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This interdisciplinary study critically analyzes current research, establishing a profound connection between sea water, sea ice, sea temperature, and surface temperature through a 4D hyperchaotic Caputo fractional differential equation. Emphasizing the collective impact on climate, focusing on challenges from anthropogenic global warming, the study scrutinizes theoretical aspects, including existence and uniqueness. Two sliding mode controllers manage chaos in this 4D fractional system, assessed amid uncertainties and disruptions. The global stability of these controlled systems is also confirmed, considering both commensurate and non-commensurate 4D fractional order. To demonstrate the intricate chaotic motion within the system, we employ the Lyapunov exponent and Poincare sections. Numerical simulations are conducted by using the predictor-corrector method. The effects of surface temperature on chaotic dynamics are discussed. The crucial role of sea ice reflection in climate stability is highlighted in two scenarios. Correlation graphs, comparing model and observational data using the predictor-corrector method, enhance the proposed 4D hyperchaotic model's credibility. Subsequently, numerical simulations validate theoretical assertions about the controllers' influence. These controllers indicate which variable significantly contributes to controlling the chaos. [ABSTRACT FROM AUTHOR]

The widespread use of computer hardware and software in society has led to the emergence of a type of criminal conduct known as cybercrime, which has become a major worldwide concern in the 21st century spanning multiple domains. As a result, in the present setting, academics and practitioners are showing a great deal of interest in conducting research on cybercrime. In this work, a fractional-order model was replaced by involving three sorts of human populations: online computer users, hackers, and cyber security professionals, in order to examine the online computer user-hacker system. The existence, uniqueness and boundedness were studied. To support our theoretical conclusions, a numerical analysis of the influence of the various logical parameters was conducted and we derived the necessary conditions for the different equilibrium points to be locally stable. We examined the effects of the fear level and refuge factor on the equilibrium densities of prey and predators in order to explore and understand the dynamics of the system in a better way. Using some special circumstances, the model was examined. Our theoretical findings and logical parameters were validated through a numerical analysis utilizing the generalized Adams-Bashforth-Moulton technique. [ABSTRACT FROM AUTHOR]

In the marine ecosystem, phytoplankton plays a vital role as the primary supplier of oxygen, contributing to 50% of the total oxygen production. Not only does it serve as a significant source of food for other species, but it also sustains life in the ocean. However, the rising ocean temperatures caused by global warming have severely hindered the ability of phytoplankton to generate oxygen. Furthermore, fishes are crucial consumers of oxygen within the marine ecosystem. This research paper presents a model that intricately connects the dynamics of oxygen, phytoplankton, zooplankton, and fish using the Caputo fractional derivative. The model aims to examine the impact of global warming on the collective dynamics by considering the relationship between the rate of oxygen generation, temperature, and time. The paper establishes the existence and uniqueness of solutions and also analyzes the stability of equilibrium points. Numerical simulations are conducted to demonstrate the impact of fractional derivatives and global warming on oxygen depletion and species extinction. [ABSTRACT FROM AUTHOR]

The earth's surface is heated by the large-scale movement of air known as atmospheric circulation, which works in conjunction with ocean circulation. More than 10 5 variables are involved in the complexity of the weather system. In this work, we analyze the dynamical behavior and chaos control of an atmospheric circulation model known as the Hadley circulation model, in the frame of Caputo and Caputo–Fabrizio fractional derivatives. The fundamental novelty of this paper is the application of the Caputo derivative with equal dimensionality to models that includes memory. A sliding mode controller (SMC) is developed to control chaos in this fractional-order atmospheric circulation system with uncertain dynamics. The proposed controller is applied to both commensurate and non-commensurate fractional-order systems. To demonstrate the intricacy of the models, we plot some graphs of various fractional orders with appropriate parameter values. We have observed the influence of thermal forcing on the dynamics of the system. The outcome of the analytical exercises is validated using numerical simulations. [ABSTRACT FROM AUTHOR]

In this work, we analyze plankton–fish dynamics in the presence of toxicity, refuge, and combine-harvesting efforts by a considering Holling type-II functional response. We have considered phytoplankton, zooplankton, and fish populations, and the interdependent evolution is presented with the help of the Caputo fractional derivative. Since toxicity in phytoplankton spreads to zooplankton and hence to fish, we have introduced toxic terms in all the populations. On the other hand, to save the population from extinction harvesting is an essential tool. Theoretical aspects are studied in terms of nonnegativity, boundedness, existence, and uniqueness of the solution. Sufficient conditions are derived for the stability of various points of equilibrium. The composite behavior is studied by varying the values of different parameters and fractional derivatives. Numerical simulations are carried out to strengthen the theoretical findings. [ABSTRACT FROM AUTHOR]

Wireless sensor networks (WSNs) are subject to cyber attacks. Security of such networks is a significant priority for everyone. Due to the network's frail defense mechanisms, WSNs are easy targets for worm attacks. A single unsecured node via contact can effectively propagate the worm across the entire network. Mathematical epidemic models are helpful in analyzing worm propagation in WSNs. To understand the attacking and spreading dynamics of worms in WSNs, a fractional‐order compartmental epidemic model is investigated with susceptible, exposed, infected, recovered, and vaccinated nodes. Dynamical aspects such as boundedness, existence, and uniqueness of the solutions are presented with the help of fractional calculus theory. Global stability of the points of equilibrium are established. The projected nonlinear structure is examined numerically via the generalized Adams–Bashforth–Moulton method. This study demonstrates the influence of the fractional operator on WSNs dynamics and the efficiency of the numerical method. [ABSTRACT FROM AUTHOR]

In this paper, we have derived a new operational matrix of fractional integration by using the independence polynomial of a complete bipartite graph and applied it to solve the Poisson equation with Dirichlet boundary conditions. While deriving the operational matrix, the Caputo sense fractional derivatives are considered. Series solutions are found by using the collocation matrix method. The main characteristic of the approach is that it reduces a complex fractional differential equation to a system of algebraic equations. The error bound and computational complexity of the projected algorithm are also investigated. Solutions obtained for the Poisson equation have established the relevance and applicability of the method described. Comparative analysis of the obtained results with the exact solutions convinces that the present method can be considered an efficient numerical tool for solving fractional differential equations. [ABSTRACT FROM AUTHOR]