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Noise-free stochastic resonance is investigated in two chaotic maps with periodically modulated control parameter close to a boundary crisis: the Hénon map and the kicked spin model. Response of the maps to the periodic signal at the fundamental frequency and its higher harmonics is examined. The systems show noise-free stochastic multiresonance, i.e. multiple maxima of the signal-to-noise ratio at the fundamental frequency as a function of the control parameter. The maxima are directly related to the fractal structure of the attractors and basins of attraction colliding at the crisis point. The signal-to-noise ratios at higher harmonics show more maxima, as well as dips where the signal-to-noise ratio is zero. This opens a way to use noise-free stochastic resonance to probe the fractal structure of colliding sets by a method which can be called "fractal spectroscopy". Using stochastic resonance at higher harmonics can reveal smaller details of the fractal structures, but the interpretation of results becomes more difficult. Quantitative theory based on a model of a colliding fractal attractor and a fractal basin of attraction is derived which agrees with numerical results for the signal-to-noise ratio at the fundamental frequency and at the first two harmonics, quantitatively for the Hénon map, and qualitatively for the kicked spin model. It is also argued that the maps under study belong to a more general class of threshold-crossing stochastic resonators with a modulated control parameter, and qualitative discussion of conditions under which stochastic multiresonance appears in such systems is given.

The behavior of a damped pendulum parametrically excited by a periodic string of symmetric pulses of finite width and amplitude is investigated. Analytical (Melnikov method) and numerical (bifurcation diagrams) results show that chaos and crises are reliably controlled over a wide range of parameters by hump-doubling of a parametric excitation which is initially formed by a periodic string of single-humped symmetric pulses. In particular, the analysis reveals that the chaotic threshold amplitude when altering solely the pulse shape presents a minimum as a single-humped pulse transforms into a double-humped pulse, the remaining parameters being held constant. Additionally, the mechanism underlying the hump-doubling-induced crises is discussed with the help of a two-dimensional map.

In this paper, we consider compact, invariant sets in Hamiltonian systems in order to extend the concept of crisis to such systems. We focus on crisis-induced intermittency in several systems where two invariant sets merge, obtaining scaling laws for the residence times and for the probability distribution decay as a function of a critical parameter. The connection to hitherto known crisis-induced intermittency in dissipative systems is discussed.

This paper is devoted to the dynamical behavior of a parametrically driven double-well Duffing (PDWD) system. Despite the invariant property of symmetry, this simple model exhibits a large diversity of patterns which can be observed in different situations. The transitions between symmetric forms of system responses often lead to bifurcation or crisis and complicated behaviors, such as the coexistence of different kinds of attractors. The bifurcations and crises are discussed, especially those inside the main periodic window. In particular, the role of chaotic saddles and their intrinsic links with the basin of attraction and transient chaos is studied.

A macroeconomic model based on the economic variables (i) assets, (ii) leverage (defined as debt over asset) and (iii) trust (defined as the maximum sustainable leverage) is proposed to investigate the role of credit in the dynamics of economic growth, and how credit may be associated with both economic performance and confidence. Our first notable finding is the mechanism of reward/penalty associated with patience, as quantified by the return on assets. In regular economies where the EBITA/Assets ratio is larger than the cost of debt, starting with a trust higher than leverage results in the highest long-term return on assets (which can be seen as a proxy for economic growth). Therefore, patient economies that first build trust and then increase leverage are positively rewarded. Our second main finding concerns a recommendation for the reaction of a central bank to an external shock that affects negatively the economic growth. We find that late policy intervention in the model economy results in the highest long-term return on assets. However, this comes at the cost of suffering longer from the crisis until the intervention occurs. The phenomenon that late intervention is most effective to attain a high long-term return on assets can be ascribed to the fact that postponing intervention allows trust to increase first, and it is most effective to intervene when trust is high. These results are derived from two fundamental assumptions underlying our model: (a) trust tends to increase when it is above leverage; (b) economic agents learn optimally to adjust debt for a given level of trust and amount of assets. Using a Markov Switching Model for the EBITA/Assets ratio, we have successfully calibrated our model to the empirical data of the return on equity of the EURO STOXX 50 for the time period 2000–2013. We find that dynamics of leverage and trust can be highly nonmonotonous with curved trajectories, as a result of the nonlinear coupling between the variables. This has an important implication for policy makers, suggesting that simple linear forecasting can be deceiving in some regimes and may lead to inappropriate policy decisions.

