Periodic Behavior Research Articles

Recovering Quasi-Biennial Oscillations from Chaos

Abstract The Quasi-Biennial Oscillation (QBO) is understood to result from wave-mean-flow interactions, but the reasons for its relative stability remain a subject of ongoing debate. In addition, consensus has yet to be reached regarding the respective roles of different equatorial wave types in shaping the QBO’s characteristics. Here we employ Holton-Lindzen-Plumb’s quasilinear model to shed light on the robustness of periodic behavior in the presence of multiple wave forcings. A comprehensive examination of the various dynamical regimes in this model reveals that increased vertical wave propagation at higher altitudes favors periodicity. In the case of single standing wave forcing, enhanced vertical propagation is controlled by the wave attenuation length scale. The occurrence of non-periodic states at high forcing amplitudes is explained by the excitation of high vertical unstable modes. Increasing the attenuation length scale prevents the emergence of such modes. When multiple wave forcing is considered, the mean flow generated by a dominant primary wave facilitates greater vertical propagation of a perturbation wave. Raising the altitude where most of the wave damping occurs favors periodicity by preventing the development of secondary jets responsible for the aperiodic behavior. This mechanism underscores the potential role of internal gravity waves in supporting the periodicity of a QBO primarily driven by planetary waves.

System code validation in periodic two-phase flow from low-pressure oscillations

Low-pressure two-phase flow instabilities can potentially challenge start-up transients of water-cooled nuclear reactors. Predicting oscillations caused by flow instability is therefore paramount to reactor safety. While many thermal hydraulics system codes are developed as general-purpose tools, their performance is not guaranteed outside their optimized application ranges. The current study therefore performs validations of two thermal hydraulics system analysis codes, the Adaptive SYStem Thermal-hydraulics Version 3 (ASYST VER3) and Reactor Excursion and Leak Analysis Program MOD3.3 (RELAP5/MOD3.3), in reproducing oscillatory two-phase flow induced by low-pressure flashing instability. Benchmark conditions are selected from a novel dataset resolving void fraction with radial resolution and quantifying periodic behavior with ensemble averaging. Focusing on the prediction of transient two-phase phenomena rather than the determination of stability, validations are conducted by simulating a single channel under prescribed periodic boundary conditions. ASYST exhibits a systematic underprediction of void fraction in an adiabatic chimney downstream of a heated section. The prominent causal discrepancy is identified as a lost travelling void wave due to overpredicted condensation. RELAP5 noticeably overpredicts subcooled boiling and underpredicts flashing, which is consistent with existing validations against steady-state separate-effect tests under low pressure. This degraded code performance also suggests that in low-pressure transients prone to flashing instability, the confidence in RELAP5 derived from existing validations against integral-effect tests shall be carefully limited to validated capabilities in integral systems. In general, the current study fills the previous gap of knowledge about the performance of ASYST and RELAP5 in predicting detailed transient two-phase phenomena in low-pressure flashing-induced oscillations. The identified code defects reveal future directions for code improvements eventually towards reliable application of ASYST and RELAP5 under low pressure beyond their original calibration.

The Demographic-Wealth model for cliodynamics.

Cliodynamics is a still a relatively new research area with the purpose of investigating and modelling historical processes. One of its first important mathematical models was proposed by Turchin and called "Demographic-Fiscal Model" (DFM). This DFM was one of the first and is one of a few models that link population with state dynamics. In this work, we propose a possible alternative to the classical Turchin DFM, which contributes to further model development and comparison essential for the field of cliodynamics. Our "Demographic-Wealth Model" (DWM) aims to also model link between population and state dynamics but makes different modelling assumptions, particularly about the type of possible taxation. As an important contribution, we employ tools from nonlinear dynamics, e.g., existence theory for periodic orbits as well as analytical and numerical bifurcation analysis, to analyze the DWM. We believe that these tools can also be helpful for many other current and future models in cliodynamics. One particular focus of our analysis is the occurrence of Hopf bifurcations. Therefore, a detailed analysis is developed regarding equilibria and their possible bifurcations. Especially noticeable is the behavior of the so-called coexistence point. While changing different parameters, a variety of Hopf bifurcations occur. In addition, it is indicated, what role Hopf bifurcations may play in the interplay between population and state dynamics. There are critical values of different parameters that yield periodic behavior and limit cycles when exceeded, similar to the "paradox of enrichment" known in ecology. This means that the DWM provides one possible avenue setup to explain in a simple format the existence of secular cycles, which have been observed in historical data. In summary, our model aims to balance simplicity, linking to the underlying processes and the goal to represent secular cycles.

