Public debt dynamics under ambiguity by means of iterated function systems on density functions

Previous Article
Stability switching and its directions in cournot duopoly game with three delays
DCDS-B Home
This Issue
Next Article
Transitions between metastable long-run consumption behaviors in a stochastic peer-driven consumer network

November 2021, 26(11): 5873-5903. doi: 10.3934/dcdsb.2021070

Public debt dynamics under ambiguity by means of iterated function systems on density functions

Davide La Torre ^1, , Simone Marsiglio ^2, , Franklin Mendivil ^3, and Fabio Privileggi ^4,,

1.	SKEMA Business School and Université Côte d'Azur, Sophia Antipolis, France
2.	Department of Economics and Management, University of Pisa, Pisa, Italy
3.	Department of Mathematics and Statistics, Acadia University, Wolfville, Canada
4.	Department of Economics and Statistics "Cognetti de Martiis", University of Turin, 10153 Torino, Italy

^* Corresponding author: Fabio Privileggi

Received September 2020 Revised November 2020 Published November 2021 Early access March 2021

We analyze a purely dynamic model of public debt stabilization under ambiguity. We assume that the debt to GDP ratio is described by a random variable, and thus it can be characterized by investigating the evolution of its density function through iteration function systems on mappings. Ambiguity is associated with parameter uncertainty which requires policymakers to respond to such an additional layer of uncertainty according to their ambiguity attitude. We describe ambiguity attitude through a simple heuristic rule in which policymakers adjust the available vague information (captured by the empirical distribution of the debt ratio) with a measure of their ignorance (captured by the uniform distribution). We show that such a model generates fractal-type objects that can be characterized as fixed-point solutions of iterated function systems on mappings. Ambiguity is a source of unpredictability in the long run outcome since it introduces some singularity features in the steady state distribution of the debt ratio. However, the presence of some ambiguity aversion removes such unpredictability by smoothing out the singularities in the steady state distribution.

Keywords: Dynamical systems, global attractor, generalized fractal transform, fixed point equation, iterated function systems on density functions, ambiguity, public debt.

Mathematics Subject Classification: Primary: 28A80; Secondary: 91B64.

Citation: Davide La Torre, Simone Marsiglio, Franklin Mendivil, Fabio Privileggi. Public debt dynamics under ambiguity by means of iterated function systems on density functions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (11) : 5873-5903. doi: 10.3934/dcdsb.2021070

References:

