scholarly journals Hierarchical scale dependence associated with the extension of the nonlinear feedback loop in a seven-dimensional Lorenz model

2016 ◽  
Author(s):  
Bo-Wen Shen
Keyword(s):  
Stability Analysis ◽  
Prandtl Number ◽  
Analytical Solutions ◽  
Feedback Loop ◽  
Critical Value ◽  
Scale Dependence ◽  
Nonlinear Feedback ◽  
Lorenz Model ◽  
Onset Of Chaos ◽  
Hierarchical Scale

Abstract. In this study, we construct a seven-dimensional Lorenz Model (7DLM) to discuss the impact of an extended nonlinear feedback loop on solutions' stability and illustrate the hierarchical scale dependence of chaotic solutions. Compared to the 5DLM, the 7DLM includes two additional high wavenumber modes that are selected based on an analysis of the nonlinear temperature advection term, a Jacobian term (J(ψ, θ)), where, ψ and θ represent the streamfunction and temperature perturbations, respectively. Fourier modes that represent temperature in the 7DLM can be categorized into three major scales as the primary (the largest scale), secondary, and tertiary (the smallest scale) modes. Further extension of the nonlinear feedback loop within the 7DLM can provide negative nonlinear feedback to stabilize solutions, thus leading to a much larger critical value for the Rayleigh parameter (rc ∼ 116.9) for the onset of chaos, as compared to an rc of 42.9 for the 5DLM as well as an rc of 24.74 for the 3DLM. The rc is determined by an analysis of ensemble Lyapunov exponents (eLEs) with Prandtl number (σ) of 10. To examine the dependence of rc on the value of the Prandtl number, a linear stability analysis is performed near the nontrivial critical point using a wide range of Rayleigh parameter (40 ≤ r ≤ 195) and Prandtl number (5 ≤ σ ≤ 25). Then an eLE analysis is conducted using selected values of the Prandtl number. The linear stability analysis is done by solving for the analytical solutions of the critical points, by linearizing the 7DLM with respect to the analytical solutions, and by calculating the eigenvalues of the linearized system. Within the range of (5 ≤ σ ≤ 25), the 7DLM requires a larger rc for the onset of chaos than the 5DLM. In addition to the negative nonlinear feedback illustrated and emulated by the quasi-equilibrium state solutions for high wavenumber modes, the 7DLM reveals the hierarchical scale dependence of chaotic solutions. For chaotic solutions with r = 120, the Pearson correlation coefficients (PCCs) between the primary and secondary modes (i.e., Z and Z1) and between the secondary and tertiary modes (i.e., Z1 and Z2) are 0.988 and 0.998, respectively. Here, Z, Z1, and Z2 represent the time-varying amplitudes of the primary, secondary, and tertiary modes, respectively. High PCCs indicate a strong linear relationship among the modes at various scales and a hierarchy of scale dependence. Future work will be undertaken to examine how higher dimensional LMs may produce a larger critical value for the Rayleigh parameter for the onset of chaos and reveal stronger hierarchical scale dependence.

scholarly journals Hierarchical scale dependence associated with the extension of the nonlinear feedback loop in a seven-dimensional Lorenz model

2016 ◽  
Vol 23 (4) ◽  
pp. 189-203 ◽  
Author(s):  
Bo-Wen Shen
Keyword(s):  
Stability Analysis ◽  
Prandtl Number ◽  
Analytical Solutions ◽  
Feedback Loop ◽  
Critical Value ◽  
Scale Dependence ◽  
Nonlinear Feedback ◽  
Lorenz Model ◽  
Onset Of Chaos ◽  
Hierarchical Scale

