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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.


Related in: MedlinePlus

Transition between regimes of stable origin and escaping trajectories.Bifurcation diagram (black dots) along with the largest eigenvalue (red symbols) as a function of ε in the top row.  marks the coupling beyond which all initial conditions lead to escaping trajectories in region E. Transient trajectories shown for ε(= 0.5) < ε* (left bottom) and ε(= 1.75) > ε* (right bottom).
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pone.0142238.g006: Transition between regimes of stable origin and escaping trajectories.Bifurcation diagram (black dots) along with the largest eigenvalue (red symbols) as a function of ε in the top row. marks the coupling beyond which all initial conditions lead to escaping trajectories in region E. Transient trajectories shown for ε(= 0.5) < ε* (left bottom) and ε(= 1.75) > ε* (right bottom).

Mentions: The roots of the auxilliary equation in this case are m± = α ± β with α = −ε2/2 and . For imaginary β, the transient solution for Eq 25 can be expressed as,xt(t)=Ax1(t)+Bx2(t),(26)which is independent of the forcing term U(ε, k, t) with x1(t) = exp(αt)cosβt, and x2(t) = exp(αt)sinβt. Consequently, the steady state solution can be obtained by using the Laplace and inverse Laplace transformations giving,xst(t)=1Ω∫0te-γ(t-x)sin(Ω(t-x))U(ε,k,x)dx,(27)where , and γ = ε2/2. Now at this point, we do not know the exact expression for U(ε, k, t). Considering the transient behavior of trajectories in partially/fully augmented system, we clearly observe that they possess an exponentially decaying/diverging envelop (see Fig 1 (inset) and Fig 6 bottom row). Based on these observations, assuming U(ε, k, t) = a0 exp (kmt) where both a0, km are functions of ε and k, and solving Eq 27 gives the particular solutionxp(t)=exp(kmt)a0(km+α)2-β2.(28)


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

Karnatak R - PLoS ONE (2015)

Transition between regimes of stable origin and escaping trajectories.Bifurcation diagram (black dots) along with the largest eigenvalue (red symbols) as a function of ε in the top row.  marks the coupling beyond which all initial conditions lead to escaping trajectories in region E. Transient trajectories shown for ε(= 0.5) < ε* (left bottom) and ε(= 1.75) > ε* (right bottom).
© Copyright Policy
Related In: Results  -  Collection

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

pone.0142238.g006: Transition between regimes of stable origin and escaping trajectories.Bifurcation diagram (black dots) along with the largest eigenvalue (red symbols) as a function of ε in the top row. marks the coupling beyond which all initial conditions lead to escaping trajectories in region E. Transient trajectories shown for ε(= 0.5) < ε* (left bottom) and ε(= 1.75) > ε* (right bottom).
Mentions: The roots of the auxilliary equation in this case are m± = α ± β with α = −ε2/2 and . For imaginary β, the transient solution for Eq 25 can be expressed as,xt(t)=Ax1(t)+Bx2(t),(26)which is independent of the forcing term U(ε, k, t) with x1(t) = exp(αt)cosβt, and x2(t) = exp(αt)sinβt. Consequently, the steady state solution can be obtained by using the Laplace and inverse Laplace transformations giving,xst(t)=1Ω∫0te-γ(t-x)sin(Ω(t-x))U(ε,k,x)dx,(27)where , and γ = ε2/2. Now at this point, we do not know the exact expression for U(ε, k, t). Considering the transient behavior of trajectories in partially/fully augmented system, we clearly observe that they possess an exponentially decaying/diverging envelop (see Fig 1 (inset) and Fig 6 bottom row). Based on these observations, assuming U(ε, k, t) = a0 exp (kmt) where both a0, km are functions of ε and k, and solving Eq 27 gives the particular solutionxp(t)=exp(kmt)a0(km+α)2-β2.(28)

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.


Related in: MedlinePlus