[1]	S. R. Baker, N. Bloom and S.J. Davis, Measuring economic policy uncertainty, Quarterly Journal of Economics, 131 (2015), 1593-1636. doi: 10.3386/w21633. Google Scholar
[2]	S. Banach, Sur les opérations dans les ensembles abstraits et leurs applications aux équationsintégrales, Fundamenta Mathematicae, 3 (1922), 133-181. doi: 10.4064/fm-3-1-133-181. Google Scholar
[3]	M. F. Barnsley, Fractals Everywhere, Academic Press, New York, 1988. Google Scholar
[4]	B. Born and J. Pfeifer, Policy risk and the business cycle, Journal of Monetary Economics, 68 (2014), 68-85. doi: 10.1016/j.jmoneco.2014.07.012. Google Scholar
[5]	W. Brainard, Uncertainty and the effectiveness of policy, American Economic Review, 57 (1967), 411-425. Google Scholar
[6]	W. A. Brock and S. Durlauf, Macroeconomics and Model Incertainty, in D. Colander (Ed.), Post Walrasian Macroeconomics: Beyond the Dynamic Stochastic General Equilibrium Model, Cambridge University Press, Cambridge, 2006. Google Scholar
[7]	W. A. Brock and L. J. Mirman, Optimal economic growth and uncertainty: The discounted case, Journal of Economic Theory, 4 (1972), 479-513. doi: 10.1016/0022-0531(72)90135-4. Google Scholar
[8]	C. Camerer and M. Weber, Recent developments in modeling preferences: uncertainty and ambiguity, Journal of Risk and Uncertainty, 5 (1992), 325-370. doi: 10.1007/BF00122575. Google Scholar
[9]	F. Caprioli, Optimal fiscal policy under learning, Jouirnal of Economics Dynamics & Control, 58 (2015), 101-124. doi: 10.1016/j.jedc.2015.05.008. Google Scholar
[10]	G. Cozzi and P. E. Giordani, Ambiguity attitude, R & D investments and economic growth, Journal of Evolutionary Economics, 21 (2011), 303-319. doi: 10.1007/s00191-010-0217-x. Google Scholar
[11]	D. Ellsberg, Risk, ambiguity and the savage axioms, Quarterly Journal of Economics, 75 (1961), 643-669. doi: 10.2307/1884324. Google Scholar
[12]	J. Etner, M. Jeleva and J.-M. Tallon, Decision theory under ambiguity, Journal of Economic Surveys, 26 (2012), 234-270. doi: 10.1111/j.1467-6419.2010.00641.x. Google Scholar
[13]	B. Forte and E. R. Vrscay, Solving the inverse problem for function and image approximation using iterated function systems, Dynamics of Continuous, Discrete and Impulsive Systems, 1 (1995), 177-232. Google Scholar
[14]	B. Forte and E. R. Vrscay, Theory of Generalized Fractal Transforms, in Y. Fisher (Ed.), Fractal Image Encoding and Analysis, NATO ASI Series F, Springer Verlag, New York, 1998. Google Scholar
[15]	D. Frisch and J. Baron, Ambiguity and rationality, Journal of Behavioral Decision Making, 1 (1988), 149-157. doi: 10.1002/bdm.3960010303. Google Scholar
[16]	P. Ghirardato, F. Maccheroni and M. Marinacci, Differentiating ambiguity and ambiguity attitude, Journal of Economic Theory, 118 (2004), 133-173. doi: 10.1016/j.jet.2003.12.004. Google Scholar
[17]	L. P. Hansen and T. J. Sargent, Robustness, Princeton University Press, Princeton, 2008. doi: 10.1515/9781400829385. Google Scholar
[18]	J. Hollmayr and C. Matthes, Learning about fiscal policy and the effects of policy uncertainty, Journal of Economic Dynamics & Control, 59 (2015), 142-162. doi: 10.1016/j.jedc.2015.08.002. Google Scholar
[19]	A. G. Karantounias, Managing pessimistic expectations and fiscal policy, Theorertical Economics, 8 (2013), 193-231. doi: 10.3982/TE899. Google Scholar
[20]	H. Kunze, D. La Torre, F. Mendivil and E. R. Vrscay, Fractal-Based Methods in Analysis, Springer, New York, 2012. doi: 10.1007/978-1-4614-1891-7. Google Scholar
[21]	D. La Torre, S. Marsiglio and F. Privileggi, Fractals and self-similarity in economics: the case of a stochastic two-sector growth model, Image Analysys and Stereology, 30 (2011), 143-151. doi: 10.5566/ias.v30.p143-151. Google Scholar
[22]	D. La Torre, S. Marsiglio, F. Mendivil and F. Privileggi, Self-similar measures in multi-sector endogenous growth models, Chaos, Solitons and Fractals, 79 (2015), 40-56. doi: 10.1016/j.chaos.2015.05.019. Google Scholar
[23]	D. La Torre, S. Marsiglio and F. Privileggi, Fractal attractors in economic growth models with random pollution externalities, Chaos, 28 (2018), 055916, 12 pp. doi: 10.1063/1.5023782. Google Scholar
[24]	D. La Torre, S. Marsiglio, F. Mendivil and F. Privileggi, Fractal attractors and singular invariant measures in two-sector growth models with random factor shares, Communications in Nonlinear Science and Numerical Simulation, 58 (2018), 185-201. doi: 10.1016/j.cnsns.2017.07.008. Google Scholar
[25]	D. La Torre and S. Marsiglio, A note on optimal debt reduction policies, Macroeconomic Dynamics, 24 (2020), 1850-1860. doi: 10.1017/S1365100519000014. Google Scholar
[26]	T. Mitra, L. Montrucchio and F. Privileggi, The nature of the steady state in models of optimal growth under uncertainty, Economic Theory, 23 (2004), 39-71. doi: 10.1007/s00199-002-0340-5. Google Scholar
[27]	T. Mitra and F. Privileggi, Cantor Type Invariant Distributions in the Theory of Optimal Growth under Uncertainty, Journal of Difference Equations and Applications, 10 (2004), 489-500. doi: 10.1080/1023619042000193649. Google Scholar
[28]	T. Mitra and F. Privileggi, Cantor type attractors in stochastic growth models, Chaos, Solitons and Fractals, 29 (2006), 626-637. doi: 10.1016/j.chaos.2005.08.094. Google Scholar
[29]	T. Mitra and F. Privileggi, On Lipschitz continuity of the iterated function system in a stochastic optimal growth model, Journal of Mathematical Economics, 45 (2009), 185-198. doi: 10.1016/j.jmateco.2008.08.003. Google Scholar
[30]	L. Montrucchio and F. Privileggi, Fractal steady states in stochastic optimal control models, Annals of Operations Research, 88 (1999), 183-197. doi: 10.1023/A:1018978213041. Google Scholar
[31]	L. J. Olson and S. Roy, Theory of stochastic optimal economic growth, in R. A. Dana, C. Le Van, T. Mitra and K. Nishimura (Eds.), Handbook on optimal growth 1: discrete time, Springer, New York (2005), 297–335. doi: 10.1007/3-540-32310-4_11. Google Scholar
[32]	F. Privileggi and S. Marsiglio, Environmental shocks and sustainability in a basic economy-environment model, International Journal of Applied Nonlinear Science, 1 (2013), 67-75. doi: 10.1504/IJANS.2013.052755. Google Scholar
[33]	D. Rodrik, Policy uncertainty and private investment, Journal of Development Economics, 36 (1991), 229-242. Google Scholar

