Spiral Chaos Research Articles

Modified crayfish optimization algorithm with adaptive spiral elite greedy opposition-based learning and search-hide strategy for global optimization

Abstract Crayfish optimization algorithm (COA) is a novel bionic metaheuristic algorithm with high convergence speed and solution accuracy. However, in some complex optimization problems and real application scenarios, the performance of COA is not satisfactory. In order to overcome the challenges encountered by COA, such as being stuck in the local optimal and insufficient search range, this paper proposes four improvement strategies: search-hide, adaptive spiral elite greedy opposition-based learning, competition-elimination, and chaos mutation. To evaluate the convergence accuracy, speed, and robustness of the modified crayfish optimization algorithm (MCOA), some simulation comparison experiments of 10 algorithms are conducted. Experimental results show that the MCOA achieved the minor Friedman test value in 23 test functions, CEC2014 and CEC2020, and achieved average superiority rates of 80.97%, 72.59%, and 71.11% in the WT, respectively. In addition, MCOA shows high applicability and progressiveness in five engineering problems in actual industrial field. Moreover, MCOA achieved 80% and 100% superiority rate against COA on CEC2020 and the fixed-dimension function of 23 benchmark test functions. Finally, MCOA owns better convergence and population diversity.

Model Map and Multistability for a Two Predator-One Prey System

This work contains a review of some important results on a known two predators - one prey system. We also add essential new numerical results on multiple attractors. We consider the case when the predators coexist. We distinguish two possibilities. The first is when the dynamics is well described by the dynamics of a one dimensional map. We discuss the main behaviour of this map. For small parameter regions the map can have two attractors but no more than two. We give a numerical example, when these two attractors exist in the original three-dimensional model. The second case is when this model map is not working and something like spiral chaos often occurs. We give numerical results showing that in this case there can be at least four different attractors and discuss the behaviour of these.

Existence of homoclinic orbit of Shilnikov type and the application in Rössler system

In this paper, we modify the methods of Zhou et al. (2004) and Shang and Han (2005) associated with proving the existence of a homoclinic orbit of Shilnikov type. We construct the series expressions of the solution based at a saddle-focus on stable and unstable manifolds, and give the sufficient conditions of the existence of homoclinic orbit and spiral chaos. Then, we consider the Rössler system with the typical parameters under which the system exhibits chaotic behavior. Using our modified method, we verify that there exists a homoclinic orbit of Shilnikov type in the Rössler system with a group of typical parameters, and prove the existence of spiral chaos by using the Shilnikov criterion, and we carry out numerical simulations to support the analytic results.

Taming cardiac arrhythmias: Terminating spiral wave chaos by adaptive deceleration pacing.

Sequences of weak electrical pulses are considered a promising alternative for terminating ventricular and atrial fibrillations while avoiding strong defibrillation shocks with adverse side effects. In this study, using numerical simulations of four different 2D excitable media, we show that pulse trains with increasing temporal intervals between successive pulses (deceleration pacing) provide high success rates at low energies. Furthermore, we propose a simple and robust approach to calculate inter-pulse spacing directly from the frequency spectrum of the dynamics (for instance, computed based on the electrocardiogram), which can be practically used in experiments and clinical applications.

Non-monotonous dose response function of the termination of spiral wave chaos

The conventional termination technique of life threatening cardiac arrhythmia like ventricular fibrillation is the application of a high-energy electrical defibrillation shock, coming along with severe side-effects. In order to improve the current treatment reducing these side-effects, the application of pulse sequences of lower energy instead of a single high-energy pulse are promising candidates. In this study, we show that in numerical simulations the dose-response function of pulse sequences applied to two-dimensional spiral wave chaos is not necessarily monotonously increasing, but exhibits a non-trivial frequency dependence. This insight into crucial phenomena appearing during termination attempts provides a deeper understanding of the governing termination mechanisms in general, and therefore may open up the path towards an efficient termination of cardiac arrhythmia in the future.

