Limits...
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.


Different dynamical regimes in fully augmented Duffing oscillator.Figs a), b) and c) show the bifurcation diagrams (black dots) and the largest eigenvalues (red symbols) for (x*, y*) = (1, 0), (0, 0), and (−1, 0) respectively. Different dynamical regimes are marked as: A shows the regimes of bistability between different stationary solutions, B shows the regime where the desired stationary solution is the only dynamical attractor, and C marks the regime where other stationary solutions are stable. Circles mark the solution branchs (x*+, y*+, z*+) and (x*−, y*−, z*−) from Eq 13 in a) and c) respectively. Related Figs a.1: for ε = 0.4, the system is bistable and the two related transient behaviors (in blue and green and likewise for other cases), a.2: for ε = 1, the trajectory approaching the stable stationary solution (1, 0), and a.3 shows an arbitrary time series for ε = 2.5. Similarly in b.1: bistability, and in b.2: the system approaching the stable stationary solution (0, 0) is shown. Identically, c.1, c.2, and c.3 show bistability (ε = 0.4), stabilization of (−1, 0) (ε = 1) and an arbitrary time series at ε = 2.5 respectively.
© Copyright Policy
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC4636295&req=5

pone.0142238.g003: Different dynamical regimes in fully augmented Duffing oscillator.Figs a), b) and c) show the bifurcation diagrams (black dots) and the largest eigenvalues (red symbols) for (x*, y*) = (1, 0), (0, 0), and (−1, 0) respectively. Different dynamical regimes are marked as: A shows the regimes of bistability between different stationary solutions, B shows the regime where the desired stationary solution is the only dynamical attractor, and C marks the regime where other stationary solutions are stable. Circles mark the solution branchs (x*+, y*+, z*+) and (x*−, y*−, z*−) from Eq 13 in a) and c) respectively. Related Figs a.1: for ε = 0.4, the system is bistable and the two related transient behaviors (in blue and green and likewise for other cases), a.2: for ε = 1, the trajectory approaching the stable stationary solution (1, 0), and a.3 shows an arbitrary time series for ε = 2.5. Similarly in b.1: bistability, and in b.2: the system approaching the stable stationary solution (0, 0) is shown. Identically, c.1, c.2, and c.3 show bistability (ε = 0.4), stabilization of (−1, 0) (ε = 1) and an arbitrary time series at ε = 2.5 respectively.

Mentions: Now considering identical augmentation with ε1 = ε2 = ε and we will see that this system has some interesting properties. Fig 3 shows the bifurcation diagrams of the system as we try targeting the different desired stationary solutions: For (x*, y*) = (1, 0), the bifurcation diagram (black dots) is shown in Fig 3a. Appropriate transient trajectories in different coupling regimes are also shown in related Fig 3(a.1), 3(a.2) and 3(a.3). It is observed that even for very small coupling values, the system quickly gets into a stable stationary state regime, although, for smaller values of ε, it exhibits bistability. The transient trajectories in this parameter regime are shown in Fig 3(a.1). We observe that the augmentation is stabilizing our desired stationary solution at (x*, y*) = (1, 0), but along with it, other stationary solutions which are ε dependent are also getting stabilized on starting with different initial conditions. These other stationary solutions for the augmented system here are given by,x*±=12-1±1-4ε2k-ε2,y*±=x*±-x*±3,u*±=-y*±ε,(13)and solutions (x*−, y*−, z*−) are observed to coexist along with (x*, y*) = (1, 0). For higher coupling values, bistability terminates via a saddle node bifurcation when the stable branch of stationary solutions (x*−, y*−, u*−) collides with the unstable branch of (x*+, y*+, u*+)(circles) as shown in Fig 3a at . The system also exhibits hysteresis in this bistable regime and a brief discussion regarding this observation is available in Sec. Duffing system: hysteresis. Beyond this regime for a range of values in ε > εSN, (x*, y*) = (1, 0) remains as the only stable attractor as shown in Fig 3(a) and 3(a.2).