Abstract. In this study, we construct a seven-dimensional Lorenz model (7DLM) to discuss the impact of an extended nonlinear feedback loop on solutions' stability and illustrate the hierarchical scale dependence of chaotic solutions. Compared to the 5DLM, the 7DLM includes two additional high wavenumber modes that are selected based on an analysis of the nonlinear temperature advection term, a Jacobian term (J(ψ, θ)), where, ψ and θ represent the streamfunction and temperature perturbations, respectively. Fourier modes that represent temperature in the 7DLM can be categorized into three major scales as the primary (the largest scale), secondary, and tertiary (the smallest scale) modes. Further extension of the nonlinear feedback loop within the 7DLM can provide negative nonlinear feedback to stabilize solutions, thus leading to a much larger critical value for the Rayleigh parameter (rc  ∼  116.9) for the onset of chaos, as compared to an rc of 42.9 for the 5DLM as well as an rc of 24.74 for the 3DLM. The rc is determined by an analysis of ensemble Lyapunov exponents (eLEs) with a Prandtl number (σ) of 10. To examine the dependence of rc on the value of the Prandtl number, a linear stability analysis is performed near the nontrivial critical point using a wide range of the Rayleigh parameter (40  ≤  r  ≤  195) and the Prandtl number (5  ≤  σ  ≤  25). Then an eLE analysis is conducted using selected values of the Prandtl number. The linear stability analysis is done by solving for the analytical solutions of the critical points, by linearizing the 7DLM with respect to the analytical solutions, and by calculating the eigenvalues of the linearized system. Within the range of (5  ≤  σ  ≤  25), the 7DLM requires a larger rc for the onset of chaos than the 5DLM. In addition to the negative nonlinear feedback illustrated and emulated by the quasi-equilibrium state solutions for high wavenumber modes, the 7DLM reveals the hierarchical scale dependence of chaotic solutions. For chaotic solutions with r  =  120, the Pearson correlation coefficients (PCCs) between the primary and secondary modes (i.e., Z and Z1) and between the secondary and tertiary modes (i.e., Z1 and Z2) are 0.988 and 0.998, respectively. Here, Z, Z1, and Z2 represent the time-varying amplitudes of the primary, secondary, and tertiary modes, respectively. High PCCs indicate a strong linear relationship among the modes at various scales and a hierarchy of scale dependence. Future work will be undertaken to examine how higher-dimensional LMs may produce a larger critical value for the Rayleigh parameter for the onset of chaos and reveal stronger hierarchical scale dependence.

scholarly journals Nonlinear feedback in a six-dimensional Lorenz Model: impact of an additional heating term

2015 ◽  
Vol 2 (2) ◽  
pp. 475-512
Author(s):  
B.-W. Shen
Keyword(s):  
Rayleigh Number ◽  
Numerical Solutions ◽  
Critical Rayleigh Number ◽  
Critical Value ◽  
Small Scale ◽  
Solution Stability ◽  
Nonlinear Feedback ◽  
Lorenz Model ◽  
Additional Heating ◽  
The Impact

Abstract. In this study, a six-dimensional Lorenz model (6DLM) is derived, based on a recent study using a five-dimensional (5-D) Lorenz model (LM), in order to examine the impact of an additional mode and its accompanying heating term on solution stability. The new mode added to improve the representation of the steamfunction is referred to as a secondary streamfunction mode, while the two additional modes, that appear in both the 6DLM and 5DLM but not in the original LM, are referred to as secondary temperature modes. Two energy conservation relationships of the 6DLM are first derived in the dissipationless limit. The impact of three additional modes on solution stability is examined by comparing numerical solutions and ensemble Lyapunov exponents of the 6DLM and 5DLM as well as the original LM. For the onset of chaos, the critical value of the normalized Rayleigh number (rc) is determined to be 41.1. The critical value is larger than that in the 3DLM (rc ~ 24.74), but slightly smaller than the one in the 5DLM (rc ~ 42.9). A stability analysis and numerical experiments obtained using generalized LMs, with or without simplifications, suggest the following: (1) negative nonlinear feedback in association with the secondary temperature modes, as first identified using the 5DLM, plays a dominant role in providing feedback for improving the solution's stability of the 6DLM, (2) the additional heating term in association with the secondary streamfunction mode may destabilize the solution, and (3) overall feedback due to the secondary streamfunction mode is much smaller than the feedback due to the secondary temperature modes; therefore, the critical Rayleigh number of the 6DLM is comparable to that of the 5DLM. The 5DLM and 6DLM collectively suggest different roles for small-scale processes (i.e., stabilization vs. destabilization), consistent with the following statement by Lorenz (1972): If the flap of a butterfly's wings can be instrumental in generating a tornado, it can equally well be instrumental in preventing a tornado. The implications of this and previous work, as well as future work, are also discussed.