On scenarios of the onset of homoclinic attractors in three-dimensional non-orientable maps

We study scenarios of the appearance of strange homoclinic attractors (which contain only one fixed point of saddle type) for one-parameter families of three-dimensional non-orientable maps. We describe several types of such scenarios that lead to the appearance of discrete homoclinic attractors including Lorenz-like and figure-8 attractors (which contain a saddle fixed point) as well as two types of attractors of spiral chaos (which contain saddle-focus fixed points with the one-dimensional and two-dimensional unstable manifolds, respectively). We also emphasize peculiarities of the scenarios and compare them with the known scenarios in the orientable case. Examples of the implementation of the non-orientable scenarios are given in the case of three-dimensional non-orientable generalized Hénon maps.

On the Organization of Homoclinic Bifurcation Curves in Systems with Shilnikov Spiral Attractors

We study spiral chaos in the classical Rössler and Arneodo – Coullet – Tresser systems. Special attention is paid to the analysis of bifurcation curves that correspond to the appearance of Shilnikov homoclinic loop of saddle-focus equilibrium states and, as a result, spiral chaos. To visualize the results, we use numerical methods for constructing charts of the maximal Lyapunov exponent and bifurcation diagrams obtained using the MatCont package.

Control of spiral wave and spatiotemporal chaos in two-layer aging cardiac tissues

Cardiac arrhythmias can be caused by the occurrence of electrical spiral waves and spatiotemporal chaos in the cardiac tissues, as well as by the changes of cardiac tissues resulting from the electrical coupling of cardiomyocytes to fibroblasts (M-F coupling). How to control the arrhythmia induced by spiral wave or spatiotemporal chaos is the problem which has attracted much attention of scientists. In this paper, a two-layer composite medium is constructed by using cardiomyocytes and fibroblasts. Luo-Rudy phase I cardiac model and passive model of fibroblast are used to study the effects of the M-F coupling on the formation of spiral wave and the control of spiral wave and spatiotemporal chaos in a two-layer composite medium. A control scheme that the spiral waves and spatiotemporal chaos are controlled by increasing the coupling strength between cells is proposed. The numerical results show that the M-F coupling has an important influence on the dynamics of spiral wave. With the increase of the density of fibroblasts, the M-F coupling may result in spiral wave meandering and spiral wave breaking into spatiotemporal chaos, and even induce the transition from spatiotemporal chaos (or spiral wave) to no wave. The eliminating spiral wave and spatiotemporal chaos in the composite medium by increasing the coupling strength between cells is only effective in most of cases, depending on the role played by fibroblasts. When fibroblasts act as current sinks for the cardiomyocyte, the spiral wave and spatiotemporal chaos are effectively eliminated only in most of cases by increasing the coupling strength between cells, and the controlled area is small. When fibroblasts act as a current source for the cardiomyocyte, increasing the coupling strength between cells to a value higher than a critical value can effectively terminate spiral wave and spatiotemporal chaos, and the controllable area is greatly increased compared with the former. Increasing the coupling strength between cardiomyocytes is a key factor in controlling the spiral waves and spatiotemporal chaos.

A physical memristor based Muthuswamy\u2013Chua\u2013Ginoux system

In 1976, Leon Chua showed that a thermistor can be modeled as a memristive device. Starting from this statement we designed a circuit that has four circuit elements: a linear passive inductor, a linear passive capacitor, a nonlinear resistor and a thermistor, that is, a nonlinear “locally active” memristor. Thus, the purpose of this work was to use a physical memristor, the thermistor, in a Muthuswamy–Chua chaotic system (circuit) instead of memristor emulators. Such circuit has been modeled by a new three-dimensional autonomous dynamical system exhibiting very particular properties such as the transition from torus breakdown to chaos. Then, mathematical analysis and detailed numerical investigations have enabled to establish that such a transition corresponds to the so-called route to Shilnikov spiral chaos but gives rise to a “double spiral attractor”.

