Chaotic Case Research Articles

Mixed-mode oscillations and chaos in a complex chemical reaction network involving heterogeneous catalysis.

The nonlinear dynamical behavior in a complex isothermal reaction network involving heterogeneous catalysis is studied. The method first determines the multiple steady states in the reaction network. This is followed by an analysis of bifurcation continuations to identify several kinds of bifurcations, including limit point, Bogdanov-Takens, generalized Hopf, period doubling, and generalized period doubling. Numerical simulations are performed around the period doubling and generalized period doubling bifurcations. Rich nonlinear behaviors are observed, including simple sustained oscillations, mixed-mode oscillations, non-mixed-mode chaotic oscillations, and mixed-mode chaotic oscillations. Concentration-time plots, 2D phase portraits, Poincaré maps, maximum Lyapunov exponents, frequency spectra, and cascade of bifurcations are reported. Period-doubling and period-adding routes leading to chaos are observed. Maximum Lyapunov exponents are positive for all the chaotic cases, but they are also positive for some non-chaotic orbits. This result diminishes the reliability of using maximum Lyapunov exponents as a tool for determining chaos in the network under study.

Asymptotic behaviour of the confidence region in orbit determination for hyperbolic maps with a parameter

When dealing with an orbit determination problem, uncertainties naturally arise from intrinsic errors related to observation devices and approximation models. Following the Least Squares Method and applying approximation schemes such as the differential correction, uncertainties can be geometrically summarized in confidence regions and estimated by confidence ellipsoids. We investigate the asymptotic behaviour of the confidence ellipsoids while the number of observations and the time span over which they are performed simultaneously increase. Numerical evidences suggest that, in the chaotic scenario, the uncertainties decay at different rates whether the orbit determination is set up to recover the initial conditions alone or along with a dynamical or kinematical parameter, while in the regular case there is no distinction. We show how to improve some of the results in Marò and Bonanno (2021), providing conditions that imply a non-faster-than-polynomial rate of decay in the chaotic case with the parameter, in accordance with the numerical experiments. We also apply these findings to well known examples of chaotic maps, such as piecewise expanding maps of the unit interval or affine hyperbolic toral transformations. We also discuss the applicability to intermittent maps.

Spread complexity and quantum chaos for periodically driven spin chains.

The complexity of quantum states under dynamical evolution can be investigated by studying the spread with time of the state over a predefined basis. It is known that this complexity is minimized by choosing the Krylov basis, thus defining the spread complexity. We study the dynamics of spread complexity for quantum maps using the Arnoldi iterative procedure. The main illustrative quantum many-body model we use is the periodically kicked Ising spinchain with nonintegrable deformations, a chaotic system where we look at both local and nonlocal interactions. In the various cases, we find distinctive behavior of the Arnoldi coefficients and spread complexity for regular versus chaotic dynamics: suppressed fluctuations in the Arnoldi coefficients as well as larger saturation value in spread complexity in the chaotic case. We compare the behavior of the Krylov measures with that of standard spectral diagnostics of chaos. We also study the effect of changing the driving frequency on the complexity saturation.

A pressure-free long-time stable reduced-order model for two-dimensional Rayleigh-Bénard convection.

The present work presents a stable proper orthogonal decomposition (POD)-Galerkin based reduced-order model (ROM) for two-dimensional Rayleigh-Bénard convection in a square geometry for three Rayleigh numbers: 104 (steady state), 3×105 (periodic), and 6×106 (chaotic). Stability is obtained through a particular (staggered-grid) full-order model (FOM) discretization that leads to a ROM that is pressure-free and has skew-symmetric (energy-conserving) convective terms. This yields long-time stable solutions without requiring stabilizing mechanisms, even outside the training data range. The ROM's stability is validated for the different test cases by investigating the Nusselt and Reynolds number time series and the mean and variance of the vertical temperature profile. In general, these quantities converge to the FOM when increasing the number of modes, and turn out to be a good measure of accuracy. However, for the chaotic case, convergence with increasing numbers of modes is relatively difficult and a high number of modes is required to resolve the low-energy structures that are important for the global dynamics.

