Spatiotemporal Chaos Research Articles

Pattern dynamics in a predator–prey model with Smith growth function and prey refuge in predator poisoned environment

In this article, we investigate temporal as well as spatiotemporal dynamics of a predator–prey system where prey grows according to Smith growth function and prey refuges in predator poisoned environment. Self as well as cross diffusion are taken into consideration to describe spatial moment of the species which make the system more realistic. This study aims to investigate the role of prey refuge, toxin-carrying carcass and the impact of cross-diffusion on the spatial distribution of species densities and associated patterns. The existence of equilibria and their stability conditions for non-spatial system are derived. We discuss the occurrence of Hopf bifurcation and detect the stability of limit cycle by computing first Lyapunov number. Temporal dynamics of the system is studied by varying the significant parameters. Study reveals that both prey refuge and fatality rate of predator due to toxin-carrying carcass, initially makes the system stable for low refuge and fatality rate but predator population goes to extinction as those effects increase. Turing instability conditions are listed and different instability regions so formed are discussed. In Turing region, pattern selection with the help of amplitude equation via weakly nonlinear analysis are performed which also numerically supported. Intensity of refuge and fatality rate due to toxin carrying carcass result paradox with real ecological world due to the presence of cross diffusion. Various kind of patterns such as mixture of stripe and spot, spot, irregular chaotic patterns emerge due to Hopf and Turing instability. Particularly, spatiotemporal chaos is a controversial topic because of its significant consequences for population dynamics. The results of this work are expected to contribute to a better understanding the level of prey refuge, use of toxin-carrying carcass considering movement in population and effect of cross-diffusion.

Accurate deep learning-based filtering for chaotic dynamics by identifying instabilities without an ensemble.

We investigate the ability to discover data assimilation (DA) schemes meant for chaotic dynamics with deep learning. The focus is on learning the analysis step of sequential DA, from state trajectories and their observations, using a simple residual convolutional neural network, while assuming the dynamics to be known. Experiments are performed with the Lorenz 96 dynamics, which display spatiotemporal chaos and for which solid benchmarks for DA performance exist. The accuracy of the states obtained from the learned analysis approaches that of the best possibly tuned ensemble Kalman filter and is far better than that of variational DA alternatives. Critically, this can be achieved while propagating even just a single state in the forecast step. We investigate the reason for achieving ensemble filtering accuracy without an ensemble. We diagnose that the analysis scheme actually identifies key dynamical perturbations, mildly aligned with the unstable subspace, from the forecast state alone, without any ensemble-based covariances representation. This reveals that the analysis scheme has learned some multiplicative ergodic theorem associated to the DA process seen as a non-autonomous random dynamical system.

A new encryption algorithm for image data based on two-way chaotic maps and iterative cellular automata

Due to their simplicity of implementation and compliance with the encryption issue, chaotic models are often utilized in picture encryption applications. Despite having many benefits, this approach still has a crucial space issue that makes encryption algorithms based on it susceptible to brute-force assaults. This research’s proposed novel picture encryption technique has a vast key space and great key sensitivity. To achieve this goal, the proposed method combines two-way chaotic maps and reversible cellular automata (RCA). First, this approach uses a two-way chaotic model named spatiotemporal chaos for image confusion. This step includes permuting the image pixels using a chaotic map at the byte level. Then, the RCA model is utilized for image diffusion. In this step, the RCA model iterates over image pixels to modify them at the bit level. The method’s performance in encrypting grayscale images was evaluated using various analysis methods. According to the results, the proposed method is a compelling image encryption algorithm with high robustness against brute-force, statistical, and differential attacks.