Development and transition of target waves in the network of Hindmarsh–Rose neurons under electromagnetic radiation

The pattern transition of target waves in Hindmarsh–Rose neuron network exposed to fixed and periodic electromagnetic radiation is reported in this paper. Our numerical results confirm that local periodical excitation can induce stable propagating target waves from the network. It is found that fixed electromagnetic radiation has great effect on the propagating target waves, and that these target waves can be obviously blocked by increasing the intensity of fixed electromagnetic radiation. We find that the periodic electromagnetic radiation with appropriate amplitude and frequency can break the target waves and induce spatiotemporal turbulence and spiral waves from the broken target waves. Our numerical simulations show that the influence of periodic electromagnetic radiation on the dynamics of target waves is complex, and that although both increasing the amplitude and decreasing the frequency can break the target waves and induce spiral waves and chaos from the network, extensive numerical results find that lower frequency is more easy to terminate the target waves and generate spiral waves and spatiotemporal chaos. The numerical simulations also show that fixed and periodic electromagnetic radiation have influence on the pattern transition of the target waves in the network, but periodic electromagnetic radiation is more helpful to develop spiral waves and turbulence from the network.

Homoclinic orbits and an invariant chaotic set in a new 4D piecewise affine systems

It is a challenging task to prove mathematically the existence of homoclinic orbits in a high-dimensional dynamical system. Here we first introduce a new four-dimensional (4D) piecewise affine system and then establishes useful yet general conditions on the existence of an orbit homoclinic to a spiral saddle-foci and chaos in the system. Rigorously mathematical analysis is also provided. A 4D example with both a homoclinic orbit and an invariant chaotic set is used to depict the effectiveness of analysis and prediction.

Spiral wave breakup manner in the excitable system with early afterdepolarizations

Early afterdepolarization (EAD) is an important cause of lethal ventricular arrhythmias in heart failure because afterdepolarizations can promote the transition from ventricular tachycardia to fibrillation, which is related to the transition from spiral wave to spatiotemporal chaos. However, it remains unclear about how the EAD results in the breakup of spiral wave. In this paper, we explore the manner of spiral wave breakup induced by EADs under evenly distributed cells. The two-dimensional tissue is simulated with the Greenberg-Hasting cellular automaton model. The normal cells and aging cells are introduced into the model, in which the EAD only occurs in aging cells and can excite the resting cells. The numerical results show that the EAD can produce backward waves as well as forward waves. The EAD has no influence on the behavior of spiral wave in a few cases. The ratio of the number of unaffected spiral waves to the number of all tests is about 26.4%. The EAD can have various effects on spiral wave in other cases. The small influences on spiral wave are that the EAD leads to the meander, drift, and the deformation of spiral wave. The serious influences on spiral wave are that the EAD results in the disappearance and breakup of spiral wave. We find that spiral wave can disappear through the conduction block and transition from spiral wave to target wave. We observe the eight kinds of spiral wave breakups in connection with the excitation of EADs, such as symmetry breaking-induced breakup, nonsymmetry breaking-induced breakup, asymmetric excitation-induced breakup, conduction block-induced breakup, double wave-induced breakup, etc. Spiral wave generally breaks up into multiple spiral waves and spatiotemporal chaos. The ratio of the number of spiral wave breakup to the number of all tests is about 13.8%. However, the ratio of spiral wave breakup can reach about 32.4% under appropriately chosen parameters. The results are basically consistent with the survey results of arrhythmia-induced death rate. Furthermore, we also find that the excitation of EAD can prevent the spiral wave from disappearing and promote the breakup of spiral wave. The physical mechanisms underlying those phenomena are also briefly analyzed.

Memory effects, transient growth, and wave breakup in a model of paced atrium.

The mechanisms underlying cardiac fibrillation have been investigated for over a century, but we are still finding surprising results that change our view of this phenomenon. The present study focuses on the transition from normal rhythm to spiral wave chaos associated with a gradual increase in the pacing rate. While some of our findings are consistent with existing experimental, numerical, and theoretical studies of this problem, one result appears to contradict the accepted picture. Specifically we show that, in a two-dimensional model of paced homogeneous atrial tissue, transition from discordant alternans to conduction block, wave breakup, reentry, and spiral wave chaos is associated with the transient growth of finite amplitude disturbances rather than a conventional instability. It is mathematically very similar to subcritical, or bypass, transition from laminar fluid flow to turbulence, which allows many of the tools developed in the context of fluid turbulence to be used for improving our understanding of cardiac arrhythmias.