On Krylov complexity in open systems: an approach via bi-Lanczos algorithm

Continuing the previous initiatives [1, 2], we pursue the exploration of operator growth and Krylov complexity in dissipative open quantum systems. In this paper, we resort to the bi-Lanczos algorithm generating two bi-orthogonal Krylov spaces, which individually generate non-orthogonal subspaces. Unlike the previously studied Arnoldi iteration, this algorithm renders the Lindbladian into a purely tridiagonal form, thus opening up a possibility to study a wide class of dissipative integrable and non-integrable systems by computing Krylov complexity at late times. Our study relies on two specific systems, the dissipative transverse-field Ising model (TFIM) and the dissipative interacting XXZ chain. We find that, for the weak coupling, initial Lanczos coefficients can efficiently distinguish integrable and chaotic evolution before the dissipative effect sets in, which results in more fluctuations in higher Lanczos coefficients. This results in the equal saturation of late-time complexity for both integrable and chaotic cases, making the notion of late-time chaos dubious.

Application of Symmetric Explicit Symplectic Integrators in Non-Rotating Konoplya and Zhidenko Black Hole Spacetime

In this study, we construct symmetric explicit symplectic schemes for the non-rotating Konoplya and Zhidenko black hole spacetime that effectively maintain the stability of energy errors and solve the tangent vectors from the equations of motion and the variational equations of the system. The fast Lyapunov indicators and Poincaré section are calculated to verify the effectiveness of the smaller alignment index. Meanwhile, different algorithms are used to separately calculate the equations of motion and variation equations, resulting in correspondingly smaller alignment indexes. The numerical results indicate that the smaller alignment index obtained by using a global symplectic algorithm is the fastest method for distinguishing between regular and chaotic cases. The smaller alignment index is used to study the effects of parameters on the dynamic transition from order to chaos. If initial conditions and other parameters are appropriately chosen, we observe that an increase in energy E or the deformation parameter η can easily lead to chaos. Similarly, chaos easily occurs when the angular momentum L is small enough or the magnetic parameter Q stays within a suitable range. By varying the initial conditions of the particles, a distribution plot of the smaller alignment in the X–Z plane of the black hole is obtained. It is found that the particle orbits exhibit a remarkably rich structure. Researching the motion of charged particles around a black hole contributes to our understanding of the mechanisms behind black hole accretion and provides valuable insights into the initial formation process of an accretion disk.

FPGA realization of four chaotic interference cases in a terrestrial trajectory model and application in image transmission

This article presents a technique to integrate two dynamical models, a four-wing spherical chaotic oscillator and the elliptical path described by the planet Earth during its translation movement around the sun. Four application cases are derived from the system by varying the dynamics of the chaotic oscillator and these can be applied in information encryption to transmit RGB and grayscale images modulated by CSK. Consequently, the three main contributions of this work are (1) the emulation of the trajectories of the planet Earth with chaotic interference, (2) the CSK modulation and image encryption in a master-slave synchronization topology, and (3) the CSK demodulation for decryption without loss of information with respect to the original information. The three contributions are based on VHDL code implementation. The results of the synchronization, encryption and decryption technique were verified by means of time series and the encrypted images showed a correlation less than − 0.000142 and − 0.0003439 for RGB and grayscale format, respectively, while the retrieved image shows a complete correlation with the image original. In this work, the co-simulations were performed between MATLAB/Simulink and Vivado, using the VHDL language on two FPGA boards from different manufacturers, namely, Xilinx Artix-7 AC701 and Intel Cyclone IV.

Time evolution of nonadditive entropies: The logistic map

Due to the second principle of thermodynamics, the time dependence of entropy for all kinds of systems under all kinds of physical circumstances always thrives interest. The logistic map xt+1=1−axt2∈[−1,1](a∈[0,2]) is neither large, since it has only one degree of freedom, nor closed, since it is dissipative. It exhibits, nevertheless, a peculiar time evolution of its natural entropy, which is the additive Boltzmann–Gibbs-Shannon one, SBG=−∑i=1Wpilnpi, for all values of a for which the Lyapunov exponent is positive, and the nonadditive one Sq=1−∑i=1Wpiqq−1 with q=0.2445… at the edge of chaos, where the Lyapunov exponent vanishes, W being the number of windows of the phase space partition. We numerically show that, for increasing time, the phase-space-averaged entropy overshoots above its stationary-state value in all cases. However, when W→∞, the overshooting gradually disappears for the most chaotic case (a=2), whereas, in remarkable contrast, it appears to monotonically diverge at the Feigenbaum point (a=1.4011…). Consequently, the stationary-state entropy value is achieved from above, instead of from below, as it could have been a priori expected. These results raise the question whether the usual requirements – large, closed, and for generic initial conditions – for the second principle validity might be necessary but not sufficient.