A spatiotemporal chaos based deep learning model watermarking scheme

With deep learning techniques achieving great results in modern industry, the intellectual property (IP) protection for deep learning models has attracted the attention of academics and engineers. However, training a commercially viable deep learning model usually needs professional resources and time. Once a malicious user clones, illegally distributes and uses the model, it can infringe on the model owner's IP and even steal its market share. Among the existing IP protection methods, scholars prefer the black-box watermarking approaches, of which the content of the trigger set and the label are the key part of the watermarking technique. However, most schemes do not consider the security and invisibility of the trigger set, which allows attackers to easily trigger the model by creating a fake trigger set, thereby committing a fraudulent ownership claim attack and claiming the ownership belongs to themselves. To overcome these drawbacks, we proposed a spatiotemporal chaotic data annotation method. Firstly, the unpredictability and acyclicity of chaos make the model resistant to fraudulent ownership claim attacks, statistical inference and other common machine learning attacks; Secondly, the trigger set and parameters are independent of each other, guaranteeing the security of the key; Thirdly, the spatiotemporal chaotic system provides a large key space, which meets the commercialization needs of deep learning models. Theoretical analysis and experimental results show that our scheme has security, practicality and robustness. To further validate the superiority of the proposed method, we also compare it with the Logistic chaotic annotation watermarking-based method, and the results show that our method performs better in terms of robustness, effectiveness, completeness, fidelity, security and practicality.

Complex Ginzburg–Landau equation for time‐varying anisotropic media

AbstractWhen extending the complex Ginzburg–Landau equation (CGLE) to more than one spatial dimension, there is an underlying question of whether one is capturing all the interesting physics inherent in these higher dimensions. Although spatial anisotropy is far less studied than its isotropic counterpart, anisotropy is fundamental in applications to superconductors, plasma physics, and geology, to name just a few examples. We first formulate the CGLE on anisotropic, time‐varying media, with this time variation permitting a degree of control of the anisotropy over time, focusing on how time‐varying anisotropy influences diffusion and dispersion within both bounded and unbounded space domains. From here, we construct a variety of exact dissipative nonlinear wave solutions, including analogs of wavetrains, solitons, breathers, and rogue waves, before outlining the construction of more general solutions via a dissipative, nonautonomous generalization of the variational method. We finally consider the problem of modulational instability within anisotropic, time‐varying media, obtaining generalizations to the Benjamin–Feir instability mechanism. We apply this framework to study the emergence and control of anisotropic spatiotemporal chaos in rectangular and curved domains. Our theoretical framework and specific solutions all point to time‐varying anisotropy being a potentially valuable feature for the manipulation and control of waves in anisotropic media.

Resonancelike emergence of chaos in complex networks of damped-driven nonlinear systems.

Characterizing the emergence of chaotic dynamics of complex networks is an essential task in nonlinear science with potential important applications in many fields such as neural control engineering, microgrid technologies, and ecological networks. Here, we solve a critical outstanding problem in this multidisciplinary research field: the emergence and persistence of spatiotemporal chaos in complex networks of damped-driven nonlinear oscillators in the significant weak-coupling regime, while they exhibit regular behavior when uncoupled. By developing a comprehensive theory with the aid of standard analytical methods, a hierarchy of lower-dimensional effective models, and extensive numerical simulations, we uncover and characterize the basic physical mechanisms concerning both heterogeneity-induced and impulse-induced emergence, enhancement, and suppression of chaos in starlike and scale-free networks of periodically driven, dissipative nonlinear oscillators.

Spatiotemporal nonlinear dynamics and chaos in a mechanical Duffing-type system

This paper investigates spatiotemporal nonlinear dynamics and chaos in a dissipative mechanical Duffing-type system subjected to external stimulus. A nonlinear wave equation with cubic nonlinearity governs the system dynamics. A perturbation description is employed to build mathematical tools that represent different aspects of system dynamics, from local to global behaviors. Lyapunov exponents are defined from the different perturbations allowing the evaluation of local, convective and mean exponents. Different dynamical regimes are investigated considering homogeneous and heterogeneous spatial stimuli. Distinct dynamical responses are observed including periodic, quasi-periodic and chaotic behaviors. A novel concept of chaotic wave is employed to explain the spatial transport of chaos through the media considering heterogeneous conditions. Chaotic wave velocity is measured by the convective Lyapunov exponents.