Spiral chaos in Lotka-Volterra like models

Spiral chaos in Lotka-Volterra like models

Numerical simulations of chaotic and complex spatiotemporal patterns in fractional reaction–diffusion systems

The generalized fractional reaction–diffusion equations which exist in the form of noninteger order partial differential equations have now found wide application for illustrating important and useful physical phenomena, such as subdiffusive and superdiffusive scenarios. The space fractional derivatives are defined in the Riesz sense on the intervals $$0<\alpha <1$$ and $$1<\alpha \le 2$$ . We propose robust numerical techniques, such as a spectral representation of the fractional Laplacian operator in conjunction with the exponential time differencing method, and present the equivalent relationship between the Riesz fractional derivative and fractional Laplacian operator. We apply these techniques to numerically solve a range of chaotic processes, such as the Chua’s equations, Rossler system, Lorenz and Lorenz-type systems. Simulation results revealed various complex and spatiotemporal chaos, spiral chaos, intermittent chaos and spots patterns in two-dimensional space fractional reaction–diffusion problems.

Approximated Spiral and Target Patterns in Bazykin’s Prey–Predator Model: Multiscale Perturbation Analysis

Spatio-temporal pattern formation has opened up a wide area of research to understand the dynamics between various interacting populations such as a prey and a predator, competing species etc. The governing equations are typically modeled by a reaction–diffusion system. The commonly known patterns, namely, traveling wave, periodic traveling wave, spot, labyrinthine, mixture of spot and stripe, spatio-temporal chaos and interacting spiral chaos can be observed in the spatio-temporal extension of various interacting population models. Apart from these, the other two types of patterns, namely, spiral and target patterns also evolve under suitable parametric conditions near the Turing–Hopf threshold though there is no systematic approach to determine the exact formalism of their emergence. In this paper, we have used a multiscale perturbation analysis to determine these patterns in the spatio-temporal extension of Bazykin’s prey–predator model. An important finding of this work is the use of approximated analytical solution as the initial condition to obtain spiral and target patterns with the help of numerical simulations. Analytical results are general in nature and hence can be used for any spatio-temporal model of interacting population as well as other pattern forming systems which are capable of producing spiral and target patterns.

Mechanical Motion Induced by Spatially Distributed Limit-Cycle Oscillators

Spatially distributed limited-cycle oscillators are seen in various physical and biological systems. In internal organs, mechanical motions are induced by the stimuli of spatially distributed limit-cycle oscillators. We study several mechanical motions by limit-cycle oscillators using simple model equations. One problem is deformation waves of radius oscillation induced by desynchronized limit-cycle oscillators, which is motivated by peristaltic motion of the small intestine. A resonance-like phenomenon is found in the deformation waves, and particles can be transported by the deformation waves. Another is the beating motion of the heart. The expansion and contraction motion is realized by a spatially synchronized limit-cycle oscillation; however, the strong beating disappears by spiral chaos, which is closely related to serious arrhythmia in the heart.

Terminating spiral wave and spatiotemporal chaos in cardiac tissues by using late sodium current

