# American Institute of Mathematical Sciences

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

 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.

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 and 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. [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. [3] M. F. Barnsley, Fractals Everywhere, Academic Press, New York, 1988. [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. [5] W. Brainard, Uncertainty and the effectiveness of policy, American Economic Review, 57 (1967), 411-425. [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. [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. [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. [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. [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. [11] D. Ellsberg, Risk, ambiguity and the savage axioms, Quarterly Journal of Economics, 75 (1961), 643-669.  doi: 10.2307/1884324. [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. [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. [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. [15] D. Frisch and J. Baron, Ambiguity and rationality, Journal of Behavioral Decision Making, 1 (1988), 149-157.  doi: 10.1002/bdm.3960010303. [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. [17] L. P. Hansen and T. J. Sargent, Robustness, Princeton University Press, Princeton, 2008.  doi: 10.1515/9781400829385. [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. [19] A. G. Karantounias, Managing pessimistic expectations and fiscal policy, Theorertical Economics, 8 (2013), 193-231.  doi: 10.3982/TE899. [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. [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. [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. [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. [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. [25] D. La Torre and S. Marsiglio, A note on optimal debt reduction policies, Macroeconomic Dynamics, 24 (2020), 1850-1860.  doi: 10.1017/S1365100519000014. [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. [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. [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. [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. [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. [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. [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. [33] D. Rodrik, Policy uncertainty and private investment, Journal of Development Economics, 36 (1991), 229-242.

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##### 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. [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. [3] M. F. Barnsley, Fractals Everywhere, Academic Press, New York, 1988. [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. [5] W. Brainard, Uncertainty and the effectiveness of policy, American Economic Review, 57 (1967), 411-425. [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. [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. [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. [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. [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. [11] D. Ellsberg, Risk, ambiguity and the savage axioms, Quarterly Journal of Economics, 75 (1961), 643-669.  doi: 10.2307/1884324. [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. [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. [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. [15] D. Frisch and J. Baron, Ambiguity and rationality, Journal of Behavioral Decision Making, 1 (1988), 149-157.  doi: 10.1002/bdm.3960010303. [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. [17] L. P. Hansen and T. J. Sargent, Robustness, Princeton University Press, Princeton, 2008.  doi: 10.1515/9781400829385. [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. [19] A. G. Karantounias, Managing pessimistic expectations and fiscal policy, Theorertical Economics, 8 (2013), 193-231.  doi: 10.3982/TE899. [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. [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. [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. [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. [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. [25] D. La Torre and S. Marsiglio, A note on optimal debt reduction policies, Macroeconomic Dynamics, 24 (2020), 1850-1860.  doi: 10.1017/S1365100519000014. [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. [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. [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. [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. [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. [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. [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. [33] D. Rodrik, Policy uncertainty and private investment, Journal of Development Economics, 36 (1991), 229-242.
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 $u_{0}\left(x\right) = 12\left(x-\frac{1}{2}\right)^2$
cumulative distribution functions associated to the densities $u_{t}$ in Figure 1
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 $u_{0}\left(x\right)\equiv1$
cumulative distribution functions associated to the densities $u_{t}$ in 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 $u_{0}\left(x\right) = 3.3852e^{-\left (6x-3\right)^2}$
cumulative distribution functions associated to the densities $u_{t}$ in Figure 5
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 $u_{7}$
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
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 $u_{0}\left(x\right) \equiv1$
cumulative distribution functions associated to the densities $u_{t}$ in 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 $u_{0}\left(x\right) = 3.3852e^{-\left(6x-3\right)^2}$
cumulative distribution functions associated to the densities $u_{t}$ in Figure 11
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
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
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
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
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
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