Short message encryption technology of Beidou communication based on SM9

AbstractTo reduce the security authentication time cost, forwarding delay, and enhance message security of Beidou communication short messages, this paper proposes a Beidou communication short message encryption technology based on SM9 (Mobile Code Secure Communication). This technology uses split hierarchical clustering algorithm to cluster the short message data of Beidou Communication, in order to unify the processing of similar short message data and improve the encryption efficiency of subsequent short messages. Based on the data clustering results, this paper constructs a short message forwarding verification model using the SM9 identification password algorithm, and provides the message content of the communication application protocol and the communication receiving information protocol to achieve low‐cost and efficient forwarding verification. In addition, this article improves the initial value selection of Logistic chaotic mapping by coupling the tent map with coupled mapping lattice (CML) in the spatiotemporal chaos model to generate a chaotic initial value sequence, thereby expanding the key space, enhancing the ability to resist exhaustive attacks, enhancing the randomness of the generated key parameters, and optimizing the encryption effect of short messages. The experimental results show that the short message security authentication time of the technology proposed in this article is basically controlled within 25 ms, and the minimum forwarding delay is only 23.9 ms, which can effectively resist attacks and improve the security of Beidou communication.

Spatiotemporal and Trade-Off Dynamics in Prey–Predator Model with Domed Functional Response and Fear Effect

In the ecological scenario, predators often risk their lives pursuing dangerous prey, potentially reducing their chances of survival due to injuries. Prey, on the other hand, try to strike a balance between reproduction rates and safety. In our study, we introduce a two-dimensional prey–predator model inspired by Tostowaryk’s work, specifically focusing on the domed-shaped functional response observed in interactions between pentatomid predators and neo-diprionid sawfly larvae. To account for the varying effectiveness of larval group defense, we incorporate a new component into the response equation. Our investigation delves into predator trade-off dynamics by adjusting the predator’s mortality rate to reflect losses incurred during encounters with dangerous prey and prey’s trade-off between safety and reproduction rate incorporating this domed-shaped functional response. Our model demonstrates bistability and undergoes various bifurcations, including transcritical, saddle-node, Hopf, Bogdanov–Takens, and Homoclinic bifurcations. Critical parameters impact both predator and prey populations, potentially leading to predator extinction if losses due to dangerous prey encounters become excessive, highlighting the risks predators face for their survival. Furthermore, the efficacy of group defense mechanisms can further endanger predators. Expanding our analysis to a spatially extended model under different perturbations, we explore Turing instability to explain the relationship between diffusion and encounter parameters through both stationary and dynamic pattern formation. Sensitivity to initial conditions uncovers spatiotemporal chaos. These findings provide valuable insights into comprehending the intricate dynamics of prey–predator interactions within ecological systems.

Influence of Raman gain on dynamics of spatiotemporal chaos in optical ring microresonators

We theoretically investigate on the evolution of spatiotemporal chaos under the influence of Raman gain, in an optical ring microresonator. The dynamics of the system is shown to be governed by the Lugiator–Lefever equation, with an additional real and imaginary Raman gain term. In the absence of pump field intensity and Raman gain, we observe an increase in the amplitude and frequency of the traveling optical field as the pump detuning frequency increases. However this field gradually vanishes as time increases, owing to the intra-cavity loss. By exploring an appropriate ansatz which incorporate a chirping parameter, the modified Lugiator–Lefever equation is transformed to a set of four coupled first-order nonlinear ordinary differential equations. Fixed point stability analysis strongly suggest that variation in Raman gain has the propensity of inducing the co-existence of bright and dark solitons in the anomalous dispersion regime. Results of numerical simulations underscore that the structural profiles of the optical field amplitude are greatly modified by the Raman effects. Variation in Raman gain equally leads to a bifurcation analysis of the system, in which the Lyapunov exponent diagrams reveal regions of period doubling and chaos. We finally observe that for a fixed value of the real part of Raman gain coefficient, an increase in the effects of soliton self-frequency shift generally drives the system to chaotic regime.

Transition to period-3 synchronized state in coupled gauss maps.

We study coupled Gauss maps in one dimension with nearest-neighbor interactions. We observe transitions from spatiotemporal chaos to period-3 states in a coarse-grained sense and synchronized period-3 states. Synchronized fixed points are frequently observed in one dimension. However, synchronized periodic states are rare. The obvious reason is that it is very easy to create defects in one dimension. We characterize all transitions using the following order parameter. Let x∗ be the fixed point of the map. The values above (below) x∗ are classified as +1 (-1) spins. We expect all sites to return to the same band after three time steps for a coarse-grained periodic or three-period state. We define the flip rate F(t) as the fraction of sites i such that si(3t-3)≠si(t). It is zero in the coarse-grained periodic state. This state may or may not be synchronized. We observe three different transitions. (a) If different sites reach different bands, the transition is in the directed-percolation universality class. (b) If all sites reach the same band, we find an Ising-type transition. (c) A synchronized period-3 state where a new exponent is observed. We also study the finite-size scaling at critical points. The exponents obtained indicate that the synchronized period-3 transition is in a new universality class.

