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Chaos is the most complex dynamic behavior a system can show. Yet, it is produced by two simple, though nonlinear, deterministic mechanisms, hence the more proper name Deterministic Chaos. The first mechanism, stretching, acts locally and linearly, expanding small sets in the system's state space in (at least) one direction and contracting in the others. The second mechanism, folding, acts globally and is the nonlinear operation of bending the elongated sets to form the forward image of the original sets. These two simple operations are invertible and form a chaotic set for the system's forward and backward map, containing an uncountable infinity of astonishingly complex system's trajectories. The simplest explanation of chaos I ever heard is James A. Yorke “pizza machine” (plenary lecture at the SIAM DS, Snowbird, 2003): kneading the pizza dough is nothing but stretching and folding. The mathematical formalization is Stephen Smale's horseshoe map. Using this nice tool, I revisit the hallmarks of chaos, one of which (in the title) still misses a constructive proof.

This paper addresses the design of controllers for systems modelled as recurrent neural networks (RNNs). A novel data-based procedure for the design of RNN-based regulators is proposed, guaranteeing closed-loop stability properties and desired performances, conferred by virtual reference feedback tuning. The approach is tested on a realistic nonlinear system.

This paper is about the emergence of technological variety arising from market interaction and technological innovation. Existing products in the market compete with innovative ones resulting in a slow and continuous evolution of the underlying technological characteristics of successful products. When technological evolution reaches an equilibrium, it can either be an evolutionarily stable strategy (ESS), where marginally innovative products do not penetrate the market, or a branching point, where new products coexist along with established ones. Thus, technological branching can give rise to product variety. In the paper we first introduce adaptive dynamics (AD), a recently proposed theory of evolutionary processes, aiming at modeling various features of technological change. By separating the timescale typical of market competition processes from that on which marginal innovations drive technological change, AD formally describes products coevolution by means of ordinary differential equations. Then, a first application of AD in economics is presented and discussed in detail. The problem we discuss is intentionally very simple, in order to clearly exemplify all the steps of the analysis, but allows to draw the following intuitive conclusion: product variety is expected in market sectors characterized by a wide capacity to absorb different technologies and by competition-safe niches even for relatively similar products. The limitations of the AD approach, as well as some promising further applications in economics and social sciences, are briefly discussed at the concluding section. Keywords: Adaptive dynamics; Market dynamics; Innovation dynamics; Characteristic trait; Technological branching; Product variety

Recent advances in nonlinear time-series prediction demonstrated the ability of recurrent neural network to forecast chaotic time series on a multi-step horizon, outperforming previous approaches. Researches considered chaotic systems with different degree of complexity, but the analysis was mainly limited to the noise-free case. In this work, we extend the analysis to a noisy environment, in order to fill the gap between deterministic and real-world time series. We consider four archetypal deterministic chaotic systems each with different levels of additive noise, representing the observation uncertainty always affecting practical applications. A time series of solar irradiance is also taken into account as a real-world case study. Various neural architectures, including feed-forward and recurrent networks, are adopted as predictors. LSTM cells are used as recurrent neurons, with a special focus on the training approach. As in the noise-free case, LSTM trained without the traditional teacher forcing, i.e., with a training that replicates the forecasting conditions, proved to be the best architecture. The experiments on the archetypal systems also shows that the error due to the model identification is negligible if compared to the one caused by a small observation noise. In other words, system identification and predictions are well distinct tasks.

We present a direct approach to study the stability of discrete-time switched linear systems that can be applied to arbitrary switching, as well as when switching is constrained by a switching automaton. We explore the tree of possible matrix products, by pruning the subtrees rooted at contractions and looking for unstable repeatable products. Generically, this simple strategy either terminates with all contracting leafs—showing the system's asymptotic stability—or finds the shortest unstable and repeatable matrix product. Although it behaves in the worst case as the exhaustive search, we show that its performance is greatly enhanced by measuring contractiveness w.r.t. sum-of-squares polynomial norms, optimized to minimize the largest expansion among the system's modes. [ABSTRACT FROM AUTHOR]

The qualitative resonance of feedback-controlled chaotic oscillators is the ability of the control system to qualitatively synchronize with a reference signal similar to one of the unstable periodic orbits embedded in the open-loop attractor. This property, discovered by O. De Feo (2004a; 2004b) while studying Shilnikov-type attractors, was explained in terms of the random-like rephasing mechanism characterizing the oscillator’s dynamics, so to guarantee the eventual in-phase looking with the reference forcing. We experimentally show that the phenomenon works more in general, even in the absence of a rephasing mechanism. Intuitively, the forcing by the target cycle, or by a qualitative approximation of it, is sufficient to bring in the in-phase condition. Our results can make chaos control more practicable than so far imagined, as a qualitative control can be achieved with no a-priori knowledge about the target solution.