Bursting Phenomenon and Chaos Phase Control in Plant Dynamics

New complex features of a Pinus family plant subjected to wind load by proposing an analog electronic simulator with an active R C realization that mimics the real-time dynamics of the system was examined in this paper. Findings reveal that the periodic or chaotic dynamics of the system depends on the favored initial conditions. We see the dynamical behavior of the plant showing transition from periodic to chaotic states. This is very beneficial because it allows for the observation that when the value of the wind amplitude ratio changes continuously, the plant behavior may change in a discontinuous way. It also helps to understand the impact of wind on the plant dynamics, in particular, and in forest in general, which is very crucial for understanding several plant communities. We show that plants exhibit multiple forms (periodic/chaotic) of repetitive spiking oscillations reflecting the phenomenon of bursting. For the periodic bursting oscillations, we observe that after each peak of excitation due to the wind load, the plant returns directly to its state of rest, whereas the chaotic bursting oscillations break the energy of the wind into decreasing value packets and its occurrence has the merit of reducing the effects of wind energy on the plant; this action ensures the stability of the plant even in chaotic cases. We also observe that the numerical study based on the isospike technique permits efficient discovery and separation between periodic and chaotic orbits visited during the temporal evolution. We also establish through the chaos phase-control techniques that various chaotic oscillations disappeared after applying the chaos phase control strategy in the system compared to the uncontrolled case. Moreover, the chaos phase-control strategy to the system is introduced to instantly inform how the correct choice of the phase minimizes the effects of the wind and thus eliminates chaotic behavior of the plant by simply driving it towards a variety of periodic orbits.

Climate System: A Global Sensitivity Approach

This article is a first attempt to develop a numerical approach to solving differential equations based on Galerkin projections and extensions of polynomial chaos to analyze the sensitivity of input parameters in the Lorenz–Stenflo climate model (Leonov in Commun Pure Appl Anal 16(6):2253–2267, 2017). The sensitivity analysis was undertaken to measure the influence of key parameters (chemical properties of the atmosphere, rotation, temperature gradient, convection motion). In addition, we do simulations of the climate model in the non-chaotic case and in the chaotic case and we calculate the Sobol indices when the parameters follow the uniform law.

Die juristische Argumentation in den böhmischen Robotpatenten 1680–1738

The article discusses the legal reasoning in the “Robotpatente”, which were issued for serfs in Bohemia between 1680 and 1738. Of special interest is the position of ‘natural equity’ (natürliche Billigkeit) in the hierarchy of legal norms within the text and the position of the whole “Robotpatente” within the hierarchy of norms in Habsburg Bohemia. Reflections on natural equity were based on Roman law, rather than on secular natural law theories. I argue that the “Robotpatente” were not aimed at changing present customary law and practical rules, their aim was to step in when laws and customs were silent. In the hierarchy of legal norms in the “Robotpatente” of 1680, 1717 and 1738 itself, ‘natural equity’ always came last. Its role was not to change or exclude the current rules, but to fill in the gaps between them. It is only in the great “Restored Robotpatent” of 1738 that the role of natural equity is differentiated. It is divided into four sections – while natural equity is used in the traditional way in the sections one and two, which regulate the relations between peasants and their landlords, it is almost missing in the section three, which is on the relations to the sovereign. Moreover, it is differentiated in section four, which discusses the implementation. Here, the role of ʺnatural equity” is foreseen only in chaotic accidental cases which defy all known rules. All written laws, urbaria and customary rules took precedence over natural equity. Even though the “Robotpatente” adjudicated all “ancient privileges” to be cancelled, it put forward “legal urbaria”’ as the best safeguard of justice for the peasants.

