sliding: Analysis of the non-smooth dynamics of the dc-dc buck converter, Nonlinearity, 11 (1998)
, 859-890.
doi: 10.1088/0951-7715/11/4/007.![]() ![]() ![]() |
|
M. di Bernardo
, C. J. Budd
and A. R. Champneys
, Corner collision implies border-collision bifurcation, Physica D, 154 (2001)
, 171-194.
doi: 10.1016/S0167-2789(01)00250-0.![]() ![]() ![]() |
|
M. di Bernardo
, P. Kowalczyk
and A. B. Nordmark
, Bifurcations of dynamical systems with sliding: Derivation of normal-form mappings, Physica D, 170 (2002)
, 175-205.
doi: 10.1016/S0167-2789(02)00547-X.![]() ![]() ![]() |
|
G.I. Bischi
, C. Mira
and L. Gardini
, Unbounded sets of attraction, International Journal of Bifurcation and Chaos, 10 (2000)
, 1437-1469.
doi: 10.1142/S0218127400000980.![]() ![]() ![]() |
|
H. Dankowicz
and A. B. Nordmark
, On the origin and bifurcations of stick-slip oscillations, Physica D, 136 (2000)
, 280-302.
doi: 10.1016/S0167-2789(99)00161-X.![]() ![]() ![]() |
|
L. Gardini
, I. Sushko
, V. Avrutin
and M. Schanz
, Critical homoclinic orbits lead to snap-back repellers, Chaos Solitons Fractals, 44 (2011)
, 433-449.
doi: 10.1016/j.chaos.2011.03.004.![]() ![]() ![]() |
|
C. Halse
, M. Homer
and M. di Bernardo
, C-bifurcations and period-adding in one-dimensional piecewise-smooth maps, Chaos, Solitons Fractals, 18 (2003)
, 953-976.
doi: 10.1016/S0960-0779(03)00066-3.![]() ![]() ![]() |
|
M. Jakobson
, Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Commun Math Phys, 81 (1981)
, 39-88.
doi: 10.1007/BF01941800.![]() ![]() ![]() |
|
A. Kumar
, S. Banerjee
and D. P. Lathrop
, Dynamics of a piecewise smooth map with singularity, Physics Letters A, 337 (2005)
, 87-92.
doi: 10.1016/j.physleta.2005.01.046.![]() ![]() ![]() |
|
Y. L. Maistrenko
, V. L. Maistrenko
and L. O. Chua
, Cycles of chaotic intervals in a time-delayed Chua's circuit, Int. J. Bifurcat. Chaos, 3 (1993)
, 1557-1572.
doi: 10.1142/S0218127493001215.![]() ![]() ![]() |
|
R. Makrooni and L. Gardini, Bifurcation Structures in a Family of One-Dimensional LinearPower Discontinuous Maps, Gecomplexity Discussion Paper N. 7,2015, ISSN: 2409–7497. http://econpapers.repec.org/paper/cstwpaper/
![]() |
|
R. Makrooni
, N. Abbasi
, M. Pourbarat
and L. Gardini
, Robust unbounded chaotic attractors in 1D discontinuous maps, Chaos, Solitons Fractals, 77 (2015)
, 310-318.
doi: 10.1016/j.chaos.2015.06.012.![]() ![]() ![]() |
|
R. Makrooni
, F. Khellat
and L. Gardini
, Border collision and fold bifurcations in a family of piecesiwe smooth maps. Part Ⅰ: Unbounded chaotic sets, J. Difference Equ. Appl., 21 (2015)
, 660-695.
doi: 10.1080/10236198.2015.1045893.![]() ![]() ![]() |
|
R. Makrooni
, F. Khellat
and L. Gardini
, Border collision and fold bifurcations in a family of piecesiwe smooth maps: divergence and bounded dynamics, J. Difference Equ. Appl., 21 (2015)
, 791-824.
doi: 10.1080/10236198.2015.1046855.![]() ![]() ![]() |
|
R. Makrooni
, L. Gardini
and I. Sushko
, Bifurcation structures in a family of 1D discontinuos linear-hyperbolic invertible maps, Int. J. Bifurcation and Chaos, 25 (2015)
, 1530039 (21 pages).
doi: 10.1142/S0218127415300396.![]() ![]() ![]() |
|
N. Metropolis
, M. L. Stein
and P. R. Stein
, On finite limit sets for transformations on the unit interval, J Comb Theory, 15 (1973)
, 25-44.
doi: 10.1016/0097-3165(73)90033-2.![]() ![]() ![]() |
|
A. B. Nordmark
, Non-periodic motion caused by grazing incidence in an impact oscillator, Journal of Sound and Vibration, 145 (1991)
, 279-297.
