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March  2018, 23(2): 701-729. doi: 10.3934/dcdsb.2018039

Cascades of alternating smooth bifurcations and border collision bifurcations with singularity in a family of discontinuous linear-power maps

Laura Gardini 1,, , Roya Makrooni 2, and  Iryna Sushko 3,

1. 

Department of Economics, Society and Politics, University of Urbino, via Saffi 42,61029 Urbino (PU), Italy
2. 

Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran, Azadi St., Tehran, P.O.Box 11365-11155, Iran
3. 

Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivska st., 01601 Kyiv, Ukraine

* Corresponding author: Laura Gardini

Received  September 2016 Revised  March 2017 Published  March 2018 Early access  December 2017

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.

Keywords: Piecewise smooth maps, border collision bifurcations, cascade of smooth and nonsmooth bifurcations, skew tent map as a normal form, unbounded chaotic sets.
Mathematics Subject Classification: Primary: 37E05, 37G35; Secondary: 37N30.
Citation: Laura Gardini, Roya Makrooni, Iryna Sushko. Cascades of alternating smooth bifurcations and border collision bifurcations with singularity in a family of discontinuous linear-power maps. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 701-729. doi: 10.3934/dcdsb.2018039
References:
[1]

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.

[2]

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.

[3]

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.

[4]

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.

[5]

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.

[6]

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.

[7]

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.

[8]

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.

[9]

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.

[10]

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.

[11]

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.

[12]

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.

[13]

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.

[14]

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.

[15]

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/

[16]

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.

[17]

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.

[18]

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.

[19]

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.

[20]

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.

[21]

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.

[22]

A. B. Nordmark, Universal limit mapping in grazing bifurcations, Physical Review E, 55 (1997), 266-270.  doi: 10.1103/PhysRevE.55.266.

[23]

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.

[24]

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.

[25]

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.

[26]

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.

[27]

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.

[28]

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.

[29]

W. T. Shi, C. L. Gooderidge and D. P. Lathrop, Viscous effects in droplet-ejecting capillary waves, Phys. Rev. E, 56 (1997), 41-57. 

[30]

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.

[31]

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.

[32]

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.

[33]

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.

[34]

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.

[35]

H. Thunberg, Periodicity versus chaos in one-dimensional dynamics, SIAM Rev, 43 (2001), 3-30.  doi: 10.1137/S0036144500376649.

[36]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, SpringerVerlag, New York, 2003.

show all references

References:
[1]

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.

[2]

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.

[3]

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.

[4]

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.

[5]

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.

[6]

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.

[7]

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.

[8]

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.

[9]

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.

[10]

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.

[11]

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.

[12]

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.

[13]

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.

[14]

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.

[15]

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/

[16]

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.

[17]

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.

[18]

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.

[19]

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.

[20]

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.

[21]

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.

[22]

A. B. Nordmark, Universal limit mapping in grazing bifurcations, Physical Review E, 55 (1997), 266-270.  doi: 10.1103/PhysRevE.55.266.

[23]

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.

[24]

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.

[25]

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.

[26]

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.

[27]

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.

[28]

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.

[29]

W. T. Shi, C. L. Gooderidge and D. P. Lathrop, Viscous effects in droplet-ejecting capillary waves, Phys. Rev. E, 56 (1997), 41-57. 

[30]

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.

[31]

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.

[32]

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.

[33]

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.

[34]

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.

[35]

H. Thunberg, Periodicity versus chaos in one-dimensional dynamics, SIAM Rev, 43 (2001), 3-30.  doi: 10.1137/S0036144500376649.

[36]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, SpringerVerlag, New York, 2003.

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
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Figure 2.  Bifurcation diagrams $a$ vs $S(x) $ at $b = -5$, where $S(x) = \arctan (x)$. In (a) $\gamma = 0.5, $ in (b) $\gamma = 2$ and in (c) $\gamma = 1$, associated with the subcritical, supercritical and degenerate flip bifurcations of the 2-cycle $LR$, respectively
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Figure 3.  Graph of map $g(x)$ given in (21). In (a): $a = -1.3$, $b = 1.5$ and $\gamma = 0.5<1;$ in (b) $a = -1.7$, $b = 10$ and $\gamma = 1.5>1$
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Figure 4.  2D bifurcation diagrams of map $f$ in the $(a, b)$-parameter plane for $\gamma = 1.5$ in (a) and $\gamma = 3$ in (b)
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Figure 5.  Maps $g(x)$ and $g^{2}(x)$ at $a = -1.7$, $b = 10$ and $\gamma = 1.5$. In the right panel the generic case starting a period doubling sequence is schematically shown
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Figure 6.  Attracting 4-cycle $(ML)^{2}$ of map $g(x)$ corresponding to the 6-cycle $R^{2}LR^{2}L$ of map $f(x)$. The cycle is close to its border collision. In (b) the enlargement of the small rectangle indicated in (a). Here $a = -2.155$, $b = 10$, $\gamma = 1.5.$
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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 $a = -2.16$, $b = 10$, $\gamma = 1.5.$
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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 $x = 0.$
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Figure 9.  In (a): fold-BCB leading to a pair of 3-cycles of map $g$; In (b): fold-BCB leading to a pair of 4-cycles of map $g$. Here $a = -2.5$, $b = 10$, $\gamma = 1.5.$
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Figure 10.  1D bifurcation diagram as a function of $a$ at $b = 4$, $\gamma = 1.5$ for map $g$ in (a), and for map $f$ in (b). The values of $x$ are scaled as $y = \arctan (x)$ in order to show the values tending to $+\infty $
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Figure 11.  Fold bifurcation of the pair of 2-cycles $LR$ at $a = -3$, $\gamma = 1.5 $ and $b = 0.35$ in (a), $b = 0.2$ in (b)
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Figure 12.  1D bifurcation diagram of map $f$ as a function of $a$ at fixed $b = 1.9$ and $\gamma = 0.5.$ The variable $x$ is scaled by $y = \arctan (x)$. In (b) an enlargement of (a) is shown for $-1.1<a<-1$
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Figure 13.  Graphs of map $g(x)$ at $\gamma = 0.5$ and $b = 0.5<b_{R}^{f}$. In (a) $a = -1.1$; in (b) $a = -2$; in (c) $a = -4.5.$
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Figure 14.  Graphs of map $g$ at $\gamma = 0.5$ and $b = 2.2>b_{R}^{f}$. In (a) $a = -1.2$; in (b) $a = -1.6.$ The fixed point is the unique attractor
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