^{1,a)}, Erik Mosekilde

^{2,b)}, Somnath Maity

^{3}, Srijith Mohanan

^{3}and Soumitro Banerjee

^{3,c)}

### Abstract

Numerical studies of higher-dimensional piecewise-smooth systems have recently shown how a torus can arise from a periodic cycle through a special type of border-collision bifurcation. The present article investigates this new route to quasiperiodicity in the two-dimensional piecewise-linear normal form map. We have obtained the chart of the dynamical modes for this map and showed that border-collision bifurcations can lead to the birth of a stable closed invariant curve associated with quasiperiodic or periodic dynamics. In the parameter regions leading to the existence of an invariant closed curve, there may be transitions between an ergodic torus and a resonance torus, but the mechanism of creation for the resonance tongues is distinctly different from that observed in smooth maps. The transition from a stable focus point to a resonance torus may lead directly to a new focus of higher periodicity, e.g., a period-5 focus. This article also contains a discussion of torus destruction via a homoclinic bifurcation in the piecewise-linear normal map. Using a dc–dc converter with two-level control as an example, we report the first experimental verification of the direct transition to quasiperiodicity through a border-collision bifurcation.

Application of nonlinear dynamics and chaos theory in practice, particularly in engineering, often leads to the analysis of piecewise-smooth systems. Low-dimensional models of such systems have shown that they can exhibit behavioral transitions, referred to as border-collision bifurcations, that are qualitatively different from the bifurcations we know for smooth dynamical systems. In particular, these bifurcations can produce direct transitions from periodicity to chaos or, for instance, from period-2 to period-3 dynamics. The purpose of this article is to show that torus birth bifurcations (transitions to quasiperiodicity) can also occur via border-collision bifurcations. In this case a pair of complex conjugate Floquet multipliers jump from the inside to the outside of the unit circle. We also examine the border-collision bifurcations through which the ergodic torus is transformed into a resonance torus. Torus destruction represents one of the most complicated routes to chaos, and the possible mechanisms for torus destruction in nonsmooth systems have not yet been examined in detail. A second purpose of the present article is to initiate this analysis. Finally, we illustrate our results through a practical example from power electronics and present the first experimental verification of torus birth via a border-collision bifurcation.

This work was supported in part by the Department of Atomic Energy, The Government of India under Project No. 2003/37/11/BRNS and in part by Danish Natural Science Foundation through the Center for Modelling, Nonlinear Dynamics and Irreversible Thermodynamics (MIDIT). Figures 4, 6, 7(a), and 8 were generated with DsTool.^{59–61}

I. INTRODUCTION

II. THE PIECEWISE LINEAR NORMAL FORM

III. TRANSITIONS BETWEEN MODE-LOCKED DYNAMICS AND QUASIPERIODICITY

IV. EXPERIMENTAL CONFIRMATION

V. CONCLUSION

## Figures

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 .

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 .

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

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

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

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

(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 .

(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 .

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 .

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 .

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

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

(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).

(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).

(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.

(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.

(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.

(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.

(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.

(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.

Experimental waveforms of the converter under regular periodic operation at .

Experimental waveforms of the converter under regular periodic operation at .

(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).

(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).

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.

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.

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, .

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, .

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.

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.

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