doi: 10.1016/0022-460X(91)90592-8.![]() ![]() |
|
A. B. Nordmark
, Universal limit mapping in grazing bifurcations, Physical Review E, 55 (1997)
, 266-270.
doi: 10.1103/PhysRevE.55.266.![]() ![]() |
|
A. B. Nordmark
, Existence of priodic orbits in grazing bifurcations of impacting mechanical oscillators, Nonlinearity, 14 (2001)
, 1517-1542.
doi: 10.1088/0951-7715/14/6/306.![]() ![]() ![]() |
|
H. E. Nusse
and J. A. Yorke
, Border-collision bifurcations including period two to period three for piecewise smooth systems, Physica D, 57 (1992)
, 39-57.
doi: 10.1016/0167-2789(92)90087-4.![]() ![]() ![]() |
|
H. E. Nusse
and J. A. Yorke
, Border-collision bifurcation for piecewise smooth one-dimensional maps, Int. J. Bifurcation Chaos, 5 (1995)
, 189-207.
doi: 10.1142/S0218127495000156.![]() ![]() ![]() |
|
H. Nusse
, E. Ott
and J. Yorke
, Border collision bifurcations: An explanation for observed bifurcation phenomena, Phys. Rev. E, 49 (1994)
, 1073-1076.
doi: 10.1103/PhysRevE.49.1073.![]() ![]() ![]() |
|
Z. Qin
, J. Yang
, S. Banerjee
and G. Jiang
, Border collision bifurcations in a generalized piecewise linear-power map, Discrete and Continuous Dynamical System, Series B, 16 (2011)
, 547-567.
doi: 10.3934/dcdsb.2011.16.547.![]() ![]() ![]() |
|
Z. Qin
, Z. Yuejing
, J. Yang
and Y. Jichen
, Nonsmooth and smooth bifurcations in a discontinuous piecewise map, Int. J. Bifurcation and Chaos, 22 (2012)
, 1250112 (7 pages).
doi: 10.1142/S021812741250112X.![]() ![]() ![]() |
|
W. T. Shi
, C. L. Gooderidge
and D. P. Lathrop
, Viscous effects in droplet-ejecting capillary waves, Phys. Rev. E, 56 (1997)
, 41-57.
![]() |
|
I. Sushko
, A. Agliari
and L. Gardini
, Bistability and bifurcation curves for a unimodal piecewise smooth map, Discrete and Continuous Dynamical Systems, Serie B, 5 (2005)
, 881-897.
doi: 10.3934/dcdsb.2005.5.881.![]() ![]() ![]() |
|
I. Sushko
, A. Agliari
and L. Gardini
, Bifurcation structure of parameter plane for a family of unimodal piecewise smooth maps: border-collision bifurcation curves, Chaos Solitons Fractals, 29 (2006)
, 756-770.
doi: 10.1016/j.chaos.2005.08.107.![]() ![]() ![]() |
|
I. Sushko
and L. Gardini
, Degenerate bifurcations and border collisions in piecewise smooth 1D and 2D maps, Int. J. Bif. and Chaos, 20 (2010)
, 2045-2070.
doi: 10.1142/S0218127410026927.![]() ![]() ![]() |
|
I. Sushko
, L. Gardini
and K. Matsuyama
, Superstable credit cycles and U-sequence, Chaos Solitons Fractals, 59 (2014)
, 13-27.
doi: 10.1016/j.chaos.2013.11.006.![]() ![]() ![]() |
|
I. Sushko
, V. Avrutin
and L. Gardini
, Bifurcation structure in the skew tent map and its application as a border collision normal form, Journal of Difference Equations and Applications, 22 (2016)
, 582-629.
doi: 10.1080/10236198.2015.1113273.![]() ![]() ![]() |
|
H. Thunberg
, Periodicity versus chaos in one-dimensional dynamics, SIAM Rev, 43 (2001)
, 3-30.
doi: 10.1137/S0036144500376649.![]() ![]() ![]() |
|
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, SpringerVerlag, New York, 2003.
![]() ![]() |
2D bifurcation diagrams of
Bifurcation diagrams
Graph of map
2D bifurcation diagrams of map
Maps
Attracting 4-cycle
In (a), repelling 4-cycle
Qualitative representation of the S-fold and BCBs of map
In (a): fold-BCB leading to a pair of 3-cycles of map
1D bifurcation diagram as a function of
Fold bifurcation of the pair of 2-cycles
1D bifurcation diagram of map
Graphs of map
Graphs of map