show all references

References:

[1]	S. R. Baker, N. Bloom and S.J. Davis, Measuring economic policy uncertainty, Quarterly Journal of Economics, 131 (2015), 1593-1636. doi: 10.3386/w21633. Google Scholar
[2]	S. Banach, Sur les opérations dans les ensembles abstraits et leurs applications aux équationsintégrales, Fundamenta Mathematicae, 3 (1922), 133-181. doi: 10.4064/fm-3-1-133-181. Google Scholar
[3]	M. F. Barnsley, Fractals Everywhere, Academic Press, New York, 1988. Google Scholar
[4]	B. Born and J. Pfeifer, Policy risk and the business cycle, Journal of Monetary Economics, 68 (2014), 68-85. doi: 10.1016/j.jmoneco.2014.07.012. Google Scholar
[5]	W. Brainard, Uncertainty and the effectiveness of policy, American Economic Review, 57 (1967), 411-425. Google Scholar
[6]	W. A. Brock and S. Durlauf, Macroeconomics and Model Incertainty, in D. Colander (Ed.), Post Walrasian Macroeconomics: Beyond the Dynamic Stochastic General Equilibrium Model, Cambridge University Press, Cambridge, 2006. Google Scholar
[7]	W. A. Brock and L. J. Mirman, Optimal economic growth and uncertainty: The discounted case, Journal of Economic Theory, 4 (1972), 479-513. doi: 10.1016/0022-0531(72)90135-4. Google Scholar
[8]	C. Camerer and M. Weber, Recent developments in modeling preferences: uncertainty and ambiguity, Journal of Risk and Uncertainty, 5 (1992), 325-370. doi: 10.1007/BF00122575. Google Scholar
[9]	F. Caprioli, Optimal fiscal policy under learning, Jouirnal of Economics Dynamics & Control, 58 (2015), 101-124. doi: 10.1016/j.jedc.2015.05.008. Google Scholar
[10]	G. Cozzi and P. E. Giordani, Ambiguity attitude, R & D investments and economic growth, Journal of Evolutionary Economics, 21 (2011), 303-319. doi: 10.1007/s00191-010-0217-x. Google Scholar
[11]	D. Ellsberg, Risk, ambiguity and the savage axioms, Quarterly Journal of Economics, 75 (1961), 643-669. doi: 10.2307/1884324. Google Scholar
[12]	J. Etner, M. Jeleva and J.-M. Tallon, Decision theory under ambiguity, Journal of Economic Surveys, 26 (2012), 234-270. doi: 10.1111/j.1467-6419.2010.00641.x. Google Scholar
[13]	B. Forte and E. R. Vrscay, Solving the inverse problem for function and image approximation using iterated function systems, Dynamics of Continuous, Discrete and Impulsive Systems, 1 (1995), 177-232. Google Scholar
[14]	B. Forte and E. R. Vrscay, Theory of Generalized Fractal Transforms, in Y. Fisher (Ed.), Fractal Image Encoding and Analysis, NATO ASI Series F, Springer Verlag, New York, 1998. Google Scholar
[15]	D. Frisch and J. Baron, Ambiguity and rationality, Journal of Behavioral Decision Making, 1 (1988), 149-157. doi: 10.1002/bdm.3960010303. Google Scholar
[16]	P. Ghirardato, F. Maccheroni and M. Marinacci, Differentiating ambiguity and ambiguity attitude, Journal of Economic Theory, 118 (2004), 133-173. doi: 10.1016/j.jet.2003.12.004. Google Scholar
[17]	L. P. Hansen and T. J. Sargent, Robustness, Princeton University Press, Princeton, 2008. doi: 10.1515/9781400829385. Google Scholar
[18]	J. Hollmayr and C. Matthes, Learning about fiscal policy and the effects of policy uncertainty, Journal of Economic Dynamics & Control, 59 (2015), 142-162. doi: 10.1016/j.jedc.2015.08.002. Google Scholar