Analysis of dynamics of a map-based neuron model via Lorenz maps.

Modeling nerve cells can facilitate formulating hypotheses about their real behavior and improve understanding of their functioning. In this paper, we study a discrete neuron model introduced by Courbage et al. [Chaos 17, 043109 (2007)], where the originally piecewise linear function defining voltage dynamics is replaced by a cubic polynomial, with an additional parameter responsible for varying the slope. Showing that on a large subset of the multidimensional parameter space, the return map of the voltage dynamics is an expanding Lorenz map, we analyze both chaotic and periodic behavior of the system and describe the complexity of spiking patterns fired by a neuron. This is achieved by using and extending some results from the theory of Lorenz-like and expanding Lorenz mappings.

Multi-dimensional phase portraits of stochastic fractional derivatives for nonlinear dynamical systems with solitary wave formation

This manuscript delves into the examination of the stochastic fractional derivative of Drinfel’d-Sokolov-Wilson equation, a mathematical model applicable in the fields of electromagnetism and fluid mechanics. In our study, the proposed equation is through examined through various viewpoints, encompassing soliton dynamics, bifurcation analysis, chaotic behaviors, and sensitivity analysis. A few dark and bright shaped soliton solutions, including the unperturbed term, are also examined, and the various 2D and 3D solitonic structures are computed using the Tanh-method. It is found that a saddle point bifurcation causes the transition from periodic behavior to quasi-periodic behavior in a sensitive area. Further analysis reveals favorable conditions for the multidimensional bifurcation of dynamic behavioral solutions. Different types of wave solutions are identified in certain solutions by entering numerous values for the parameters, demonstrating the effectiveness and precision of Tanh-methods. A planar dynamical system is then created using the Galilean transformation, with the actual model serving as a starting point. It is observed that a few physical criteria in the discussed equation exhibit more multi-stable properties, as many multi-stability structures are employed by some individuals. Moreover, sensitivity behavior is employed to examine perturbed dynamical systems across diverse initial conditions. The techniques and findings presented in this paper can be extended to investigate a broader spectrum of nonlinear wave phenomena.

The formation of invariant optical soliton structures to electric-signal in the telegraph lines on basis of the tunnel diode and chaos visualization, conserved quantities: Lie point symmetry approach

This study critically examines the nonlinear Lonngren wave model, a mathematical representation used to describe waves in shallow water environments such as beaches, lakes, and rivers, as well as electrical signals in semiconductor materials like tunnel diodes. The symmetry generators take into account the Lie invariance criteria. A three-dimensional Lie algebra can be obtained by using the suggested methodology. It links translation symmetries in space and time, respectively, with conservation of mass and energy. The third symmetry pertains to scaling or dilation. Recognizing the constraints of the inverse scattering transform in addressing the Cauchy problem for this equation, the investigation adopts new auxiliary equation approach to ordinary differential equations. This approach yields closed-form analytical solutions for the wave model. The obtained solutions include a plane solution, mixed hyperbolic solution, periodic and mixed periodic solutions, mixed trigonometric solution, smooth soliton, trigonometric solution, mixed singular solution, complex solitary and singular solution. Furthermore, the study develops the Hamiltonian function of the dynamical system to discuss the total energy of the system in terms of its position and momentum. Chaos analysis is performed to describe periodic, quasi-periodic, and chaotic behaviours, and a sensitivity analysis highlights the model’s sensitivity to different wave numbers. Graphical representations of selected solutions are presented in both 3-D and 2-D, illustrating the behaviour under specific physical conditions. The study concludes by identifying a comprehensive set of local conservation laws for the nonlinear Lonngren wave equation, applicable to arbitrary constant coefficients, using conservation laws multipliers.