Economic unions are international agreements oriented to increase economic efficiency and establishing political and cultural ties between the member countries. Becoming a member of an existing union usually requires the approval of both the candidate and members, while leaving it may require only the unilateral will of the exiting country. There are many examples of accession of states to previously consolidated economic unions, and a recent example of leaving is the withdrawal of the United Kingdom from the European Union. Motivated by the Brexit process, in this paper we propose an agent-based model to study the determinant factors driving withdrawals from an economic union. We show that both Union and local taxes promote the exits, whereas customs fees out of the Union boost cohesion. Furthermore, heterogeneity in both business conditions and wealth distribution promotes withdrawals, while countries' size diversity does not have a significant effect on them. We also deep into the individual causes that lead to dissatisfaction and, ultimately, to exits. We found that, for low Union taxes, the wealth inequality within the country is the leading cause of anti-Union opinion spreading. Conversely, for high Union taxes, the country's performance turns out to be the main driving force, resulting in a risk of wealthier countries leaving the Union. These findings will be helpful for the design of economic policies and effective informative campaigns. • Agent-based modeling is helpful to study opinion dynamics in economic unions. • Taxes promote economic unions withdrawals, whereas customs fees boost cohesion. • Heterogeneity in both business conditions and wealth distribution promotes exits. • For low taxes, within-country wealth inequality fosters anti-Union opinion spreading. • High Union taxes involves a risk of wealthier countries leaving the Union. [ABSTRACT FROM AUTHOR]

In many applications in different fields of science and engineering only two extreme values of the control variable can be easily applied, and a threshold on a suitable output variable is used to discriminate among the two control actions. This relaycontrol introduces a discontinuity, so the closed-loop system is discontinuous piecewise smooth (called Filippov system). When a standard equilibrium attains the threshold, as a system or control parameter is varied, two generic scenarios are possible: the standard equilibrium turns into a pseudo-equilibrium on the discontinuity boundary (persistence), so a stationary solution persists trough the bifurcation; the collision and disappearance of the standard equilibrium and a coexisting pseudo-equilibrium (nonsmooth-fold). In this paper we analyze the degenerate situation separating these two scenarios, and we apply our results to a four-dimensional SISO system describing the ecological dynamics of a protected natural resource (a resource that cannot be harvested when below threshold). We show that while profitable exploitation is guaranteed (though often at the threshold) in the persistence scenario, the food chain collapses after a nonsmooth-fold.

• Feed-forward and LSTM networks are compared in the forecasting of noisy chaotic time series. • Observation noise strongly affects the performance of all the neural predictors. • Recurrent architectures are more suitable for learning the non-stationary dynamics caused by structural noise. • LSTM trained without teacher forcing shows the greatest ability of mitigating the different sources of noise. Recurrent neural networks have recently proved the state-of-the-art approach in forecasting complex oscillatory time series on a multi-step horizon. Researchers in the field investigated different machine learning techniques and training approaches on dynamical systems with different degrees of complexity. Still, these analyses are usually limited to noise-free chaotic time series. This paper extends the analysis from a deterministic to a noisy environment, by considering both observation and structural noise. Observation noise is evaluated by adding different levels of artificially-generated random values on deterministic processes obtained from the simulation of four archetypal chaotic systems. A case of structural noise is implemented through a time-varying version of the logistic map, which exhibits a slow structural change of the system's dynamic that makes the system non-stationary. Finally, a time series of ozone concentration in Northern Italy is considered to test the theoretical findings on a real-world case study in which both forms of noise play a significant role. Recurrent neural networks formed by LSTM cells are compared with two benchmark feed-forward architectures. LSTM trained without the standard teacher forcing approach, i.e., with training that replicates the setting used in inference mode, proved to have the best performance in compensating the stochasticity generated by the observation noise and reproducing the structural non-stationarity of the process. [ABSTRACT FROM AUTHOR]