[19]	A. G. Karantounias, Managing pessimistic expectations and fiscal policy, Theorertical Economics, 8 (2013), 193-231. doi: 10.3982/TE899. Google Scholar
[20]	H. Kunze, D. La Torre, F. Mendivil and E. R. Vrscay, Fractal-Based Methods in Analysis, Springer, New York, 2012. doi: 10.1007/978-1-4614-1891-7. Google Scholar
[21]	D. La Torre, S. Marsiglio and F. Privileggi, Fractals and self-similarity in economics: the case of a stochastic two-sector growth model, Image Analysys and Stereology, 30 (2011), 143-151. doi: 10.5566/ias.v30.p143-151. Google Scholar
[22]	D. La Torre, S. Marsiglio, F. Mendivil and F. Privileggi, Self-similar measures in multi-sector endogenous growth models, Chaos, Solitons and Fractals, 79 (2015), 40-56. doi: 10.1016/j.chaos.2015.05.019. Google Scholar
[23]	D. La Torre, S. Marsiglio and F. Privileggi, Fractal attractors in economic growth models with random pollution externalities, Chaos, 28 (2018), 055916, 12 pp. doi: 10.1063/1.5023782. Google Scholar
[24]	D. La Torre, S. Marsiglio, F. Mendivil and F. Privileggi, Fractal attractors and singular invariant measures in two-sector growth models with random factor shares, Communications in Nonlinear Science and Numerical Simulation, 58 (2018), 185-201. doi: 10.1016/j.cnsns.2017.07.008. Google Scholar
[25]	D. La Torre and S. Marsiglio, A note on optimal debt reduction policies, Macroeconomic Dynamics, 24 (2020), 1850-1860. doi: 10.1017/S1365100519000014. Google Scholar
[26]	T. Mitra, L. Montrucchio and F. Privileggi, The nature of the steady state in models of optimal growth under uncertainty, Economic Theory, 23 (2004), 39-71. doi: 10.1007/s00199-002-0340-5. Google Scholar
[27]	T. Mitra and F. Privileggi, Cantor Type Invariant Distributions in the Theory of Optimal Growth under Uncertainty, Journal of Difference Equations and Applications, 10 (2004), 489-500. doi: 10.1080/1023619042000193649. Google Scholar
[28]	T. Mitra and F. Privileggi, Cantor type attractors in stochastic growth models, Chaos, Solitons and Fractals, 29 (2006), 626-637. doi: 10.1016/j.chaos.2005.08.094. Google Scholar
[29]	T. Mitra and F. Privileggi, On Lipschitz continuity of the iterated function system in a stochastic optimal growth model, Journal of Mathematical Economics, 45 (2009), 185-198. doi: 10.1016/j.jmateco.2008.08.003. Google Scholar
[30]	L. Montrucchio and F. Privileggi, Fractal steady states in stochastic optimal control models, Annals of Operations Research, 88 (1999), 183-197. doi: 10.1023/A:1018978213041. Google Scholar
[31]	L. J. Olson and S. Roy, Theory of stochastic optimal economic growth, in R. A. Dana, C. Le Van, T. Mitra and K. Nishimura (Eds.), Handbook on optimal growth 1: discrete time, Springer, New York (2005), 297–335. doi: 10.1007/3-540-32310-4_11. Google Scholar
[32]	F. Privileggi and S. Marsiglio, Environmental shocks and sustainability in a basic economy-environment model, International Journal of Applied Nonlinear Science, 1 (2013), 67-75. doi: 10.1504/IJANS.2013.052755. Google Scholar
[33]	D. Rodrik, Policy uncertainty and private investment, Journal of Development Economics, 36 (1991), 229-242. Google Scholar