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

Karnatak R - PLoS ONE (2015)

Different dynamical regimes in fully augmented Duffing oscillator.Figs a), b) and c) show the bifurcation diagrams (black dots) and the largest eigenvalues (red symbols) for (x*, y*) = (1, 0), (0, 0), and (−1, 0) respectively. Different dynamical regimes are marked as: A shows the regimes of bistability between different stationary solutions, B shows the regime where the desired stationary solution is the only dynamical attractor, and C marks the regime where other stationary solutions are stable. Circles mark the solution branchs (x*+, y*+, z*+) and (x*−, y*−, z*−) from Eq 13 in a) and c) respectively. Related Figs a.1: for ε = 0.4, the system is bistable and the two related transient behaviors (in blue and green and likewise for other cases), a.2: for ε = 1, the trajectory approaching the stable stationary solution (1, 0), and a.3 shows an arbitrary time series for ε = 2.5. Similarly in b.1: bistability, and in b.2: the system approaching the stable stationary solution (0, 0) is shown. Identically, c.1, c.2, and c.3 show bistability (ε = 0.4), stabilization of (−1, 0) (ε = 1) and an arbitrary time series at ε = 2.5 respectively.
© Copyright Policy
Related In: Results  -  Collection

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

pone.0142238.g003: Different dynamical regimes in fully augmented Duffing oscillator.Figs a), b) and c) show the bifurcation diagrams (black dots) and the largest eigenvalues (red symbols) for (x*, y*) = (1, 0), (0, 0), and (−1, 0) respectively. Different dynamical regimes are marked as: A shows the regimes of bistability between different stationary solutions, B shows the regime where the desired stationary solution is the only dynamical attractor, and C marks the regime where other stationary solutions are stable. Circles mark the solution branchs (x*+, y*+, z*+) and (x*−, y*−, z*−) from Eq 13 in a) and c) respectively. Related Figs a.1: for ε = 0.4, the system is bistable and the two related transient behaviors (in blue and green and likewise for other cases), a.2: for ε = 1, the trajectory approaching the stable stationary solution (1, 0), and a.3 shows an arbitrary time series for ε = 2.5. Similarly in b.1: bistability, and in b.2: the system approaching the stable stationary solution (0, 0) is shown. Identically, c.1, c.2, and c.3 show bistability (ε = 0.4), stabilization of (−1, 0) (ε = 1) and an arbitrary time series at ε = 2.5 respectively.
Mentions: Now considering identical augmentation with ε1 = ε2 = ε and we will see that this system has some interesting properties. Fig 3 shows the bifurcation diagrams of the system as we try targeting the different desired stationary solutions: For (x*, y*) = (1, 0), the bifurcation diagram (black dots) is shown in Fig 3a. Appropriate transient trajectories in different coupling regimes are also shown in related Fig 3(a.1), 3(a.2) and 3(a.3). It is observed that even for very small coupling values, the system quickly gets into a stable stationary state regime, although, for smaller values of ε, it exhibits bistability. The transient trajectories in this parameter regime are shown in Fig 3(a.1). We observe that the augmentation is stabilizing our desired stationary solution at (x*, y*) = (1, 0), but along with it, other stationary solutions which are ε dependent are also getting stabilized on starting with different initial conditions. These other stationary solutions for the augmented system here are given by,x*±=12-1±1-4ε2k-ε2,y*±=x*±-x*±3,u*±=-y*±ε,(13)and solutions (x*−, y*−, z*−) are observed to coexist along with (x*, y*) = (1, 0). For higher coupling values, bistability terminates via a saddle node bifurcation when the stable branch of stationary solutions (x*−, y*−, u*−) collides with the unstable branch of (x*+, y*+, u*+)(circles) as shown in Fig 3a at . The system also exhibits hysteresis in this bistable regime and a brief discussion regarding this observation is available in Sec. Duffing system: hysteresis. Beyond this regime for a range of values in ε > εSN, (x*, y*) = (1, 0) remains as the only stable attractor as shown in Fig 3(a) and 3(a.2).

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.