Most Na+ channels open transiently upon depolarization of cardiac cell membrane and then are quickly inactivated. However, some Na+ channels remain active, which generate the late sodium current during the action potential plateau. So far, late sodium current has been regarded as a relevant contributor to arrhythmias and its inhibition can suppress re-entrant and multifocal ventricular fibrillation so that its inhibition may become a novel therapeutic strategy to treat cardiac arrhythmias in the future. Therefore, how to inhibit late sodium current has received special attention. Since both the late sodium current and defibrillation shocks can lead to the increase of action potential duration, the late sodium current can be used to terminate ventricular fibrillation. However, the suppression of spiral wave and spatiotemporal chaos in cardiac tissues via late sodium current has been neglected. In this paper, we use the model of human heart to study the suppression of spiral wave and spatiotemporal chaos in two-dimensional cardiac tissue by generating late sodium current. We suggest that such a control strategy to induce late sodium current. The slow inactivation gate of sodium channel is clamped to 0.7 while the threshold voltage of corresponding fast inactivation gate is real-timely modulated. We first reduce the threshold voltage from 71.55 mV to 50.55 mV within the time interval T1, and then increase it from 50.55 mV to 71.55 mV within the time interval T2. When the threshold voltage returns to 71.55 mV, the changes of the relevant inactivation gates of sodium channel go back to normal dynamic state. Numerical simulation results show that when the control parameters are properly chosen, the control-induced late sodium current can effectively suppress spiral wave and spatiotemporal chaos even if there are some cardiac cells with spontaneous late sodium current. The advantage of the control scheme is that the control-induced late sodium current is small. The control duration is short because the spiral wave and spatiotemporal chaos disappear mainly due to the conduction obstacle. In a few cases, the spatiotemporal chaos disappears through the transition from spiral wave to target wave. We hope that these results may provide a new strategy to treat heart disease.

Controlling spiral wave and spatiotemporal chaos in cardiac tissues by slowing sodium channel activation and inactivation

Much evidence shows that the appearance and instability of the spiral wave in cardiac tissue can be linked to a kind of heart disease. Therefore there needs a method of controlling spiral wave more safely and effectively. The intelligent modification of specific ion channel to achieve desired control is the future direction of gene therapy in heart disease. The key question that has to be answered is which ion channel is the best candidate for controlling spiral wave. Modern biological technology has been able to make the mutation of sodium channel gene to change its relaxation time constant. In this paper, we adopt the Luo-Rudy phase I model to investigate how to regulate the relaxation time constant of sodium channel gate to control spiral wave and spatiotemporal chaos in cardiac tissues. We suggest a control strategy which slows down the rate of sodium current activation and inactivation by increasing the relaxation time constant of the sodium activation gate by up to times while its fast inactivation gate is clamped to 0.77. Numerical simulation results show that a gradual increase of will cause the activation gate of sodium current to reach maximum more slowly, and its amplitude is gradually reduced, so that the amplitude and duration of the action potential of cardiomyocyte are gradually reduced. When the factor is large enough, the spiral wave and spatiotemporal chaos cannot propagate in the medium except planar wave with low frequency. The reason is that the excitabilities of medium and wave speed significantly decrease. Therefore, the spiral waves and spatiotemporal chaos can be effectively eliminated when the control time is properly selected and the factor is large enough. Spiral wave and spatiotemporal chaos disappear mainly due to conduction obstacle. In some cases, spiral wave can disappear through the transition from spiral wave to target wave or tip retraction. Spatiotemporal chaos disappears after spatiotemporal chaos has evolved into meandering spiral wave. When the parameters are chosen properly, the phenomenon that spiral wave evolves into a self-sustained target wave is also observed. The corresponding target wave source is the pair of spiral waves with opposite rotation directions. These results can provide useful information for gene therapy in heart disease.

Exponential system-size dependence of the lifetime of transient spiral chaos in excitable and oscillatory media.

Excitable media can develop spiral chaos, in which the number of spirals changes chaotically with time. Depending on parameter values in dynamical equations, spiral chaos may permanently persist or spontaneously arrive at a steady state after a transient time, referred to as the lifetime. Previous numerical studies have demonstrated that the lifetime of transient spiral chaos increases exponentially with system size to a good approximation. In this study, using the fact that the number of spirals obeys a Gaussian distribution, we provide a general expression for the system size dependence of the lifetime for large system sizes, which is indeed exponential. We confirm that the expression is in good agreement with numerically obtained lifetimes for both excitable and oscillatory media with parameter sets near the onset of transient chaos. The expression we develop for the lifetime is expected to be useful for predicting lifetimes in large systems.