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Linear Augmentation for Stabilizing Stationary Solutions: Potential Pitfalls and Their Application.

Karnatak R - PLoS ONE (2015)

Bottom Line: The simplicity of the procedure is the main highlight of this scheme but questions related to its general applicability still need to be addressed.Focusing on the issue of targeting stationary solutions, this work demonstrates instances where the scheme fails to stabilize the required solutions and leads to other complicated dynamical scenarios.Examples from conservative as well as dissipative systems are presented in this regard and important applications in dissipative predator-prey systems are discussed, which include preventative measures to avoid potentially catastrophic dynamical transitions in these systems.

View Article: PubMed Central - PubMed

Affiliation: Nonlinear Dynamics and Time Series Analysis Research Group, Max-Planck-Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187 Dresden, Germany.

ABSTRACT
Linear augmentation has recently been shown to be effective in targeting desired stationary solutions, suppressing bistablity, in regulating the dynamics of drive response systems and in controlling the dynamics of hidden attractors. The simplicity of the procedure is the main highlight of this scheme but questions related to its general applicability still need to be addressed. Focusing on the issue of targeting stationary solutions, this work demonstrates instances where the scheme fails to stabilize the required solutions and leads to other complicated dynamical scenarios. Examples from conservative as well as dissipative systems are presented in this regard and important applications in dissipative predator-prey systems are discussed, which include preventative measures to avoid potentially catastrophic dynamical transitions in these systems.

No MeSH data available.


Behavior of the partially augmented Duffing oscillator.Largest eigenvalue estimates for stationary solutions (x*, y*) = (0, 0) (black), (x*, y*) = (±1, 0) (red).
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pone.0142238.g002: Behavior of the partially augmented Duffing oscillator.Largest eigenvalue estimates for stationary solutions (x*, y*) = (0, 0) (black), (x*, y*) = (±1, 0) (red).

Mentions: Substituting x* = 0,±1, and rearranging the terms, we can obtain the characteristic eigenvalue equations for these stationary solutions as,λ3+kλ2+(ε2-1)λ-k=0,(11)for (x*, y*) = (0, 0) (hyperbolic for ε = 0), andλ3+kλ2+(ε2+2)λ+2k=0,(12)for (x*, y*) = (±1, 0) (non hyperbolic for ε = 0) respectively. It is straightforward to check that the largest eigenvalue λ1 → 0 for larger ε values in both these cases which implies that a stable/unstable stationary solution will stay the same until a stability change (zero crossing of the eigenvalue/s) occurs in the ε → ∞ limit. Furthermore using the RHC, it is easily verifiable that Eq 11 will always have positive root/roots, whereas Eq 12 will have all negative roots ∀ ε > 0; which implies that the x* = 0 is always unstable and x* = ±1 is always stable. Therefore, we see that partial augmentation works for stabilizing (x*, y*) = (±1, 0) but fails completely to stabilize the origin (x*, y*) = (0, 0). Fig 2 shows the largest eigenvalue calculations which verify these deductions. This brings us to an important observation that there might exist situations where it is not possible to target the desired stationary solution even on using an appropriate feedback function with any combination of k and ε values.


Linear Augmentation for Stabilizing Stationary Solutions: Potential Pitfalls and Their Application.

Karnatak R - PLoS ONE (2015)

Behavior of the partially augmented Duffing oscillator.Largest eigenvalue estimates for stationary solutions (x*, y*) = (0, 0) (black), (x*, y*) = (±1, 0) (red).
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4636295&req=5

pone.0142238.g002: Behavior of the partially augmented Duffing oscillator.Largest eigenvalue estimates for stationary solutions (x*, y*) = (0, 0) (black), (x*, y*) = (±1, 0) (red).
Mentions: Substituting x* = 0,±1, and rearranging the terms, we can obtain the characteristic eigenvalue equations for these stationary solutions as,λ3+kλ2+(ε2-1)λ-k=0,(11)for (x*, y*) = (0, 0) (hyperbolic for ε = 0), andλ3+kλ2+(ε2+2)λ+2k=0,(12)for (x*, y*) = (±1, 0) (non hyperbolic for ε = 0) respectively. It is straightforward to check that the largest eigenvalue λ1 → 0 for larger ε values in both these cases which implies that a stable/unstable stationary solution will stay the same until a stability change (zero crossing of the eigenvalue/s) occurs in the ε → ∞ limit. Furthermore using the RHC, it is easily verifiable that Eq 11 will always have positive root/roots, whereas Eq 12 will have all negative roots ∀ ε > 0; which implies that the x* = 0 is always unstable and x* = ±1 is always stable. Therefore, we see that partial augmentation works for stabilizing (x*, y*) = (±1, 0) but fails completely to stabilize the origin (x*, y*) = (0, 0). Fig 2 shows the largest eigenvalue calculations which verify these deductions. This brings us to an important observation that there might exist situations where it is not possible to target the desired stationary solution even on using an appropriate feedback function with any combination of k and ε values.

Bottom Line: The simplicity of the procedure is the main highlight of this scheme but questions related to its general applicability still need to be addressed.Focusing on the issue of targeting stationary solutions, this work demonstrates instances where the scheme fails to stabilize the required solutions and leads to other complicated dynamical scenarios.Examples from conservative as well as dissipative systems are presented in this regard and important applications in dissipative predator-prey systems are discussed, which include preventative measures to avoid potentially catastrophic dynamical transitions in these systems.

View Article: PubMed Central - PubMed

Affiliation: Nonlinear Dynamics and Time Series Analysis Research Group, Max-Planck-Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187 Dresden, Germany.

ABSTRACT
Linear augmentation has recently been shown to be effective in targeting desired stationary solutions, suppressing bistablity, in regulating the dynamics of drive response systems and in controlling the dynamics of hidden attractors. The simplicity of the procedure is the main highlight of this scheme but questions related to its general applicability still need to be addressed. Focusing on the issue of targeting stationary solutions, this work demonstrates instances where the scheme fails to stabilize the required solutions and leads to other complicated dynamical scenarios. Examples from conservative as well as dissipative systems are presented in this regard and important applications in dissipative predator-prey systems are discussed, which include preventative measures to avoid potentially catastrophic dynamical transitions in these systems.

No MeSH data available.