Abstract Earth’s climate system is a highly complex and interconnected network governed by nonlinear interactions among the atmosphere, oceans, land, ice, and biosphere, where energy exchanges and feedback mechanisms play a dominant role. In recent decades, anthropogenic greenhouse gas emissions, especially carbon dioxide (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(CO_2\)</EquationSource> </InlineEquation>), have significantly disrupted this balance, resulting in accelerated ocean heat uptake and persistent temperature anomalies. Determining the long-term dynamics of these interactions remains a critical challenge for accurate climate prediction and mitigation planning. This paper examines the combined dynamics of temperature anomaly, atmospheric <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(CO_2\)</EquationSource> </InlineEquation> concentration, and ocean heat content (OHC) using a novel mathematical approach. By employing the Caputo derivative to describe the model as a fractional-order dynamical system, hereditary effects and long-term dependencies that are inherent in climatic processes can be incorporated. Boundedness, existence and uniqueness of solutions, and both local and global stability are among the fundamental qualitative characteristics of the system that are investigated. To further illustrate stability behavior, streamline graphs are plotted. To ensure an accurate approximation of the fractional dynamics, numerical simulations are conducted using the Adams Bashforth Moulton (ABM) predictor–corrector method. Bifurcation analysis and computations of the Lyapunov exponent are performed to investigate the nonlinear properties of the system, exposing parameter regimes that behave chaotically for different fractional orders. Phase portraits in 2D and 3D show the intricate history of the climate variables. Additionally, to control chaotic oscillations, a sliding mode control approach is used. The findings highlight the promise of control theoretic techniques in climate dynamics by showing that the system is stabilized and chattering is successfully eliminated with the right control parameters. The results demonstrate that the fractional-order formulation provides enhanced capability in capturing long-term dependencies and nonlinear feedback mechanisms inherent in climate dynamics. The overall results show the model’s robustness as a theoretical framework for climate analysis and offer quantitative insights into the coupled climate system’s long-term behavior. The model’s incorporation of nonlinear interactions among important variables improves the model’s interpretability and gives a more accurate picture of climate dynamics, which strengthens the foundation for assessing the effects of emissions and guiding the formulation of climate policy. Graphical Abstract Graphical Abstract of the variables ocean heat content, GHG concentration <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(CO_2\)</EquationSource> </InlineEquation>, and temperature anomaly. The main result of the suggested mathematical framework is summarized in the graphical abstract, which shows that atmospheric <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(CO_2\)</EquationSource> </InlineEquation> concentration acts as the main external force controlling the evolution of ocean heat content and temperature anomalies. By utilizing a thorough combination of analytical and computational methods, such as bifurcation analysis, stability and boundedness analysis, Lyapunov exponent characterization, and multi-dimensional phase space visualization, the study shows that long-term increases in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(CO_2\)</EquationSource> </InlineEquation> emissions can cause substantial qualitative changes in the system’s dynamics, which may result in complex transient or chaotic behavior, instability, and nonlinear oscillations. Additionally, the results highlight the coupled climate system’s delicate dependence on important parameters, with bifurcation analysis identifying significant thresholds that indicate transitions between stable and unstable regimes. Additional useful insight is provided by the use of a sliding mode control method, which shows how suitable intervention mechanisms can successfully control system trajectories and suppress chattering in the system. Overall, the results highlight how crucial it is to regulate atmospheric <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(CO_2\)</EquationSource> </InlineEquation> levels in order to preserve the climate system’s stability and boundedness. The work emphasizes that proactive abatement of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(CO_2\)</EquationSource> </InlineEquation> emissions is crucial to preventing the commencement of complicated and potentially irreversible dynamical behavior in the coupled climate system, as well as to limit temperature escalation and ocean heat buildup.

Nonlinear Dynamics and Control of \(CO_2\) Driven Climate Variability and Ocean Heat Feedbacks

期刊论文
发表日期：

2026年5月27日

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K. Sherly,
P. Veeresha,
Mahesh Bade

摘要

Abstract

Earth’s climate system is a highly complex and interconnected network governed by nonlinear interactions among the atmosphere, oceans, land, ice, and biosphere, where energy exchanges and feedback mechanisms play a dominant role. In recent decades, anthropogenic greenhouse gas emissions, especially carbon dioxide ( \(CO_2\) ), have significantly disrupted this balance, resulting in accelerated ocean heat uptake and persistent temperature anomalies. Determining the long-term dynamics of these interactions remains a critical challenge for accurate climate prediction and mitigation planning. This paper examines the combined dynamics of temperature anomaly, atmospheric \(CO_2\) concentration, and ocean heat content (OHC) using a novel mathematical approach. By employing the Caputo derivative to describe the model as a fractional-order dynamical system, hereditary effects and long-term dependencies that are inherent in climatic processes can be incorporated. Boundedness, existence and uniqueness of solutions, and both local and global stability are among the fundamental qualitative characteristics of the system that are investigated. To further illustrate stability behavior, streamline graphs are plotted. To ensure an accurate approximation of the fractional dynamics, numerical simulations are conducted using the Adams Bashforth Moulton (ABM) predictor–corrector method. Bifurcation analysis and computations of the Lyapunov exponent are performed to investigate the nonlinear properties of the system, exposing parameter regimes that behave chaotically for different fractional orders. Phase portraits in 2D and 3D show the intricate history of the climate variables. Additionally, to control chaotic oscillations, a sliding mode control approach is used. The findings highlight the promise of control theoretic techniques in climate dynamics by showing that the system is stabilized and chattering is successfully eliminated with the right control parameters. The results demonstrate that the fractional-order formulation provides enhanced capability in capturing long-term dependencies and nonlinear feedback mechanisms inherent in climate dynamics. The overall results show the model’s robustness as a theoretical framework for climate analysis and offer quantitative insights into the coupled climate system’s long-term behavior. The model’s incorporation of nonlinear interactions among important variables improves the model’s interpretability and gives a more accurate picture of climate dynamics, which strengthens the foundation for assessing the effects of emissions and guiding the formulation of climate policy.

Graphical Abstract

Graphical Abstract of the variables ocean heat content, GHG concentration \(CO_2\) , and temperature anomaly. The main result of the suggested mathematical framework is summarized in the graphical abstract, which shows that atmospheric \(CO_2\) concentration acts as the main external force controlling the evolution of ocean heat content and temperature anomalies. By utilizing a thorough combination of analytical and computational methods, such as bifurcation analysis, stability and boundedness analysis, Lyapunov exponent characterization, and multi-dimensional phase space visualization, the study shows that long-term increases in \(CO_2\) emissions can cause substantial qualitative changes in the system’s dynamics, which may result in complex transient or chaotic behavior, instability, and nonlinear oscillations. Additionally, the results highlight the coupled climate system’s delicate dependence on important parameters, with bifurcation analysis identifying significant thresholds that indicate transitions between stable and unstable regimes. Additional useful insight is provided by the use of a sliding mode control method, which shows how suitable intervention mechanisms can successfully control system trajectories and suppress chattering in the system. Overall, the results highlight how crucial it is to regulate atmospheric \(CO_2\) levels in order to preserve the climate system’s stability and boundedness. The work emphasizes that proactive abatement of \(CO_2\) emissions is crucial to preventing the commencement of complicated and potentially irreversible dynamical behavior in the coupled climate system, as well as to limit temperature escalation and ocean heat buildup.