Phase trajectories and Chaos theory for dynamical demonstration and explicit propagating wave formation

This paper is subjected to study the nonlinear integrable model which is the (3+1)-dimensional Boussinesq equation which has a lot of applications in engineering and modern sciences. To find and examine the analytical exact solitary wave solutions of (3+1)-dimensional Boussinesq equation, a modified generalized exponential rational functional method is exerted. As a result, waves, singular periodic, hyperbolic, and trigonometric type solutions are obtained. These acquired solutions are more innovative and encouraging to researchers in their endeavor to study physical marvels. To illustrate how some selected exact solutions propagate, the graphical representation in 2D, Contour, and 3D of those solutions is provided with various parametric values. The considered equation is additionally transformed into the planar dynamical structure by applying the Galilean transformation. All potential phase portraits of the dynamical system are investigated using the theory of bifurcation. The Hamiltonian function of the dynamical system of differential equations is established to see that, the system is conservative over time. The presentation of energy levels through graphics provides valuable insights, and it demonstrates that the model has solutions that can be expressed in closed form. The periodic, quasi-periodic, and chaotic behaviors of the 2D, 3D, and time series are also observable once the dynamical system is subjected to an external force. Meanwhile, the sensitivity of the derived solutions is carefully examined for a range of initial conditions.

Numerical study on the flow and heat transfer performance of SCO2/molten salt in Z-type printed circuit heat exchangers

The utilization of a PCHE in conjunction with a SCO2 Brayton cycle and molten salt thermal storage demonstrates a high efficiency. Investigating the flow and heat transfer characteristics of SCO2 and molten salt within a Z-type PCHE is essential. However, this particular aspect has not been explored by previous researches. This study employs a Z-type PCHE to analyze the heat transfer and flow characteristics of SCO2 and molten salt under different inlet temperatures, mass fluxes, and outlet pressures. Furthermore, the heat transfer efficiency of a Z-type PCHE is evaluated and compared to that of a conventional straight-type PCHE. The results indicate that the flow pattern of SCO2 exhibits periodic behavior due to the periodic bending structure of Z-type PCHE, while the flow of molten salt is minimally affected. Furthermore, different SCO2 conditions lead to divergent patterns in flow and heat transfer efficiency compared to molten salt. Z-type channels help alleviate the decline in heat transfer that typically happens in the inlet section of straight channels. Nu for a dual working fluid in a straight-type PCHE is only 1.1 times greater than that of a single working fluid. In Z-type PCHE configuration, the ratio of the dual working fluid to the single working fluid Nu is 1.4. The sensitivity of SCO2/molten salt dual working fluid to the heat exchanger structure is significant, with Z-type PCHE demonstrating superior heat transfer capabilities. The results outlined in this study offer the prospect of improving the design and deployment of Z-type PCHE in molten salt energy storage systems that operate within SCO2 Brayton cycle.

Correlated dynamics of immune network and sl(3, R) symmetry algebra

Abstract We observed the existence of periodic orbits in immune network under transitive solvable Lie algebra. In this article, we focus to develop condition of maximal Lie algebra for immune network model and use that condition to construct a vector field of symmetry to study nonlinear pathogen model. We used two methods to obtain analytical structure of solution, namely normal generator and differential invariant function. Numerical simulation of analytical structure exhibits correlated periodic pattern growth under spatiotemporal symmetry, which is similar to the linear dynamical simulation result. We used Lie algebraic method to understand correlation between growth pattern and symmetry of dynamical system. We employ idea of using one parameter point group of transformation of variables under which linear manifold is retained. In procedure, we present the method of deriving Lie point symmetries, the calculation of the first integral and the invariant solution for the ordinary differential equation (ODE). We show the connection between symmetries and differential invariant solutions of the ODE. The analytical structure of the solution exhibits periodic behavior around attractor in local domain, same behavior obtained through dynamical analysis.

On control intervals of chaos with Fréchet derivative of certain iterations as discrete dynamical system in Banach spaces

Dynamical systems are one of the interesting concepts where iteration algorithms and chaos can be considered together. In iteration algorithms, one of the basic concepts of fixed point theory, it is well known that the behavior of the iteration mechanism is chaotic if the original transformation is taken as chaotic. One of the natural ways to transform a chaotic system into a dynamical system is through control mechanisms. In this paper, we first consider an iteration class defined on Banach spaces, which is prominent in the literature in terms of both speed and convergence rate. Then, we consider the transformation constituting the iteration class as chaotic and obtain the stability and unstability behaviors of the iterations according to the operator norm by using Gâteaux and Fréchet derivatives representing the direction‐dependent derivative. In this way, we aimed to obtain the chaos control intervals obtained with functions in real space with the help of operators in Banach spaces. Using the operator norm obtained with the Fréchet derivative, we derive interesting dynamical system intervals from a chaotic system with parameter variables of iteration algorithms. It is noteworthy that the parameter variables of the studied iteration classes are the same in some transformation classes. In addition, analytical proofs are followed by computer simulations of parameter‐dependent control intervals considering the logistic operator with a chaotic structure. At the same time, the periodic behavior of the iteration algorithms used in our study is illustrated by the Lyapunov exponent with parameter‐dependent control intervals according to operator norm. Finally, as a real‐life problem, it has been shown that the chaos of the logistic population growth model with dynamic features can be controlled by chaos control mechanisms established by fixed point iteration methods.