Bifurcations and chaotic behavior of a predator-prey model with discrete time

<abstract><p>In this paper, the dynamical behavior of a predator-prey model with discrete time is discussed in terms of both theoretical analysis and numerical simulation. The existence and stability of four equilibria are analyzed. It is proved that the system undergoes Flip bifurcation and Hopf bifurcation around its unique positive equilibrium point using center manifold theorem and bifurcation theory. Additionally, by applying small perturbations to the bifurcation parameter, chaotic cases occur at some corresponding internal equilibria. Finally, numerical simulations are provided with the help of maximum Lyapunov exponent and phase diagrams, which reveal a complex dynamical behavior.</p></abstract>

Rigorous numerical study of the Colpitts oscillator with an exponential nonlinearity.

The dynamics of the Colpitts oscillator with an exponential nonlinearity is investigated using rigorous interval arithmetic based tools. The existence of various types of periodic attractors is proved using the interval Newton method. The main results involve the chaotic case for which a trapping region for the associated return map is constructed and a rigorous lower bound for the value of the topological entropy is computed, thus proving that the system is chaotic in the topological sense. A systematic search for unstable periodic orbits embedded in the chaotic attractor is carried out and the results are used to obtain an accurate approximation of the topological entropy of the system.

Probing chaos by magic monotones

There is a property of a quantum state called ``magic.'' As shown by the Gottesman-Knill theorem, so-called stabilizer states, which are composed of only Clifford gates, can be efficiently computed on a classical computer, and thus quantum computation gives no advantage. Nonstabilizer states are called magic states, which are necessary to achieve the universal quantum computation. Magic (monotone) is the measure of the amount of nonstabilizer resource, and it measures how difficult it is for a classical computer to simulate the state. We study magic of states in the integrable and chaotic regimes of the higher-spin generalization of the Ising model through two quantities: ``mana'' and ``robustness of magic'' (RoM). We find that in the chaotic regime, mana increases monotonically in time in the early-time region, and at late times these quantities oscillate around some nonzero value that increases linearly with respect to the system size. Our result also suggests that under chaotic dynamics, any state evolves to a state whose mana almost saturates the optimal upper bound; i.e., the state becomes ``maximally magical.'' We find that RoM also shows similar behaviors. On the other hand, in the integrable regime, mana and RoM behave periodically in time in contrast to the chaotic case. In addition to mana and RoM, for the early-time behavior of magic, we study the stabilizer R\'enyi entropy, which can be numerically computed for larger systems than mana and RoM. In the anti-de Sitter/conformal field theory correspondence, classical spacetime emerges from the chaotic nature of the dual quantum system. Our results suggest that magic of quantum states is strongly involved in the emergence of spacetime geometry.

Numerical Treatment on a Chaos Model of Fluid Flow Using New Iterative Method

This article treats analytically and numerically to the three dimensional Rössler system. The governing equations of the problem are derived from traditional Lorentz system of fluid mechanics to nonlinear ordinary differential equations (ODEs) for modelling. A semi-analytic solution is developed by using New Iterative Method (NIM) whereas the numerical solution is presented by Runge-Kutta order four (RK4) scheme. A comparative study of the analytical and numerical solutions are made. The results confirm clearly that the two methods coincide closely for both the chaotic and non-chaotic cases of the determined system. This observation would be helpful to apply NIM on nonlinear problems of fluid flow with different fluid parameters in future.

Discovery of interpretable structural model errors by combining Bayesian sparse regression and data assimilation: A chaotic Kuramoto-Sivashinsky test case.

