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.
Border collision route to quasiperiodicity: Numerical investigation and experimental confirmation
Rent this article for
View: Figures


Image of FIG. 1.
FIG. 1.

Bifurcation diagram for the normal form map as the parameter varies from to . The diagram shows a direct transition from a period-1 orbit to a quasiperiodic orbit. The other parameters are , , , and .

Image of FIG. 2.
FIG. 2.

(Color) Chart of dynamical modes of the normal form map in the parameter plane with the remaining parameters fixed at , , and .

Image of FIG. 3.
FIG. 3.

Bifurcation diagram calculated for the section situated along the main diagonal of Fig. 2 . The diagram shows repeated transitions between mode locking and quasiperiodicity.

Image of FIG. 4.
FIG. 4.

(Color) Phase portrait of the normal form map within the 1:5 tongue of periodicity for , , , , and . Here is the fixed point, and are the node and saddle period-5 cycles, and and are the stable and unstable manifolds, respectively, of the saddle .

Image of FIG. 5.
FIG. 5.

Bifurcation diagram where is varied from 0.9 to 1.2 whereas the other parameters are fixed at , , , and . This diagram shows a hysteretic transition between mode locking and quasiperiodicity. and are the border-collision fold bifurcation points. is the point of transition from quasiperiodic to periodic dynamics. The periodic orbit coexists with quasiperiodicity within the region , where and .

Image of FIG. 6.
FIG. 6.

(Color) (a) Homoclinic tangency appearing at . (b) Magnified part of the phase portrait outlined by the rectangle in (a).

Image of FIG. 7.
FIG. 7.

(Color) (a) Phase portrait of the map after the closed invariant curve has been destroyed. Here the stable period-5 cycle coexists with the quasiperiodic orbit . The basins of attraction of the periodic and quasiperiodic orbits are separated by the stable manifold of the period-5 saddle cycle. (b) Bifurcation diagram illustrating the birth of a pair of stable and saddle period-5 cycle together with the quasiperiodic orbit from the fixed point through a border-collision bifurcation with varying . Other parameters are the same as in (a).

Image of FIG. 8.
FIG. 8.

(Color) Phase portraits of the map in the region of multistability. The small open circle in the middle represents the unstable fixed point. (a) Stable period-4 cycle coexisting with the closed invariant curve. (b) Magnified part of the phase portrait outlined by the rectangle in (a). The closed invariant curve is the union of the unstable manifold of the saddle cycle of period 7 and the points of the stable focus and saddle period-7 cycles. (c) Phase portrait of the map for the case when the closed invariant curve does not exist. Stable period-5 cycle coexists here with a stable period-6 cycle. The numbers 5 and 6 mark period-5 and period-6 saddle points, respectively.

Image of FIG. 9.
FIG. 9.

(a) Schematic diagram of the dc–dc buck converter with two-level control. Here is the sample–hold unit. (b) Generation of switching signals and in a two-level controlled buck converter. denotes the period of the ramp function.

Image of FIG. 10.
FIG. 10.

(a) Experimental bifurcation diagram of the two-level controlled buck converter with the input voltage as a bifurcation parameter and (b) close-up of the parameter region where the transition to quasiperiodicity takes place.

Image of FIG. 11.
FIG. 11.

Experimental waveforms of the converter under regular periodic operation at .

Image of FIG. 12.
FIG. 12.

(a) Waveforms just before the transition to quasiperiodicity at and (b) just after the transition, . Note how the strictly periodic dynamics in (a) becomes modulated by a slower dynamics in (b).

Image of FIG. 13.
FIG. 13.

Phase portraits on the Poincaré section, (a) for the quasiperiodic dynamics at and (b) for the 1:8 mode-locked dynamics for . Note how the closed invariant curve for the ergodic torus turns into a set of discrete points for the resonance torus.

Image of FIG. 14.
FIG. 14.

The waveforms (a) before the transition from ergodic torus to resonance torus, , (b) after the transition, , (c) before the transition from resonance torus to ergodic torus, , and (d) after the transition, .

Image of FIG. 15.
FIG. 15.

Experimental bifurcation diagram illustrating the hysteretic transition from periodic to quasiperiodic orbit and vice versa (indicated by the punctuated lines): (a) when the input voltage decreases and (b) when the input voltage increases.


Article metrics loading...


Full text loading...

This is a required field
Please enter a valid email address
752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Border collision route to quasiperiodicity: Numerical investigation and experimental confirmation