Breather solutions for a radially symmetric curl-curl wave equation with double power nonlinearity

This paper is concerned with breather solutions of a radially symmetric curl-curl wave equation with double power nonlinearity ρ(x)utt+∇×(M(x)∇×u)+μ(x)u+vp(x)|u|p−1u+vq(x)|u|q−1u=0,where (x,t)∈R3×R, u:R3×R→R3 is the unknown function, M:R3→R3×3 and ρ,μ,vp,vq:R3→(0,+∞) are radially symmetric coefficient functions with 1<p<q. By considering the solutions with a special form u=y(|x|,t)x|x|, we obtain a family of ordinary differential equations (ODEs) parameterized by the radial variable r=|x|. Then we characterize periodic behaviors and analyze the joint effects of the double power nonlinear terms on the minimal period and the maximal amplitude. Under certain conditions, we construct a 2πρ(0)/μ(0)-periodic breather solution for the original curl-curl wave equation and find such a solution which can generate a continuum of phase-shifted breathers.

A healthy lifestyle can support future sexual satisfaction: results from a 9-year longitudinal survey.

Previous follow-up studies have demonstrated the association between good health behavior and good sexual functioning for men, but the longitudinal relationship between multiple health behaviors and satisfaction with sex life remains understudied. The aim of the study was to explore whether good health behavior associates with improved satisfaction with sex life for men and women in a follow-up of 9 years. This cohort study utilized survey data from the population-based Health and Social Support study. It includes responses from 10 671 working-aged Finns. Using linear regression models, we examined a composite sum score representing 4 health behaviors (range, 0-4) in 2003 as a predictor of satisfaction with sex life in 2012. The analyses adjusted for various covariates in 2003, including satisfaction with sex life, living status, age, gender, education, number of diseases, and importance of sex life in 2012. The outcome in the study was satisfaction with sex life in the year 2012. Participants who exhibited better health behavior at baseline demonstrated improved satisfaction with sex life when compared with those with poorer health behavior (β = -0.046, P = .009), even when controlling for the aforementioned covariates. The positive effect of reporting all beneficial health behaviors vs none of them was greater than having none vs 3 chronic conditions. Furthermore, this was almost half the effect of how satisfaction with sex life in 2003 predicted its level in 2012. These findings were supported by an analysis of the congruence of health behavior in the observation period from 2003 to 2012 predicting changes in satisfaction with sex life. The results could serve as a motivator for a healthy lifestyle. The current study used a longitudinal large sample and a consistent survey procedure, and it explored the personal experience of satisfaction instead of sexual function. However, the study is limited in representing today's diversity of gender, since the options for gender at the time of survey were only male and female. These findings indicate that engaging in healthy behaviors contributes to the maintenance and enhancement of satisfaction with sex life over time.

Amplifying Sine Unit: An Oscillatory Activation Function for Deep Neural Networks to Recover Nonlinear Oscillations Efficiently

Many industrial and real-life problems exhibit highly nonlinear periodic behaviors and the conventional methods may fall short of finding their analytical or closed-form solutions. Such problems demand cutting-edge computational tools with increased functionality and reduced cost. Recently, deep neural networks have gained massive research interest due to their ability to handle large data and universality to learn complex functions. In this work, we put forward a methodology based on deep neural networks with responsive layers structure to deal nonlinear oscillations in microelectromechanical systems. We incorporated some oscillatory and non-oscillatory activation functions such as growing cosine unit known as GCU, Sine, Mish and Tanh in our designed network to have a comprehensive analysis on their performance for highly nonlinear problems. Integrating oscillatory activation functions with deep neural networks definitely outperform in predicting the periodic patterns of underlying systems. To support oscillatory actuation for nonlinear systems, we have proposed a novel oscillatory activation function called Amplifying Sine Unit denoted as ASU which is more efficient than GCU for complex vibratory systems such as microelectromechanical systems. Experimental results show that the designed network with our proposed activation function ASU is more reliable and robust to handle the challenges posed by oscillations

Comparison of conducted emissions due to electric vehicle charging processes under isolated and on-line conditions in the 9–500 kHz frequency range