Prediction and control of spatiotemporal chaos by learning conjugate tubular neighborhoods

I present a data-driven predictive modeling tool that is applicable to high-dimensional chaotic systems with unstable periodic orbits. The basic idea is using deep neural networks to learn coordinate transformations between the trajectories in the periodic orbits’ neighborhoods and those of low-dimensional linear systems in a latent space. I argue that the resulting models are partially interpretable since their latent-space dynamics is fully understood. To illustrate the method, I apply it to the numerical solutions of the Kuramoto–Sivashinsky partial differential equation in one dimension. Besides the forward-time predictions, I also show that these models can be leveraged for control.

Mutil-medical image encryption by a new spatiotemporal chaos model and DNA new computing for information security

Mutil-medical image encryption by a new spatiotemporal chaos model and DNA new computing for information security

Research on pattern dynamics of a class of predator-prey model with interval biological coefficients for capture

<abstract><p>Due to factors such as climate change, natural disasters, and deforestation, most measurement processes and initial data may have errors. Therefore, models with imprecise parameters are more realistic. This paper constructed a new predator-prey model with an interval biological coefficient by using the interval number as the model parameter. First, the stability of the solution of the fractional order model without a diffusion term and the Hopf bifurcation of the fractional order $ \alpha $ were analyzed theoretically. Then, taking the diffusion coefficient of prey as the key parameter, the Turing stability at the equilibrium point was discussed. The amplitude equation near the threshold of the Turing instability was given by using the weak nonlinear analysis method, and different mode selections were classified by using the amplitude equation. Finally, we numerically proved that the dispersal rate of the prey population suppressed the spatiotemporal chaos of the model.</p></abstract>

Determination of steady states of tank and recycle tubular reactors using homotopy and parametric continuation methods

Abstract This work concerns the application of the homotopy method to solve the mathematical model of a non-adiabatic chemical continuous stirred tank reactor (CSTR) and tubular reactor with mass recycle (TRR) (Berezowski 2000. Spatio-temporal chaos in tubular chemical reactors with the recycle of mass, Chaos, Solitons & Fractals, vol. 11, no. 8, pp. 1197–1204). This method was associated with the parametric multivariable continuation algorithm. Thanks to this, this method can automatically find all the multiple steady states of the reactor model without the need to use any iteration. The parametric continuation method is used to determine a curve whose each point is a solution of the tested model. Therefore, the starting point must be very precisely designated so that it lies on this curve. Otherwise, the result is a graph that deviates from the correct graph. However, this condition is not required when the homotopy method is also introduced into the calculations. The starting point can then be a point with any coordinates. Different curves are also obtained, but the homotopy method ensures that each of them passes through the point where the parameter p = 1. The solution we are looking for in the model is just such a point. This is undoubtedly a big advantage resulting from the combination of both above-mentioned methods.

Transient power dynamics in nematic electroconvection.

We have observed the transient dynamics in nematic electroconvection during a sudden change in the driving voltage. The steady state dynamics of nematic electroconvection during a constant driving voltage are well characterized examples of spatiotemporal chaos, and thus are useful for the study of chaotic transients. For a set of starting conditions, we were able to show that the system can enter a long-lived transient state with a larger power dissipation rate. This transient state decays through the skew varicose instabilities into steady state defect turbulence. We observed the onset of both Eckhaus and skew varicose instabilities, interpret this transition in the framework of dynamical crisis, and quantify the transition through the evolution of the information entropy in the images as well as the power dissipated by the sample. We identify this long-lived but transient state as originating from the structure of the convective rolls in the initial state and hypothesize that the exploitation of transient states such as these may occur in natural systems and could be leveraged for industrial processing.

