Hopf Bifurcations, Periodic Windows and Intermittency in the Generalized Lorenz Model

Wawrzaszek, Anna; Krasińska, Agata

doi:10.1142/s0218127419300428

Cited by 7 publications

(5 citation statements)

References 33 publications

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“…which occurs with w ∈ R in the Lorenz-Stenflo model from Stenflo (1996), its magnetic variant (Wawrzaszek and Krasinska 2019), and with w ∈ R 2 in the models from Molteni et al (1993); for the latter we choose α = O(ε) and shift the auxiliary variables (which gives b = 0) to obtain the form ( 47). An extension of Lorenz-Stenflo with nonlinear additional equations is considered in Moon et al (2021), but still fits into the present framework when, e.g., scaling the variables in addition to Lorenz-Stenflo with j = 3 and choosing Lewis number of order ε −2 .…”

Section: Other Lorenz-like Systemsmentioning

confidence: 99%

Time Averages and Periodic Attractors at High Rayleigh Number for Lorenz-like Models

Ovsyannikov,

Rademacher,

Welter

et al. 2023

J Nonlinear Sci

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Revisiting the Lorenz ’63 equations in the regime of large of Rayleigh number, we study the occurrence of periodic solutions and quantify corresponding time averages of selected quantities. Perturbing from the integrable limit of infinite $$\rho $$ ρ , we provide a full proof of existence and stability of symmetric periodic orbits, which confirms previous partial results. Based on this, we expand time averages in terms of elliptic integrals with focus on the much studied average ‘transport,’ which is the mode reduced excess heat transport of the convection problem that gave rise to the Lorenz equations. We find a hysteresis loop between the periodic attractors and the nonzero equilibria of the Lorenz equations. These have been proven to maximize transport, and we show that the transport takes arbitrarily small values in the family of periodic attractors. In particular, when the nonzero equilibria are unstable, we quantify the difference between maximal and typically realized values of transport. We illustrate these results by numerical simulations and show how they transfer to various extended Lorenz models.

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Section: Other Lorenz-like Systemsmentioning

confidence: 99%

Time Averages and Periodic Attractors at High Rayleigh Number for Lorenz-like Models

Ovsyannikov,

Rademacher,

Welter

et al. 2023

J Nonlinear Sci

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“…which occurs with w ∈ R in the Lorenz-Stenflo model from [24], its magnetic variant [26], and with w ∈ R 2 in the models from [12]; for the latter we choose α = O(ε) and shift the auxiliary variables (which gives b = 0) to obtain the form (47). An extension of Lorenz-Stenflo with nonlinear additional equations is considered in [13], but still fits into the present framework when, e.g., scaling the variables in addition to Lorenz-Stenflo with j = 3 and choosing Lewis number of order ε −2 .…”

Section: Other Lorenz-like Systemsmentioning

confidence: 99%

“…This range corresponds to the values λ > 2/3 on the left-hand side of (27). Also, every k 1 ∈ (k * , 1) gives a unique positive value for B via formula (26).…”

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confidence: 99%

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Time averages and periodic attractors at high Rayleigh number for Lorenz-like models

Ovsyannikov¹,

Rademacher²,

Welter³

et al. 2023

Preprint

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Revisiting the Lorenz '63 equations in the regime of large of Rayleigh number, we study the occurrence of periodic solutions and quantify corresponding time averages of selected quantities. Perturbing from the integrable limit of infinite ρ, we provide a full proof of existence and stability of symmetric periodic orbits, which confirms previous partial results. Based on this, we expand time averages in terms of elliptic integrals with focus on the much studied average 'transport', which is the mode reduced excess heat transport of the convection problem that gave rise to the Lorenz equations. We find a hysteresis loop between the periodic attractors and the non-zero equilibria of the Lorenz equations. These have been proven to maximize transport, and we show that the transport takes arbitrarily small values in the family of periodic attractors. In particular, when the non-zero equilibria are unstable, we quantify the difference between maximal and typically realized values of transport. We illustrate these results by numerical simulations and show how they transfer to various extended Lorenz models.

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“…As the theory of bifurcation is a tool to help to understand equilibrium loss and its consequences for complex behavior. There are many types of periodic orbit local bifurcation such as period-doubling bifurcation, and Hopf bifurcation [20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

Period-doubling bifurcation analysis and chaos control for load torque using FLC

Moustafa

Sobaih

Abozalam

et al. 2021

Complex Intell. Syst.

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Chaotic phenomena are observed in several practical and scientific fields; however, the chaos is harmful to systems as they can lead them to be unstable. Consequently, the purpose of this study is to analyze the bifurcation of permanent magnet direct current (PMDC) motor and develop a controller that can suppress chaotic behavior resulted from parameter variation such as the loading effect. The nonlinear behaviors of PMDC motors were investigated by time-domain waveform, phase portrait, and Floquet theory. By varying the load torque, a period-doubling bifurcation appeared which in turn led to chaotic behavior in the system. So, a fuzzy logic controller and developing the Floquet theory techniques are applied to eliminate the bifurcation and the chaos effects. The controller is used to enhance the performance of the system by getting a faster response without overshoot or oscillation, moreover, tends to reduce the steady-state error while maintaining its stability. The simulation results emphasize that fuzzy control provides better performance than that obtained from the other controller.

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Hopf Bifurcations, Periodic Windows and Intermittency in the Generalized Lorenz Model

Cited by 7 publications

References 33 publications

Time Averages and Periodic Attractors at High Rayleigh Number for Lorenz-like Models

Time Averages and Periodic Attractors at High Rayleigh Number for Lorenz-like Models

Time averages and periodic attractors at high Rayleigh number for Lorenz-like models

Period-doubling bifurcation analysis and chaos control for load torque using FLC

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