Three Kinds of Butterfly Effects within Lorenz Models

Shen, Bo-Wen; Pielke, Roger A.; Zeng, Xubin; Cui, Jialin; Faghih-Naini, Sara; Paxson, Wei; Atlas, Robert

doi:10.3390/encyclopedia2030084

Cited by 11 publications

(12 citation statements)

References 40 publications

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“…This characteristic has been observed in memristor-based systems and circuits [34,35]. Since they can show rich behaviors, they have been employed in many research branches and real-world applications [8,[36][37][38].…”

Section: Introductionmentioning

confidence: 88%

Coexisting attractors and multi-stability within a Lorenz model with periodic heating function

Ahmadi

Parthasarathy²,

Natiq³

et al. 2023

Phys. Scr.

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In this paper, the classical Lorenz model is under investigation, in which a periodic heating term replaces the constant one. Applying the variable heating term causes time-dependent behaviors in the Lorenz model. The time series produced by this model are chaotic; however, they have fixed point or periodic-like qualities in some time intervals. The energy dissipation and equilibrium points are examined comprehensively. This modified Lorenz system can demonstrate multiple kinds of coexisting attractors by changing its initial conditions and, thus, is a multi-stable system. Because of multi-stability, the bifurcation diagrams are plotted with three different methods, and the dynamical analysis is completed by studying the Lyapunov exponents and Kaplan-Yorke dimension diagrams. Also, the attraction basin of the modified system is investigated, which approves the appearance of coexisting attractors in this system.

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Section: Introductionmentioning

confidence: 88%

Coexisting attractors and multi-stability within a Lorenz model with periodic heating function

Ahmadi

Parthasarathy²,

Natiq³

et al. 2023

Phys. Scr.

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“…In a recent study by Shen et al [47], three major kinds of butterfly effects can be identified: (1) SDIC, (2) the ability of a tiny perturbation in creating an organized circulation at a large distance, and (3) the hypothetical role of small scale processes in contributing to finite predictability. While the first kind of butterfly effect with SDIC is well accepted, the concept of multistability suggests that the first kind of butterfly effect does not always appear.…”

Section: Coexisting Attractors and Multistability Within The Glmmentioning

confidence: 99%

“…As shown in the second panels of Figure 5, these solutions were obtained using tiny differences in ICs and a time varying Rayleigh parameter. The 3rd to 5th panels display the feature of SDIC [3,17,47], the first kind of attractor coexistence (i.e., coexisting steady-state and chaotic solutions), and the second kind of attractor coexistence (i.e., coexisting steady-state and periodic solutions) at different time intervals, respectively. The alternative appearance of two kinds of attractor coexistence suggests time varying multistability.…”

Section: Time Varying Multistability and Recurrent Slowly Varying Sol...mentioning

confidence: 99%

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The Dual Nature of Chaos and Order in the Atmosphere

et al. 2022

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In the past, the Lorenz 1963 and 1969 models have been applied for revealing the chaotic nature of weather and climate and for estimating the atmospheric predictability limit. Recently, an in-depth analysis of classical Lorenz 1963 models and newly developed, generalized Lorenz models suggested a revised view that “the entirety of weather possesses a dual nature of chaos and order with distinct predictability”, in contrast to the conventional view of “weather is chaotic”. The distinct predictability associated with attractor coexistence suggests limited predictability for chaotic solutions and unlimited predictability (or up to their lifetime) for non-chaotic solutions. Such a view is also supported by a recent analysis of the Lorenz 1969 model that is capable of producing both unstable and stable solutions. While the alternative appearance of two kinds of attractor coexistence was previously illustrated, in this study, multistability (for attractor coexistence) and monostability (for single type solutions) are further discussed using kayaking and skiing as an analogy. Using a slowly varying, periodic heating parameter, we additionally emphasize the predictable nature of recurrence for slowly varying solutions and a less predictable (or unpredictable) nature for the onset for emerging solutions (defined as the exact timing for the transition from a chaotic solution to a non-chaotic limit cycle type solution). As a result, we refined the revised view outlined above to: “The atmosphere possesses chaos and order; it includes, as examples, emerging organized systems (such as tornadoes) and time varying forcing from recurrent seasons”. In addition to diurnal and annual cycles, examples of non-chaotic weather systems, as previously documented, are provided to support the revised view.

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“…For this study, we apply the KdV-SIR equation and its analytical solutions in order to illustrate the dependence of prediction horizons on initial conditions and system parameters. Such sensitivities of solutions are then compared to the sensitive dependence of solutions on initial conditions (SDIC), known as the butterfly effect, within chaotic systems (e.g., Shen et al 2022a, b, c [10] , [11] , [12] ).…”

Section: Introductionmentioning

confidence: 99%