Efficient computation of coherent multimode instabilities in lasers using a spectral approach

Coherent multimode instabilities are responsible for several phenomena of recent interest in semiconductor lasers, such as the generation of frequency combs and ultrashort pulses. These techonologies have proven disruptive in optical telecommunications and spectroscopy applications. While the standard Maxwell-Bloch equations (MBEs) encompass such complex lasing phenomena, their integration is computationally expensive and offers limited analytical insight. In this paper, we demonstrate an efficient spectral approach to the simulation of multimode instabilities via a quantitative analysis of the instability of single-frequency lasing in ring lasers, referred to as the Lorenz-Haken (LH) instability or the RNGH instability in distinct parameter regimes. Our approach, referred to as CFTD, uses generally non-Hermitian Constant Flux modes to obtain projected Time Domain equations. CFTD provides excellent agreement with finite-difference integration of the MBEs across a wide range of parameters in regimes of non-stationary inversion, including frequency comb formation and spatiotemporal chaos. We also develop a modal linear stability analysis using CFTD to efficiently predict multimode instabilities in lasers. The combination of numerical accuracy, speedup, and semi-analytic insight across a variety of dynamical regimes make the CFTD approach ideal to analyze multimode instabilities in lasers, especially in more complex geometries or coupled laser arrays.

Amplitude modulation control of spatiotemporal chaos in starlike networks of damped-driven pendula

Applying amplitude modulations to a parametrically excited damped pendulum is shown to be a reliable method to control (suppress or enhance) its chaotic behaviour. Analytical (Melnikov analysis) and numerical (Lyapunov exponents and bifurcation diagrams) results show an effective control scenario for a wide range of resonances between the two excitations implicated. Different routes of regularization as the chaos-controlling parameters vary are identified, including period-doubling and crises. The method’s effectiveness at suppressing spatiotemporal chaos of starlike networks of sinusoidally coupled chaotic pendula is demonstrated where effective regularization is obtained under localized control on an increasing number of pendula.

A numerical solver for active hydrodynamics in three dimensions and its application to active turbulence

We present a higher-order convergent numerical solver for active polar hydrodynamics in three-dimensional domains of arbitrary shape, along with a scalable open-source software implementation for shared- and distributed-memory parallel computers. This enables the computational study of the nonlinear dynamics of out-of-equilibrium materials from first principles. We numerically solve the nonlinear active Ericksen–Leslie hydrodynamic equations of three-dimensional (3D) active nematics using both a meshfree and a hybrid particle-mesh method in either the Eulerian or Lagrangian frame of reference. The solver is validated against a newly derived analytical solution in 3D and implemented using the OpenFPM software library for scalable scientific computing. We then apply the presented method to studying the transition of 3D active polar fluids to spatiotemporal chaos, the emergence of coherent angular motion in a 3D annulus, and chiral vortices in symmetric and asymmetric 3D shapes resembling dividing cells. Overall, this provides a robust and efficient open-source simulation framework for 3D active matter with verified numerical convergence and scalability on parallel computers.

Dynamic properties of a reaction–diffusion predator–prey model with nonlinear harvesting: A linear and weakly nonlinear analysis

In this work, we have studied the spatiotemporal dynamics of a predator–prey model with selective nonlinear saturated harvesting of both species. The study of the non-diffusive system includes analytical and numerical analysis of local and global bifurcations. Bifurcation analysis of the non-diffusive system unveils that the system may exhibit transcritical, Hopf, saddle–node and homoclinic bifurcation with increased carrying capacity and harvesting efforts. The simulation results further reveal that the paradox of enrichment may disappear in a harvested system for a higher value of carrying capacity. The spatiotemporal study consists of an analysis of different pattern-forming instabilities. A critical ratio of the diffusivities of predator to prey for the occurrence of Turing instability is established. Further, a weakly nonlinear analysis is provided to derive the amplitude equation and predict the form of the pattern. The key highlight of the work is that prey harvesting facilitates spatiotemporal chaos, and predator diffusion controls it. Furthermore, intensive prey harvesting forces spatial segregation, whereas predator harvesting relaxes it. It is also observed that spatial segregation enriches the overall species biomass. A unique observation of this study is the occurrence and nonoccurrence of spatiotemporal chaos in the Hopf-Turing parametric space.