Figure 1. First $ 7 $ iterations of operator $ T^{*} $ defined in (11) for the only map $ w\left(x\right) = \frac{1}{2}x+\frac{1}{4} $ together with the only greyscale map $ \phi\left(y\right) = 2y $ starting from $ u_{0}\left(x\right) = 12\left(x-\frac{1}{2}\right)^2 $

Download full-size image

Download as PowerPoint slide

Figure 1">Figure 2. cumulative distribution functions associated to the densities $ u_{t} $ in Figure 1

Download full-size image

Download as PowerPoint slide

Figure 3. First $ 7 $ iterations of operator $ T^{*}_{2} $ defined in (16) for the maps $ w_{1}\left(x\right) = \frac{1}{4}x $ and $ w_{2}\left(x\right) = \frac{3}{4}x+\frac{1}{4} $ together with greyscale maps $ \phi_{1}\left(y\right) = 2y $ and $ \phi_{2}\left(y\right) = \frac{2}{3}y $ starting from $ u_{0}\left(x\right)\equiv1 $

Download full-size image

Download as PowerPoint slide

Figure 3">Figure 4. cumulative distribution functions associated to the densities $ u_{t} $ in Figure 3

Download full-size image

Download as PowerPoint slide

Figure 5. First $ 7 $ iterations of operator $ T^{*}_{2} $ defined in (16) for the maps $ w_{1}\left(x\right) = \frac{1}{4}x $ and $ w_{2}\left(x\right) = \frac{3}{4}x+\frac{1}{4} $ together with greyscale maps $ \phi_{1}\left(y\right) = 2y $ and $ \phi_{2}\left(y\right) = \frac{2}{3}y $ starting from $ u_{0}\left(x\right) = 3.3852e^{-\left (6x-3\right)^2} $

Download full-size image

Download as PowerPoint slide

Figure 5">Figure 6. cumulative distribution functions associated to the densities $ u_{t} $ in Figure 5

Download full-size image

Download as PowerPoint slide

Figure 7. a) $ 5^{th} $ and b) $ 7^{th} $ iteration of operator $ T^{*} _{2} $ defined in (16) for the wavelets maps $ w_{1}\left(x\right ) = \frac{1}{2}x $ and $ w_{2}\left(x\right) = \frac{1}{2}x+\frac{1}{2} $ together with greyscale maps $ \phi_{1}\left(y\right) = \phi_{2}\left(y\right) = y $ starting from $ u_{0}\left(x\right) = 3x^{2} $, c) and d) cumulative distribution functions associated to the densities $ u_{5} $ and $ u_{7} $

Download full-size image

Download as PowerPoint slide

Figure 8. a) $ 5^{th} $ iteration of operator $ T^{*}_{3} $ defined in (16) for the maps $ w_{1}\left(x\right) = \frac{1}{8}x $, $ w_{2}\left(x\right) = \frac{3}{8}x+\frac{1}{8} $ and $ w_{3}\left(x\right) = \frac{1}{2}x+\frac{1}{2} $ together with greyscale maps $ \phi_{1}\left(y\right) = \frac{8}{3}y $, $ \phi_{2}\left(y\right) = \frac{8}{9}y $ and $ \phi_{3}\left(y\right) = \frac{2}{3}y $ starting from $ u_{0}\left(x\right)\equiv 1 $, b) its associated cumulative distribution function

Download full-size image

Download as PowerPoint slide

Figure 9. First $ 7 $ iterations of operator $ T $ defined in (19) for the maps $ w_{1}\left(x\right) = \frac{1}{4}x $ and $ w_{2}\left(x\right) = \frac{3}{4}x+\frac{1}{4} $ together with greyscale maps $ \phi_{1}\left(y\right) = \frac{2}{5}y+\frac{8}{5} $ and $ \phi_{2}\left(y\right) = \frac{2}{3}y $ starting from $ u_{0}\left(x\right) \equiv1 $

Download full-size image

Download as PowerPoint slide

Figure 9">Figure 10. cumulative distribution functions associated to the densities $ u_{t} $ in Figure 9