Quasi-Periodic Orbits in the Five-Dimensional Nondissipative Lorenz Model: The Role of the Extended Nonlinear Feedback Loop

2018 ◽  
Vol 28 (06) ◽  
pp. 1850072 ◽  
Author(s):  
Sara Faghih-Naini ◽  
Bo-Wen Shen
Keyword(s):  
Periodic Solution ◽  
Feedback Loop ◽  
Three Dimensional ◽  
Nonlinear Feedback ◽  
Lorenz Model ◽  
Restoring Force ◽  
Oscillatory Solutions ◽  
Nonlinear Temperature

A recent study suggested that the nonlinear feedback loop (NFL) of the three-dimensional nondissipative Lorenz model (3D-NLM) serves as a nonlinear restoring force by producing nonlinear oscillatory solutions as well as linear periodic solutions near a nontrivial critical point. This study discusses the role of the extension of the NFL in producing quasi-periodic trajectories using a five-dimensional nondissipative Lorenz model (5D-NLM). An analytical solution to the locally linear 5D-NLM is first obtained to illustrate the association of the extended NFL and two incommensurate frequencies whose ratio is irrational, yielding a quasi-periodic solution. The quasi-periodic solution trajectory moves endlessly on a torus but never intersects itself. While the NFL of the 3D-NLM consists of a pair of downscaling and upscaling processes, the extended NFL within the 5D-NLM additionally introduces two new pairs of downscaling and upscaling processes that are enabled by two high wavenumber modes. One pair of downscaling and upscaling processes provides a two-way interaction between the original (primary) Fourier modes of the 3D-NLM and the newly-added (secondary) Fourier modes of the 5D-NLM. The other pair of downscaling and upscaling processes involves interactions amongst the secondary modes. By comparing the numerical simulations using one- and two-way interactions, we illustrate that the two-way interaction is crucial for producing the quasi-periodic solution. A follow-up study using a 7D nondissipative LM shows that a further extension of NFL, which may appear throughout the spatial mode-mode interactions rooted in the nonlinear temperature advection, is capable of producing one more incommensurate frequency.

scholarly journals On the nonlinear feedback loop and energy cycle of the non-dissipative Lorenz model

2014 ◽  
Vol 1 (1) ◽  
pp. 519-541 ◽  
Author(s):  
B.-W. Shen
Keyword(s):  
Saddle Point ◽  
Feedback Loop ◽  
Initial Perturbation ◽  
Half Cycle ◽  
Nonlinear Feedback ◽  
Lorenz Model ◽  
Restoring Force ◽  
Linear Heating ◽  
Nonlinear Restoring Force ◽  
Energy Cycle

Abstract. In this study, we discuss the role of the nonlinear terms and linear (heating) term in the energy cycle of the three-dimensional (X–Y–Z) non-dissipative Lorenz model (3D-NLM). (X, Y, Z) represent the solutions in the phase space. We first present the closed-form solution to the nonlinear equation d2 X/dτ2+ (X2/2)X = 0, τ is a non-dimensional time, which was never documented in the literature. As the solution is oscillatory (wave-like) and the nonlinear term (X2) is associated with the nonlinear feedback loop, it is suggested that the nonlinear feedback loop may act as a restoring force. We then show that the competing impact of nonlinear restoring force and linear (heating) force determines the partitions of the averaged available potential energy from Y and Z modes, respectively, denoted as APEY and APEZ. Based on the energy analysis, an energy cycle with four different regimes is identified with the following four points: A(X, Y) = (0,0), B = (Xt, Yt), C = (Xm, Ym), and D = (Xt, -Yt). Point A is a saddle point. The initial perturbation (X, Y, Z) = (0, 1, 0) gives (Xt, Yt) = ( √ 2σr , r) and (Xm, Ym) = (2√  σr , 0). σ is the Prandtl number, and r is the normalized Rayleigh number. The energy cycle starts at (near) point A, A+ = (0, 0+) to be specific, goes through B, C, and D, and returns back to A, i.e., A- = (0,0-). From point A to point B, denoted as Leg A–B, where the linear (heating) force dominates, the solution X grows gradually with { KE↑, APEY↓, APEZ↓}. KE is the averaged kinetic energy. We use the upper arrow (↑) and down arrow (↓) to indicate an increase and decrease, respectively. In Leg B–C (or C–D) where nonlinear restoring force becomes dominant, the solution X increases (or decreases) rapidly with {KE↑, APEY↑, APEZ↓} (or {KE↓, APEY↓, APEZ↑}). In Leg D–A, the solution X decreases slowly with {KE↓, APEY↑, APEZ↑ }. As point A is a saddle point, the aforementioned cycle may be only half of a "big" cycle, displaying the wing pattern of a glasswinged butterfly, and the other half cycle is antisymmetric with respect to the origin, namely B = (-Xt, -Yt), C = (-Xm, 0), and D = (-Xt, Yt).