This paper aims at comparing the conducted emissions generated by Electric Vehicle Charging Processes (EVCPs) under isolated and on-line conditions in the frequency and time domains, covering the 9–500 kHz frequency range. The isolated conditions correspond to the use of a Line Impedance Stabilization Network (LISN), whereas measurements in the Low Voltage (LV) grid where the EV is connected are referred to as on-line conditions. Regarding the frequency analysis, the results lead to conclude that different amplitude and spectral features of the Non-Intentional Emissions (NIEs) are measured depending on the measurement conditions, registering higher-amplitude NIEs when the measurements are conducted in the LV grid. Moreover, the paper also shows that the spectral characteristics of the NIEs correspond to tonal and narrowband emissions at specific frequencies in the 9–150 kHz frequency range, while background noise with low-amplitude emissions (in the case of isolated conditions) and colored noise (in the case of on-line conditions) are reported in the 150–500 kHz band. In the time domain, a periodic time-dependent behavior within the fundamental cycle of the mains (20 ms) is reported in all the analyzed cases, except for the tonal emissions with oscillating central frequency generated by a specific EV model. The time variability between both measurement configurations only differs by more than 3 dB for three out of the twelve analyzed EVCPs, where a higher variability is observed if the measurements are carried out using a LISN. As a conclusion, considering the differences in the frequency and time characteristics of the NIEs under isolated and on-line conditions, evaluating the emissions generated by EVCPs might lead to an underestimation of the NIEs in real LV grids.

Periodic Behavior of Selected Solar, Geomagnetic and Cosmic Activity Indices during Solar Cycle 24

In this study, we performed periodicity analyses of selected daily solar (flare index, coronal index, number of coronal mass ejections), geomagnetic (planetary equivalent range index, disturbance storm time index, interplanetary magnetic field) and cosmic ray indices for the last Solar Cycle 24 (from December 2008 to December 2019). To study the periodic variation of the above-listed datasets, the following analysis methods were applied; multi-taper method, Morlet wavelet, cross-wavelet transform and wavelet coherence analysis. The outcome of our analyses revealed the following. (i) The 25–33 days solar rotation periodicities exist in all datasets without any exception in the MTM power spectra. (ii) Except for the solar rotation periodicity, all periods show data preference, and they appear around the investigated cycle’s maximum phase. (iii) When comparing the phase relations between periodicities in the used datasets, they exhibit a gradual transition from small to large periods. For short-term periodicities, there are no phase relations but a mixed phase, whereas for high periodicities, there are complete phase/antiphase transitions. (iv) All identified flare index periodicities are common to all other datasets examined in this investigation.

Periodicity, Elliott waves, and fractals in the NFT market

Non-fungible tokens (NFTs) are unique digital assets that exist on a blockchain and have provided new revenue streams for creators. This research investigates NFT market inefficiencies to identify claimed cyclic behavior and cryptocurrency influences on NFT prices. The research found that while linear models are not useful in modeling NFT price series, models that extract periodic behavior can provide explanations and predictions of price behavior. The investigation of autocycles in cryptocurrency and NFT markets did not support the existence of Elliott Wave behavior in any of these blockchain enabled assets. Rather NFT price behavior is strongly tied to the underlying asset and its community of fans. These fans commit to periodic bouts of idiosyncratic trading which cools for a while, and then restarts. The research found no evidence supporting whole market effects across the full price series of individual NFTs. The research strongly supports prior findings that the offsetting movements significantly influence NFT prices and trading volume in Bitcoin and Ether. The research found NFT markets exhibit characteristics resembling a social media platform rather than more traditional asset markets like stock exchanges. It found that traditional linear econometric models cannot predict or explain NFT price series, only that NFT price and volume were weakly correlated. Fractal models consistent with Elliott wave theory do explain some of NFT price behavior, but are not consistent or stable over time. This research confirmed prior research findings that Bitcoin and Ether price movements are correlated with general NFT price and volume series in periods of between 24 and 48 h, with significant numbers of trades into and out of cryptocurrencies at 2 and 8 h.

Intracellular ion accumulation in the genesis of complex action potential dynamics under cardiac diseases.