Bifurcations and chaos in a network of three identical sigmoidal neurons are examined. The network consists of a two-neuron oscillator of the Wilson–Cowan type and an additional third neuron, which has a simpler structure than chaotic neural networks in the previous studies. A codimension-two fold-pitchfork bifurcation connecting two periodic solutions exists, which is accompanied by the Neimark–Sacker bifurcation. A stable quasiperiodic solution is generated and Arnold’s tongues emanate from the locus of the Neimark–Sacker bifurcation in a two-dimensional parameter space. The merging, splitting and crossing of the Arnold tongues are observed. Further, multiple chaotic attractors are generated through cascades of period-doubling bifurcations of periodic solutions in the Arnold tongues. The chaotic attractors grow and are destroyed through crises. Transient chaos and crisis-induced intermittency due to the crises are also observed. These quasiperiodic solutions and chaotic attractors are robust to small asymmetry in the output function of neurons.

In this paper, we study complex dynamics of the interaction between natural convection and thermal explosion in porous media. This process is modeled with the nonlinear heat equation coupled with the nonstationary Darcy equation under the Boussinesq approximation for a fluid-saturated porous medium in a rectangular domain. Numerical simulations with the Radial Basis Functions Method (RBFM) reveal complex dynamics of solutions and transitions to chaos after a sequence of period doubling bifurcations. Several periodic windows alternate with chaotic regimes due to intermittence or crisis. After the last chaotic regime, a final periodic solution precedes transition to thermal explosion.

The paper describes some aspects of sudden transformations of closed invariant curves in a 2D piecewise smooth map. In particular, using detailed numerically calculated phase portraits, we discuss transitions from smooth to piecewise smooth closed invariant curves. We show that such transitions may occur not only when a closed invariant curve collides with a border but also via a homoclinic bifurcation. Furthermore, we describe an unusual transformation from a closed invariant curve to a large amplitude chaotic attractor and demonstrate that this transition occurs in two steps, involving a small amplitude closed-invariant-curve-like chaotic attractor.

Birhythmicity oscillators have been extensively applied in fields such as biology, physics, and engineering. Studying their global dynamics is crucial for gaining a comprehensive understanding of the intrinsic mechanisms that govern oscillator behavior. This paper focuses on investigating the influence of memory effects on the global dynamics of the fractional birhythmic van der Pol (BVDP) oscillator. To determine the system’s global properties, we employ an improved cell mapping method. Specifically, the system’s evolution is computed by introducing additional auxiliary variables, creating a space for storing historical information. This method allows us to examine the system’s global properties without memory loss. Through a comparison of the global dynamic behaviors of BVDP oscillators with memory to those without memory, we observe that the presence of memory effects results in the emergence of chaotic attractors in the system. This, in turns, results in system instability and heightened sensitivity to initial conditions. Furthermore, our findings suggest that changes in the fractional-order can induce various crises in the oscillator and may have the potential to suppress chaotic oscillations.

This paper constructs a nonlinear dynamical model of a mixed ownership market duopoly game for environmental tax collection within the framework of bounded rationality expectations. A stability analysis of the proposed model is conducted, and a Nash equilibrium stability region is estimated using the Jury criterion. The effects of the degree of privatization and environmental tax rate on the stability are investigated through numerical simulations. Subsequently, the comprehensive global analysis is carried out using the composite cell coordinate system method. Two types of crisis phenomena, namely, interior and boundary crises, have been identified. The former is caused by the collision between chaotic or periodic attractors and interior chaotic saddle within the basin of attraction, and the latter is caused by the collision between chaotic attractor and periodic saddle on the basin boundary. Meanwhile, the global dynamic behavior of two private enterprises engaged in synchronized output adjustment is investigated. When the output adjustment speed varies, the structure of the basin of attraction undergoes a transformation as changes are made to the number of coexisting attractors. The sudden disappearance of one attractor is due to the collision of the chaotic attractor with the saddle, on the basin boundary, merging into a larger saddle and resulting in a boundary crisis. The emergence of numerous “holes” can also be observed. It is found that the presence of periodic or chaotic saddles enriches the dynamical phenomena. Finally, the chaotic phenomenon occurring in the mixed ownership market is controlled using the time-delay feedback control method, stabilizing the system in a stable state, which implies a steady-state of total market output. The study of this model can provide theoretical guidance for the decision-making process of enterprises, helping minimize profit losses while stabilizing the market.