scholarly journals Nonlinear feedback in a six-dimensional Lorenz model: impact of an additional heating term

2015 ◽  
Vol 22 (6) ◽  
pp. 749-764 ◽  
Author(s):  
B.-W. Shen
Keyword(s):  
Rayleigh Number ◽  
Numerical Solutions ◽  
Critical Rayleigh Number ◽  
Critical Value ◽  
Small Scale ◽  
Solution Stability ◽  
Nonlinear Feedback ◽  
Lorenz Model ◽  
Additional Heating ◽  
The Impact

Abstract. In this study, a six-dimensional Lorenz model (6DLM) is derived, based on a recent study using a five-dimensional (5-D) Lorenz model (LM), in order to examine the impact of an additional mode and its accompanying heating term on solution stability. The new mode added to improve the representation of the streamfunction is referred to as a secondary streamfunction mode, while the two additional modes, which appear in both the 6DLM and 5DLM but not in the original LM, are referred to as secondary temperature modes. Two energy conservation relationships of the 6DLM are first derived in the dissipationless limit. The impact of three additional modes on solution stability is examined by comparing numerical solutions and ensemble Lyapunov exponents of the 6DLM and 5DLM as well as the original LM. For the onset of chaos, the critical value of the normalized Rayleigh number (rc) is determined to be 41.1. The critical value is larger than that in the 3DLM (rc ~ 24.74), but slightly smaller than the one in the 5DLM (rc ~ 42.9). A stability analysis and numerical experiments obtained using generalized LMs, with or without simplifications, suggest the following: (1) negative nonlinear feedback in association with the secondary temperature modes, as first identified using the 5DLM, plays a dominant role in providing feedback for improving the solution's stability of the 6DLM, (2) the additional heating term in association with the secondary streamfunction mode may destabilize the solution, and (3) overall feedback due to the secondary streamfunction mode is much smaller than the feedback due to the secondary temperature modes; therefore, the critical Rayleigh number of the 6DLM is comparable to that of the 5DLM. The 5DLM and 6DLM collectively suggest different roles for small-scale processes (i.e., stabilization vs. destabilization), consistent with the following statement by Lorenz (1972): "If the flap of a butterfly's wings can be instrumental in generating a tornado, it can equally well be instrumental in preventing a tornado." The implications of this and previous work, as well as future work, are also discussed.

scholarly journals Tuning of a Linear-Quadratic Stabilization System for an Anti-Aircraft Missile

Aerospace ◽  
2021 ◽  
Vol 8 (2) ◽  
pp. 48
Author(s):  
Witold Bużantowicz
Keyword(s):  
Analytical Solutions ◽  
Feedback Loop ◽  
Linear Quadratic Regulator ◽  
Added Value ◽  
Tuning Parameter ◽  
Stabilization System ◽  
Linear Quadratic ◽  
Quadratic Stabilization ◽  
Novel Method ◽  
Selection Of

A description is given of an application of a linear-quadratic regulator (LQR) for stabilizing the characteristics of an anti-aircraft missile, and an analytical method of selecting the weighting elements of the gain matrix in feedback loop is proposed. A novel method of LQR tuning via a single parameter ς was proposed and tested. The article supplements and develops the topics addressed in the author’s previous work. Its added value includes the observation that the solutions obtained are symmetric pairs, and that the tuning parameter ς proposed for the designed linear-quadratic regulator enables the selection of suitable parameters for the airframe stabilizing loop for the majority of the analytical solutions of the considered Riccati equation.

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