A self-consistent dynamical system with multiple absolutely continuous invariant measures

Fanni M. Sélley

doi:10.3934/jcd.2021002

Article Contents

2021, Volume 8, Issue 1: 9-32. Doi: 10.3934/jcd.2021002

This issue Previous Article Homogeneous darboux polynomials and generalising integrable ODE systems Next Article The geometry of convergence in numerical analysis

A self-consistent dynamical system with multiple absolutely continuous invariant measures

Fanni M. Sélley^,

Laboratoire de Probabilités, Statistique et Modélisation (LPSM), Sorbonne Université, Université de Paris, 4 Place Jussieu, 75005 Paris, France

Received: September 2019

Revised: June 2020

Published: January 2021

^*The author was supported by ERC grant No 787304 and NKFIH OTKA grant K123782.

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Abstract

In this paper we study a class of self-consistent dynamical systems, self-consistent in the sense that the discrete time dynamics is different in each step depending on current statistics. The general framework admits popular examples such as coupled map systems. Motivated by an example of [9], we concentrate on a special case where the dynamics in each step is a $ \beta $-map with some $ \beta \geq 2 $. Included in the definition of $ \beta $ is a parameter $ \varepsilon > 0 $ controlling the strength of self-consistency. We show such a self-consistent system which has a unique absolutely continuous invariant measure (acim) for $ \varepsilon = 0 $, but at least two for any $ \varepsilon > 0 $. With a slight modification, we transform this system into one which produces a phase transition-like behavior: it has a unique acim for $ 0< \varepsilon < \varepsilon^* $, and multiple for sufficiently large values of $ \varepsilon $. We discuss the stability of the invariant measures by the help of computer simulations employing the numerical representation of the self-consistent transfer operator.

Keywords:

Self-consistent dynamics,
phase transition,
$ \beta $-map,
absolutely continuous invariant measure,
discrete transfer operator.

Mathematics Subject Classification: Primary: 37A05, 37A10, 37E05; Secondary: 37M25, 65P99.

Citation:

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Figure 1. The choice of $ \beta_k $, $ k = 3 $ is pictured

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Figure 2. Approximation of $ \psi^{\varepsilon} $ with ergodic averages, $ F(x) = x $. $ P_{x, N}(\beta) $ is plotted for $ \beta \in [2, \beta_1] $ with gridsize $ \Delta $ and for $ x $ drawn uniform randomly from $ [0, 1] $. The line $ x = y $ is plotted

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Figure 3. Approximation of $ \psi^{\varepsilon} $ with ergodic averages. $ P_{x, N}(\beta) $ is plotted for $ \beta \in [2, 3] $ with gridsize $ \Delta = 10^{-4} $, $ N = 10^6 $ and for $ x $ drawn uniform randomly from $ [0, 1] $, $ \varepsilon = 0.8 $. The line $ x = y $ is plotted

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Figure 4. Approximation of $ \psi^{\varepsilon} $ with ergodic averages, $ F(x) = x^2 $. $ P_{x, N}(\beta) $ is plotted for $ \beta \in [2, \beta_1] $ with gridsize $ \Delta = 10^{-4} $, $ N = 10^6 $ and for $ x $ drawn uniform randomly from $ [0, 1] $. The line $ x = y $ is plotted

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Figure 5. Approximation of $ \psi^{\varepsilon} $ with ergodic averages, $ P_{x, N}(\beta) $ is plotted for $ \beta \in [2, 3] $ with gridsize $ \Delta = 10^{-4} $, $ N = 10^6 $ and for $ x $ drawn uniform randomly from $ [0, 1] $. The line $ x = y $ is plotted

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Figure 6. Total variation for each $ f_t^i $ as a function of time, $ F(x) = x $, $ T = 100 $, $ K_1\times K_2 = 100 $

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Figure 7. Associated slope to each $ f_t^i $ as a function of time, $ F(x) = x $, $ T = 100 $, $ K_1\times K_2 = 100 $

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Figure 8. Approximation of the invariant density of the self-consistent system (4) with $ F(x) = x $ by a high iterate of an appropriate initial density

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Figure 9. For each $ \varepsilon = k \cdot 10^{-3} $, $ k = 1, \dots, 10^3 $ the values $ \overline{\text{var}}(t) $ and $ \overline{\beta}(t) $ are plotted for $ t = 150, \dots, 200 $, $ F(x) = x $, $ K_1 \times K_2 = 100 $

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Figure 10. For each $ \varepsilon = k \cdot \Delta $, $ k = 1, \dots, E/\Delta $ the values $ \overline{\text{var}}(t) $ are plotted for $ t = 150, \dots, 200 $, $ K_1 \times K_2 = 100 $

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Table 1. Computation of the mean total variation $ \overline{\text{var}} $ and mean slope for $ F(x) = x $. $ T = 100 $, $ M = K_1 = K_2 = 10 $. Left hand side: $ \varepsilon = 0.1 $, right hand side: $ \varepsilon = 0.2 $

