1887
banner image
No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
Geometric analysis of transient bursts
Rent:
Rent this article for
USD
10.1063/1.4826655
/content/aip/journal/chaos/23/4/10.1063/1.4826655
http://aip.metastore.ingenta.com/content/aip/journal/chaos/23/4/10.1063/1.4826655
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

Bifurcation diagram plotted in (z, , )-space of system (1) in the singular limit → 0. We used  = 0.75 and  = 1.0. The fast subsystem has a Z-shaped family of equilibria containing the saddle-node bifurcation points SN and SN. The family also contains a subcritical Hopf bifurcation point H on the upper branch that gives rise to the family of periodic orbits that exhibit a SNP before it ends in a homolinic bifurcation.

Image of FIG. 2.
FIG. 2.

Bifurcation diagram in the ()-plane near  = 0. Additional equilibria of system (1) exist due to fold bifurcations, labelled SN and SN, which can become stable due to a Hopf bifurcation, labelled H.

Image of FIG. 3.
FIG. 3.

Transient response generated for system (1) with  = 0.75 and  = 1.0. A perturbation with amplitude  = 0.02 and duration  = 15 takes the system away from its equilibrium FP (grey segment). The relaxation back to FP (black segment) exhibits three additional spikes before reaching FP. The response is shown in (, , )space in panel (a), with the underlying bifurcation diagram of the fast subsystem for reference; see also Figure 1 . Panel (b) shows the corresponding time series for with [−50, 500].

Image of FIG. 4.
FIG. 4.

As decreases, the response changes from 1 spike to more and more spikes, with each spike-adding transition characterised by a strong increase in the integral norm. Responses for  = 1 and  = 1.15 (one spike),  = 1.0 (two spikes),  = 0.85 (three spikes) and  = 0.43 (nine spikes) are highlighted and their corresponding time series for are shown in panels (b)–(e), respectively.

Image of FIG. 5.
FIG. 5.

Two viewpoints in panels (a) and (b) of the transient response at the onset from one to two spikes; we also show the bifurcation diagram of the fast subsystem, along with a subset of the family ( ) of stable manifolds associated with saddle equilibria on the middle branch in between SN and the homoclinic bifurcation (not labelled). The response for 1.07256 traces up to the fold point at SN; panel (c) shows the corresponding time series for with [−50, 500].

Image of FIG. 6.
FIG. 6.

Two viewpoints in panels (a) and (b) of the transient response at the onset from three to four spikes; we also show the bifurcation diagram of the fast subsystem, along with a subset of the family ( ) of stable manifolds associated with saddle equilibria on the middle branch in between SN and just past the homoclinic bifurcation (not labelled). The response for  0.778355 traces up to the fold point at SN; panel (c) shows the corresponding time series for with [−50, 500].

Image of FIG. 7.
FIG. 7.

Deformation of an orbit segment pair , ) through the moment of the second spike onset. Here,  = 1 is fixed and  = 1 at the start of the continuation. Note how the end point transforms from a local maximum into a local minimum, which is detected as a fold bifurcation. Panel (a) shows a waterfall diagram of the orbit segments computed as part of the continuation; panel (b) shows the variation of (black) and (grey), while panel (c) plots (black) and the end point (grey) as a function of The spike onset is detected at LP.

Image of FIG. 8.
FIG. 8.

Two-parameter bifurcation diagram indicating the regions in the ()plane with different numbers of additional spikes.

Loading

Article metrics loading...

/content/aip/journal/chaos/23/4/10.1063/1.4826655
2013-10-31
2014-04-20
Loading

Full text loading...

This is a required field
Please enter a valid email address
752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Geometric analysis of transient bursts
http://aip.metastore.ingenta.com/content/aip/journal/chaos/23/4/10.1063/1.4826655
10.1063/1.4826655
SEARCH_EXPAND_ITEM