Download full-size image

Download as PowerPoint slide

Figure 11. First $ 7 $ iterations of operator $ T $ defined in (19) for the maps $ w_{1}\left(x\right) = \frac{1}{4}x $ and $ w_{2}\left(x\right) = \frac{3}{4}x+\frac{1}{4} $ together with greyscale maps $ \phi_{1}\left(y\right) = \frac{2}{5}y+\frac{8}{5} $ and $ \phi_{2}\left(y\right) = \frac{2}{3}y $ starting from $ u_{0}\left(x\right) = 3.3852e^{-\left(6x-3\right)^2} $

Download full-size image

Download as PowerPoint slide

Figure 11">Figure 12. cumulative distribution functions associated to the densities $ u_{t} $ in Figure 11

Download full-size image

Download as PowerPoint slide

Figure 13. a) $ 7^{th} $ iteration of operator $ T $ defined in (19) for the maps $ w_{1}\left(x\right) = \frac{1}{4}x $ and $ w_{2}\left(x\right) = \frac{3}{4}x+\frac{1}{4} $ together with greyscale maps $ \phi_{1}\left(y\right) = y+1 $ and $ \phi_{2}\left(y\right) = \frac{1}{3}y+\frac{1}{3} $ starting from $ u_{0}\left(x\right)\equiv 1 $, b) its associated cumulative distribution function

Download full-size image

Download as PowerPoint slide

Figure 14. a) $ 7^{th} $ iteration of operator $ T $ defined in (19) for the maps $ w_{1}\left(x\right) = \frac{1}{2}x $ and $ w_{2}\left(x\right) = \frac{1}{2}x+\frac{1}{2} $ together with greyscale maps $ \phi_{1}\left(y\right) = \frac{2}{5}y+\frac{3}{5} $ and $ \phi_{2}\left(y\right) = \frac{4}{5}y+\frac{1}{5} $ starting from $ u_{0}\left(x\right) = 12\left(x-\frac{1}{2}\right)^2 $, b) its associated cumulative distribution function

Download full-size image

Download as PowerPoint slide

Figure 15. a) $ 7^{th} $ iteration of operator $ T $ defined in (19) for the maps $ w_{1}\left(x\right) = \frac{1}{2}x $ and $ w_{2}\left(x\right) = \frac{1}{2} x+\frac{1}{2} $ together with greyscale maps $ \phi_{1}\left(y\right) = \phi_{2}\left(y\right) = \frac{1}{2}y+\frac{1}{2} $ starting from $ u_{0}\left(x\right) = 12\left (x-\frac{1}{2}\right)^2 $, b) its associated cumulative distribution function

Download full-size image

Download as PowerPoint slide

Figure 16. a) $ 5^{th} $ iteration of operator $ T $ defined in (19) for the maps $ w_{1}\left(x\right) = \frac{1}{8}x $, $ w_{2}\left(x\right) = \frac{3}{8}x+\frac {1}{8} $ and $ w_{3}\left(x\right) = \frac{1}{2}x+\frac{1}{2} $ together with greyscale maps $ \phi_{1}\left(y\right) = \frac{8}{15}y+\frac {32}{15} $, $ \phi_{2}\left(y\right) = \frac{16}{45}y+\frac{8}{15} $ and $ \phi_{3} \left(y\right) = \frac{2}{3}y $ starting from $ u_{0}\left(x\right)\equiv1 $, b) its associated cumulative distribution function

Download full-size image

Download as PowerPoint slide

Figure 17. a) $ 5^{th} $ iteration of operator $ T $ defined in (19) for the maps $ w_{1}\left(x\right) = \frac{1}{8}x $, $ w_{2}\left(x\right) = \frac{3}{8}x+\frac {1}{8} $ and $ w_{3}\left(x\right) = \frac{1}{2}x+\frac{1}{2} $ together with greyscale maps $ \phi_{1}\left(y\right) = \frac{32}{15}y+\frac {8}{15} $, $ \phi_{2}\left(y\right) = \frac{4}{45}y+\frac{4}{5} $ and $ \phi_{3}\left(y\right) = \frac{1}{15}y+\frac{3}{5} $ starting from $ u_{0}\left(x\right)\equiv1 $, b) its associated cumulative distribution function