We show in this paper that the evolution of cannibalistic consumer populations can be a never ending story involving alternating levels of polymorphism. More precisely, we show that a monomorphic population can evolve toward high levels of cannibalism until it reaches a so-called branching point, where the population splits into two sub-populations characterized by different, but initially very close, cannibalistic traits. Then, the two traits coevolve until the more cannibalistic sub-population undergoes evolutionary extinction. Finally, the remaining population evolves back to the branching point, thus closing an evolutionary cycle. The model on which the study is based is purely deterministic and derived through the adaptive dynamics approach. Evolutionary dynamics are investigated through numerical bifurcation analysis, applied both to the ecological (resident-mutant) model and to the evolutionary model. The general conclusion emerging from this study is that branching-extinction evolutionary cycles can be present in wide ranges of environmental and demographic parameters, so that their detection is of crucial importance when studying evolutionary dynamics.

• LSTM nets outperform feed-forward competitors in forecasting noise-free chaotic time series. • Training without teacher forcing (no-TF) enhances the forecasting accuracy and the robustness of the LSTM predictors. • The robustness to the number of data samples provided as input (lags) is especially relevant for real-world applications. • LSTM-no-TF also ensure a uniform predictive power within the chaotic attractor. • LSTM-no-TF nets well predict chaotic dynamics up to 7 Lyapunov times. Recurrent neurons (and in particular LSTM cells) demonstrated to be efficient when used as basic blocks to build sequence to sequence architectures, which represent the state-of-the-art approach in many sequential tasks related to natural language processing. In this work, these architectures are proposed as general purposes, multi-step predictors for nonlinear time series. We analyze artificial, noise-free data generated by chaotic oscillators and compare LSTM nets with the benchmarks set by feed-forward, one-step-recursive and multi-output predictors. We focus on two different training methods for LSTM nets. The traditional one makes use of the so-called teacher forcing, i.e. the ground truth data are used as input for each time step ahead, rather than the outputs predicted for the previous steps. Conversely, the second feeds the previous predictions back into the recurrent neurons, as it happens when the network is used in forecasting. LSTM predictors robustly show the strengths of the two benchmark competitors, i.e., the good short-term performance of one-step-recursive predictors and greatly improved mid-long-term predictions with respect to feed-forward, multi-output predictors. Training LSTM predictors without teacher forcing is recommended to improve accuracy and robustness, and ensures a more uniform distribution of the predictive power within the chaotic attractor. We also show that LSTM architectures maintain good performances when the number of time lags included in the input differs from the actual embedding dimension of the dataset, a feature that is very important when working on real data. [ABSTRACT FROM AUTHOR]

• An ARED (adaptive rational equilibrium dynamics) asset pricing model where agents trade according to heterogeneous and adaptive beliefs on the future prices. • The effects of the "uptick rule". • The investigation underlines that the uptick rule meets its intended objective. • The analysis suggests also possible and unwanted side effects of the regulation, such as an excessive swelling of speculative bubbles. This paper investigates the effects of the "uptick rule" (a short selling regulation formally known as Exchange Act Rule 10a-1) by means of a simple stock market model, based on the ARED (adaptive rational equilibrium dynamics) modeling framework, where heterogeneous and adaptive beliefs on the future prices of a risky asset were first shown to be responsible for endogenous price fluctuations. The dynamics of stock prices generated by the model, with and without the uptick-rule restriction, are analyzed by pairing the classical fundamental prediction with beliefs based on both linear and nonlinear forecasting rules deriving from the technical analysis of the financial markets. The comparison shows a reduction of downward price movements of undervalued shares when the short selling restriction is imposed. This gives evidence that the uptick rule meets its intended objective. However, the effects of the short selling regulation fade when the intensity of choice to switch trading strategies is high. In addition, the analysis suggests possible side effects of the regulation on price dynamics, such as an excessive swelling of speculative bubbles. [ABSTRACT FROM AUTHOR]