Intracellular ions, including sodium (Na^{+}), calcium (Ca^{2+}), and potassium (K^{+}), etc., accumulate slowly after a change of the state of the heart, such as a change of the heart rate. The goal of this study is to understand the roles of slow ion accumulation in the genesis of cardiac memory and complex action-potential duration (APD) dynamics that can lead to lethal cardiac arrhythmias. We carry out numerical simulations of a detailed action potential model of ventricular myocytes under normal and diseased conditions, which exhibit memory effects and complex APD dynamics. We develop a low-dimensional iterated map (IM) model to describe the dynamics of Na^{+}, Ca^{2+}, and APD and use it to uncover the underlying dynamical mechanisms. The development of the IM model is informed by simulation results under the normal condition. We then use the IM model to perform linear stability analyses and computer simulations to investigate the bifurcations and complex APD dynamics, which depend on the feedback loops between APD and intracellular Ca^{2+} and Na^{+} concentrations and the steepness of the APD response to the ion concentrations. When the feedback between APD and Ca^{2+} concentration is positive, a Hopf bifurcation leading to periodic oscillatory behavior occurs as the steepness of the APD response to the ion concentrations increases. The negative feedback loop between APD and Na^{+} concentration is required for the Hopf bifurcation. When the feedback between APD and Ca^{2+} concentration is negative, period-doubling bifurcations leading to high periodicity and chaos occurs. In this case, Na^{+} accumulation plays little role in the dynamics. Finally, we carry out simulations of the detailed action potential model under two diseased conditions, which exhibit steep APD responses to ion concentrations. Under both conditions, Hopf bifurcations leading to slow oscillations or period-doubling bifurcations leading to high periodicity and chaotic APD dynamics occur, depending on the strength of the ion pump-Na^{+}-Ca^{2+} exchanger. Using functions reconstructed from the simulation data, the IM model accurately captures the bifurcations and dynamics under the two diseased conditions. In conclusion, besides using computer simulations of a detailed high-dimensional action-potential model to investigate the effects of slow ion accumulation and short-term memory on bifurcations and genesis of complex APD dynamics in cardiac myocytes under diseased conditions, this study also provides a low-dimensional mathematical tool, i.e., the IM model, to allow stability analyses for uncovering the underlying mechanisms.

Assessing Wet and Dry Periods Using Standardized Precipitation Index Fractal (SPIF) and Polygons: A Novel Approach

In the open literature, there are numerous studies on the normal and extreme (flood and drought) behavior of wet and dry periods based on the understanding of the standard precipitation index (SPI), which provides a series of categorizations by considering the standard normal (Gaussian) probability distribution function (PDF). The numerical meaning of each categorization assessment is quite lacking in terms of future predictions of wet and dry period duration based on historical records. This paper presents a new approach for calculating possible formations of future wet and dry period durations based on historical records through an effective fractal geometric forecasting approach. The essence of the proposed methodology is based on the number of dry periods (steps) of non-overlapping monthly duration along consecutive broken line paths in the SPI classification for wet and dry period durations. It has been observed that the plot of periods on double logarithmic paper falls along a straight line against the number of such periods, implying a power function, which is the essence of fractal geometry. Extending the empirically derived straight line provides the number of periods that may occur in the future over a range of SPI levels. This methodology is referred to as SPI fractal (SPIF), and the classic SPI classification is converted into SPIF wet and dry polygons, which provide additional information about the drought period number within a valid polygonal area, compared to the classic SPI results. The wet and dry period features of any hydro-meteorology time series are constrained in SPIF polygons. The application of the methodology was carried out on monthly rainfall records on the European side of the Istanbul Florya meteorological station in Turkey.

Pattern selection mechanism from the equilibrium point and limit cycle.

The outbreak of infectious diseases often exhibits periodicity, and this periodic behavior can be mathematically represented as a limit cycle. However, the periodic behavior has rarely been considered in demonstrating the cluster phenomenon of infection induced by diffusion (the instability modes) in the SIR model. We investigate the emergence of Turing instability from a stable equilibrium and a limit cycle to illustrate the dynamical and biological mechanisms of pattern formation. We identify the Hopf bifurcation to demonstrate the existence of a stable limit cycle using First Lyapunov coefficient in our spatiotemporal diffusion-driven SIR model. The competition between different instability modes induces different types of patterns and eventually spot patterns emerge as stable patterns. We investigate the impact of susceptible, infected, and recovered individuals on the type of patterns. Interestingly, these instability modes play a vital role in selecting the pattern formations, which is directly related to the number of observed spot patterns. Subsequently, we explain the dynamical and biological mechanisms of spot patterns to develop an effective epidemic prevention strategy.

Periodic behavior and dynamical analysis of a prey–predator model incorporating the Allee effect and fear effect

Periodic behavior and dynamical analysis of a prey–predator model incorporating the Allee effect and fear effect