Download full-size image

Download as PowerPoint slide

[1]	Pablo G. Barrientos, Abbas Fakhari, Aliasghar Sarizadeh. Density of fiberwise orbits in minimal iterated function systems on the circle. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3341-3352. doi: 10.3934/dcds.2014.34.3341
[2]	Welington Cordeiro, Manfred Denker, Michiko Yuri. A note on specification for iterated function systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3475-3485. doi: 10.3934/dcdsb.2015.20.3475
[3]	Kanji Inui, Hikaru Okada, Hiroki Sumi. The Hausdorff dimension function of the family of conformal iterated function systems of generalized complex continued fractions. Discrete & Continuous Dynamical Systems, 2020, 40 (2) : 753-766. doi: 10.3934/dcds.2020060
[4]	David Cheban, Cristiana Mammana. Continuous dependence of attractors on parameters of non-autonomous dynamical systems and infinite iterated function systems. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 499-515. doi: 10.3934/dcds.2007.18.499
[5]	Saisai Shi, Bo Tan, Qinglong Zhou. Best approximation of orbits in iterated function systems. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4085-4104. doi: 10.3934/dcds.2021029
[6]	Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809
[7]	Michael Barnsley, James Keesling, Mrinal Kanti Roychowdhury. Special issue on fractal geometry, dynamical systems, and their applications. Discrete & Continuous Dynamical Systems - S, 2019, 12 (8) : i-i. doi: 10.3934/dcdss.201908i
[8]	Thomas Jordan, Mark Pollicott. The Hausdorff dimension of measures for iterated function systems which contract on average. Discrete & Continuous Dynamical Systems, 2008, 22 (1&2) : 235-246. doi: 10.3934/dcds.2008.22.235
[9]	Noah H. Rhee, PaweŁ Góra, Majid Bani-Yaghoub. Predicting and estimating probability density functions of chaotic systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 297-319. doi: 10.3934/dcdsb.2017144
[10]	Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281
[11]	Wided Kechiche. Regularity of the global attractor for a nonlinear Schrödinger equation with a point defect. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1233-1252. doi: 10.3934/cpaa.2017060
[12]	Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 3027-3042. doi: 10.3934/dcdss.2021031
[13]	B. Coll, A. Gasull, R. Prohens. On a criterium of global attraction for discrete dynamical systems. Communications on Pure & Applied Analysis, 2006, 5 (3) : 537-550. doi: 10.3934/cpaa.2006.5.537
[14]	David Cheban. Global attractors of nonautonomous quasihomogeneous dynamical systems. Conference Publications, 2001, 2001 (Special) : 96-101. doi: 10.3934/proc.2001.2001.96
[15]	Stefano Galatolo. Global and local complexity in weakly chaotic dynamical systems. Discrete & Continuous Dynamical Systems, 2003, 9 (6) : 1607-1624. doi: 10.3934/dcds.2003.9.1607
[16]	Tiphaine Jézéquel, Patrick Bernard, Eric Lombardi. Homoclinic orbits with many loops near a $0^2 i\omega$ resonant fixed point of Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3153-3225. doi: 10.3934/dcds.2016.36.3153
[17]	João Ferreira Alves, Michal Málek. Zeta functions and topological entropy of periodic nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 465-482. doi: 10.3934/dcds.2013.33.465
[18]	Peter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein. Computing complete Lyapunov functions for discrete-time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 299-336. doi: 10.3934/dcdsb.2020331
[19]	Michael Schönlein. Asymptotic stability and smooth Lyapunov functions for a class of abstract dynamical systems. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 4053-4069. doi: 10.3934/dcds.2017172
[20]	Felipe García-Ramos, Brian Marcus. Mean sensitive, mean equicontinuous and almost periodic functions for dynamical systems. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 729-746. doi: 10.3934/dcds.2019030

2020 Impact Factor: 1.327

Tools

Metrics

Other articles
by authors

on AIMS
Davide La Torre Simone Marsiglio Franklin Mendivil Fabio Privileggi
on Google Scholar
Davide La Torre Simone Marsiglio Franklin Mendivil Fabio Privileggi

2020 Impact Factor: 1.327

Tools

Article outline

Figures and Tables

2020 Impact Factor: 1.327

Tools

Figures and Tables

2020 Impact Factor: 1.327

Tools

Metrics

Other articles
by authors

on AIMS
Davide La Torre Simone Marsiglio Franklin Mendivil Fabio Privileggi
on Google Scholar
Davide La Torre Simone Marsiglio Franklin Mendivil Fabio Privileggi

[Back to Top]