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On coupled Dirac systems
Statistical and deterministic dynamics of maps with memory
1. | Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec H3G 1M8, Canada |
2. | Department of Mathematics, Honghe University, Mengzi, Yunnan 661100, China |
3. | Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec H3G 1M8, Canada |
We consider a dynamical system to have memory if it remembers the current state as well as the state before that. The dynamics is defined as follows: $x_{n+1}=T_{\alpha }(x_{n-1}, x_{n})=\tau (\alpha \cdot x_{n}+(1-\alpha)\cdot x_{n-1}), $ where $\tau$ is a one-dimensional map on $I=[0, 1]$ and $0 < \alpha < 1$ determines how much memory is being used. $T_{\alpha }$ does not define a dynamical system since it maps $U=I\times I$ into $I$. In this note we let $\tau $ be the symmetric tent map. We shall prove that for $0 < \alpha < 0.46, $ the orbits of $\{x_{n}\}$ are described statistically by an absolutely continuous invariant measure (acim) in two dimensions. As $\alpha $ approaches $0.5 $ from below, that is, as we approach a balance between the memory state $x_{n-1}$ and the present state $x_{n}$, the support of the acims become thinner until at $\alpha =0.5$, all points have period 3 or eventually possess period 3. For $% 0.5 < \alpha < 0.75$, we have a global attractor: for all starting points in $U$ except $(0, 0)$, the orbits are attracted to the fixed point $(2/3, 2/3).$ At $%\alpha=0.75, $ we have slightly more complicated periodic behavior.
References:
[1] |
P. Góra, A. Boyarsky and Z. Li, Singular SRB measures for a non 1-1 map of the unit square, Journal of Stat. Physics, 165 (2016), 409-433, available at http://arxiv.org/abs/1607. 01658, full-text view-only version: http://rdcu.be/kod0
doi: 10.1007/s10955-016-1620-y. |
[2] |
F. Dyson,
Birds and Frogs, Notices of Amer. Math. Soc., 56 (2009), 212-223.
|
[3] |
J. Maynard Smith, Mathematical Ideas in Biology, Cambridge University Press, 1968.
doi: 10.1017/CBO9780511565144. |
[4] |
B. Saussol,
Absolutely continuous invariant measures for multidimensional expanding maps, Israel J. Math., 116 (2000), 223-248.
doi: 10.1007/BF02773219. |
[5] |
M. Tsujii,
Absolutely continuous invariant measures for piecewise real-analytic expanding maps on the plane, Commun. Math Phys., 208 (2000), 605-622.
doi: 10.1007/s002200050003. |
[6] |
G. -C. Wu and D. Baleanu,
Discrete chaos in fractional delayed logistic maps, Nonlinear Dynam., 80 (2015), 1697-1703.
doi: 10.1007/s11071-014-1250-3. |
[7] |
L. Zou,
A lower bound for the smallest singular value, J. Math. Inequal., 6 (2012), 625-629.
doi: 10.7153/jmi-06-60. |
show all references
References:
[1] |
P. Góra, A. Boyarsky and Z. Li, Singular SRB measures for a non 1-1 map of the unit square, Journal of Stat. Physics, 165 (2016), 409-433, available at http://arxiv.org/abs/1607. 01658, full-text view-only version: http://rdcu.be/kod0
doi: 10.1007/s10955-016-1620-y. |
[2] |
F. Dyson,
Birds and Frogs, Notices of Amer. Math. Soc., 56 (2009), 212-223.
|
[3] |
J. Maynard Smith, Mathematical Ideas in Biology, Cambridge University Press, 1968.
doi: 10.1017/CBO9780511565144. |
[4] |
B. Saussol,
Absolutely continuous invariant measures for multidimensional expanding maps, Israel J. Math., 116 (2000), 223-248.
doi: 10.1007/BF02773219. |
[5] |
M. Tsujii,
Absolutely continuous invariant measures for piecewise real-analytic expanding maps on the plane, Commun. Math Phys., 208 (2000), 605-622.
doi: 10.1007/s002200050003. |
[6] |
G. -C. Wu and D. Baleanu,
Discrete chaos in fractional delayed logistic maps, Nonlinear Dynam., 80 (2015), 1697-1703.
doi: 10.1007/s11071-014-1250-3. |
[7] |
L. Zou,
A lower bound for the smallest singular value, J. Math. Inequal., 6 (2012), 625-629.
doi: 10.7153/jmi-06-60. |


































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