$t $	$\overline{\text{var}}$	$\overline{\beta}$	$t $	$\overline{\text{var}}$	$\overline{\beta}$
0	7.1815	2.0065	0	5.1121	1.9993
5	1.1544	2.0039	5	0.9983	2.0130
10	0.9559	2.0020	10	0.9415	2.0117
15	0.9856	2.0015	15	0.9490	2.0134
20	0.9901	2.0012	20	0.9429	2.0149
25	0.9917	2.0011	25	0.9371	2.0160
30	0.9928	2.0010	30	0.9341	2.0165
35	0.9934	2.0009	35	0.9323	2.0168
40	0.9940	2.0008	40	0.9330	2.0171
45	0.9944	2.0008	45	0.9320	2.0174
50	0.9947	2.0007	50	0.9309	2.0176
55	0.9949	2.0007	55	0.9298	2.0178
60	0.9950	2.0007	60	0.9289	2.0179
65	0.9952	2.0007	65	0.9280	2.0180
70	0.9953	2.0007	70	0.9274	2.0181
75	0.9954	2.0007	75	0.9270	2.0181
80	0.9954	2.0007	80	0.9269	2.0181
85	0.9955	2.0007	85	0.9269	2.0181
90	0.9955	2.0006	90	0.9269	2.0181
95	0.9956	2.0006	95	0.9270	2.0181
100	0.9956	2.0006	100	0.9270	2.0181

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Table 2. Computation of the mean total variation $ \overline{\text{var}} $ and mean slope for $ F(x) = x $, $ \text{var}(f^i_0) < 10^{-4} $. $ T = 300 $, $ M = K_1 = K_2 = 10 $. Left hand side: $ \varepsilon = 0.1 $, right hand side: $ \varepsilon = 0.2 $

$t $	$\overline{\text{var}}$	$\overline{\beta}$	$t $	$\overline{\text{var}}$	$\overline{\beta}$
0	0.0000	2.0000	0	0.0000	2.0000
25	1.0000	2.0000	25	1.0000	2.0000
50	1.0000	2.0000	50	0.9993	2.0003
75	1.0000	2.0000	75	0.9783	2.0072
100	0.9998	2.0000	100	0.9362	2.0160
125	0.9995	2.0001	125	0.9307	2.0177
150	0.9987	2.0002	150	0.9268	2.0181
175	0.9978	2.0003	175	0.9269	2.0181
200	0.9969	2.0005	200	0.9269	2.0181
225	0.9966	2.0005	225	0.9269	2.0181
250	0.9966	2.0005	250	0.9269	2.0181
275	0.9964	2.0005	275	0.9269	2.0181
300	0.9962	2.0006	300	0.9269	2.0181

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Table 3. Computation of $ \overline{\text{int}}_{f_*(\varepsilon)} $ and mean slope for $ F(x) = x $, $ T = 200 $, $ \bar{T} = 5000 $. $ M = K_1 = K_2 = 10 $

$t$	$\overline{\text{int}}_{f_*(0.1)}$	$\overline{\text{int}}_{f_*(0.2)}$
0	0.4096	0.5018
10	0.0101	0.0321
20	0.0049	0.0174
30	0.0028	0.0097
40	0.0017	0.0064
50	0.0011	0.0040
60	0.0007	0.0021
70	0.0005	0.0007
80	0.0004	0.0003
90	0.0003	0.0002
100	0.0002	0.0001
110	0.0002	0.0001
120	0.0002	0.0001
130	0.0001	0.0001
140	0.0001	0.0001
150	0.0001	0.0001
160	0.0001	0.0001
170	0.0001	0.0001
180	0.0001	0.0001
190	0.0001	0.0001
200	0.0001	0.0001

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Table 4. Computation of the mean total variation $ \overline{\text{var}} $. $ T = 100 $, $ M = K_1 = K_2 = 10 $. Left hand side: $ F(x) = x^2 $, center: $ F(x) = x^4 $, right hand side: $ F(x) = x^6 $

$t $	$ \varepsilon=1 $	$ \varepsilon=2.5$	$t $	$\varepsilon=1 $	$ \varepsilon=35$	$t$	$\varepsilon=1$	$ \varepsilon=400 $
0	6.4694	5.0341	0	5.3766	6.4806	0	7.7582	5.1444
5	1.2483	1.1951	5	1.2370	1.1785	5	1.3852	1.0419
10	0.9808	0.8849	10	0.9882	0.9788	10	0.5224	0.5990
15	0.9914	0.8471	15	0.6884	0.9873	15	0.0210	0.1435
20	0.9976	0.8447	20	0.0474	0.5994	20	0.0007	0.0026
25	0.9994	0.8275	25	0.0015	0.1093	25	0.0000	0.0002
30	0.9999	0.8418	30	0.0000	0.0347	30	0.0000	0.0000
35	1.0000	0.8305	35	0.0000	0.0023	35	0.0000	0.0000
40	1.0000	0.8468	40	0.0000	0.0001	40	0.0000	0.0000
45	1.0000	0.8321	45	0.0000	0.0000	45	0.0000	0.0000
50	0.9531	0.8358	50	0.0000	0.0000	50	0.0000	0.0000
55	0.5357	0.8382	55	0.0000	0.0000	55	0.0000	0.0000
60	0.1802	0.7651	60	0.0000	0.0000	60	0.0000	0.0000
65	0.0128	0.6366	65	0.0000	0.0000	65	0	0.0000
70	0.0004	0.5117	70	0.0000	0.0000	70	0	0
75	0.0000	0.4535	75	0	0.0000	75	0	0
80	0.0000	0.4269	80	0	0.0000	80	0	0
85	0.0000	0.4255	85	0	0	85	0	0
90	0.0000	0.4015	90	0	0	90	0	0
95	0.0000	0.3966	95	0	0	95	0	0
100	0.0000	0.3813	100	0	0	100	0	0

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