Models of many engineering and natural systems are imperfect. The discrepancy between the mathematical representations of a true physical system and its imperfect model is called the model error. These model errors can lead to substantial differences between the numerical solutions of the model and the state of the system, particularly in those involving nonlinear, multi-scale phenomena. Thus, there is increasing interest in reducing model errors, particularly by leveraging the rapidly growing observational data to understand their physics and sources. Here, we introduce a framework named MEDIDA: Model Error Discovery with Interpretability and Data Assimilation. MEDIDA only requires a working numerical solver of the model and a small number of noise-free or noisy sporadic observations of the system. In MEDIDA, first, the model error is estimated from differences between the observed states and model-predicted states (the latter are obtained from a number of one-time-step numerical integrations from the previous observed states). If observations are noisy, a data assimilation technique, such as the ensemble Kalman filter, is employed to provide the analysis state of the system, which is then used to estimate the model error. Finally, an equation-discovery technique, here the relevance vector machine, a sparsity-promoting Bayesian method, is used to identify an interpretable, parsimonious, and closed-form representation of the model error. Using the chaotic Kuramoto-Sivashinsky system as the test case, we demonstrate the excellent performance of MEDIDA in discovering different types of structural/parametric model errors, representing different types of missing physics, using noise-free and noisy observations.

Application of Explicit Symplectic Integrators in a Magnetized Deformed Schwarzschild Black Spacetime

Abstract Following the latest work of Wu et al., we construct time-transformed explicit symplectic schemes for a Hamiltonian system on the description of charged particles moving around a deformed Schwarzschild black hole with an external magnetic field. Numerical tests show that such schemes have good performance in stabilizing energy errors without secular drift. Meantime, tangent vectors are solved from the variational equations of the system with the aid of an explicit symplectic integrator. The obtained tangent vectors are used to calculate several chaos indicators, including Lyapunov characteristic exponents, fast Lyapunov indicators, a smaller alignment index, and a generalized alignment index. It is found that the smaller alignment index and generalized alignment index are the fastest indicators for distinguishing between regular and chaotic cases. The smaller alignment index is applied to explore the effects of the parameters on the dynamical transition from order to chaos. When the positive deformation factor and angular momentum decrease, or when the energy, positive magnetic parameter, and the magnitude of the negative deformation parameter increase, chaos easily occurs for the appropriate choices of initial conditions and the other parameters.

Identifying localized and spreading chaos in nonlinear disordered lattices by the Generalized Alignment Index (GALI) method

Implementing the Generalized Alignment Index (GALI) method of chaos detection we investigate the dynamical behavior of the nonlinear disordered Klein–Gordon lattice chain in one spatial dimension. By performing extensive numerical simulations of single site and single mode initial excitations, for several disordered realizations and different disorder strengths, we determine the probability to observe chaotic behavior as the system is approaching its linear limit, i.e. when its total energy, which plays the role of the system’s nonlinearity strength, decreases. We find that the percentage of chaotic cases diminishes as the energy decreases leading to exclusively regular motion on multidimensional tori. We also discriminate between localized and spreading chaos, with the former dominating the dynamics for lower energy values. In addition, our results show that single mode excitations lead to more chaotic behaviors for larger energies compared to single site excitations. Furthermore, we demonstrate how the GALI method can be efficiently used to determine a characteristic chaoticity time scale for the system when strong enough nonlinearites lead to energy delocalization in both the so-called ‘weak’ and ‘strong chaos’ spreading regimes.

Numerical Solutions of Nine-Dimensional Lorenz System

In this study, the performance of the Taylor’s decomposition method in the nine-dimensional Lorenz system is investigated. The proposed method is applied to the nine-dimensional Lorenz system for both chaotic and hyperchaotic cases. Phase portraits and time series graphs of the problem are drawn. The results obtained are compared with both the fourth order Runge-Kutta method and multi-domain compact finite difference relaxation method. Comparisons show that the obtained results are consistent with the results of other methods. Thus, the accuracy and efficiency of the method for nonlinear systems were emphasized by using the nine-dimensional Lorenz system.

Predicting the chaos and solution bounds in a complex dynamical system

The competitive modes for a nonlinear chaotic complex system are studied in this paper. In hyperchaotic, chaotic, and periodic cases, we examined competitive modes that are a tool for detecting chaos in a system. Also, using an analytical method and Lagrange optimization, we were able to calculate the ultimate bound of the nonlinear chaotic complex systems. We have presented is simpler and more accurate than other methods that implicitly calculate the ultimate bound. The estimation of the explicit ultimate bound can be used to study chaos control and chaos synchronization. Numerical simulations illustrate the analytical results.