^{1}, Federico Bizzarri

^{2}, Marco Storace

^{3,a)}and Oscar De Feo

^{4,b)}

### Abstract

We investigate the families of periodic and nonperiodic behaviors admitted by a hysteresis-based circuit oscillator. The analysis is carried out by combining brute-force simulations with continuation methods. As a result of the analysis, it is shown that the existence of many different periodic solutions and of the chaotic behaviors associated with them is organized by few codimension-2 bifurcation points. This implies the possibility of switching between different periodic solutions by controlling only two bifurcation parameters, which makes the oscillator a possible generator of nontrivial periodic solutions suitable, for instance for actual radiofrequency identification systems applications.

The possibility of generating from tens up to thousands of different periodic solutions by using one simple circuit is a growing requirement for the electronic implementation of many of the newly proposed information and communications technologies, such as the low-cost radiofrequency identification systems. Here we analyze the peculiar dynamics of an electronic oscillator that, thanks to a hysteretic nonlinearity in its circuitry, is able to generate a larger variety of different periodic and chaotic stable solutions than other common electronic oscillators. The increase in periodic solutions results mainly from the presence of Shil’nikov-Hopf bifurcations and blue-sky catastrophes, the latter meant as nontransversal collisions between two limit cycles, which are uncommon phenomena in standard electronic oscillators. Given the larger number of periodic solutions, controlling the switches between them may become an arduous task in real applications. However, by combining brute-force simulations with continuation methods, here it is shown that the existence of such periodic solutions and of the chaotic behaviors associated with them is organized by few codimension-2 bifurcation points. This implies the possibility of switching between the different periodic solutions by controlling only two circuit (bifurcation) parameters, thus making the oscillator a possible generator of nontrivial periodic solutions suitable for actual applications.

The authors are grateful to Professor Y. A. Kuznetsov for his helpful comments about the orbit switch bifurcation. This work was supported by MIUR, within the FIRB framework (Project RBNE012NSW), and the University of Genoa.

I. INTRODUCTION

II. THE CIRCUIT

III. OVERALL RESULT OF BIFURCATION ANALYSIS AND FAMILIES OF POSSIBLE BEHAVIORS

IV. SPECIFIC BIFURCATIONS

A. Trivial (equilibria) bifurcations

B. Bogdanov-Takens: Existence of a global bifurcation

C. Orbit switch: Appearing of S-type solutions

D. Belyakov degeneracy: Organization of chaotic solutions

E. Shil’nikov-Hopf: Organization of the oscillations

F. Homoclinic bifurcations of : oscillations and symmetry breaking

1. Transverse homoclinic bifurcation: Symmetry breaking

2. A blue-sky-like catastrophe: Formation of the oscillations

G. Death of chaotic attractors

H. Global scenario: Ordering the periodic windows

V. CONCLUSIONS

### Key Topics

- Bifurcations
- 114.0
- Oscillators
- 33.0
- Periodic solutions
- 26.0
- Bifurcation theory
- 7.0
- Eigenvalues
- 6.0

## Figures

Electrical scheme of the hysteresis oscillator. (a) Linear part. (b) Hysteretic cell.

Electrical scheme of the hysteresis oscillator. (a) Linear part. (b) Hysteretic cell.

(Color online) Qualitative sketch of the overall bifurcation scenario. The parameter space is partitioned into six main regions by the following bifurcation curves: pitchfork of equilibria (supercritical below ); Hopf (supercritical below and ); supercritical Hopf; and fold of cycles; period doubling; supercritical pitchfork of cycles; , , and heteroclinic; and homoclinic; symmetry breaking; death of chaotic attractors.

(Color online) Qualitative sketch of the overall bifurcation scenario. The parameter space is partitioned into six main regions by the following bifurcation curves: pitchfork of equilibria (supercritical below ); Hopf (supercritical below and ); supercritical Hopf; and fold of cycles; period doubling; supercritical pitchfork of cycles; , , and heteroclinic; and homoclinic; symmetry breaking; death of chaotic attractors.

Families of periodic and chaotic solutions in the state space ; the scale is the same for all plots ( , , and ). Limit cycles:(a) small (visiting half the state space) S-type solutions (shown in black and gray, respectively); (b) large (visiting the whole state space) S-type (only one solution is shown); (c) F-type ; (d) F-type . (e)–(h) Chaotic solutions belonging to the families of periodic solutions shown in (a)–(d), respectively.

Families of periodic and chaotic solutions in the state space ; the scale is the same for all plots ( , , and ). Limit cycles:(a) small (visiting half the state space) S-type solutions (shown in black and gray, respectively); (b) large (visiting the whole state space) S-type (only one solution is shown); (c) F-type ; (d) F-type . (e)–(h) Chaotic solutions belonging to the families of periodic solutions shown in (a)–(d), respectively.

(Color) Brute-force simulations. (a) Bifurcation scenario [ vs ], with coloring proportional to . (b)–(e) One-dimensional bifurcation diagrams obtained along the black segments labeled correspondingly in (a). The Poincaré section chosen is .

(Color) Brute-force simulations. (a) Bifurcation scenario [ vs ], with coloring proportional to . (b)–(e) One-dimensional bifurcation diagrams obtained along the black segments labeled correspondingly in (a). The Poincaré section chosen is .

(Color online) Quantitative bifurcation diagram (obtained by numerical continuation) corresponding to most of the sketch in Fig. 2 .

(Color online) Quantitative bifurcation diagram (obtained by numerical continuation) corresponding to most of the sketch in Fig. 2 .

(Color online) Bifurcation curves for equilibria. (a) Qualitative bifurcation diagram with sketches of the bifurcating invariant sets for low values of (for high values of , refer to Fig. 7 ). Two bifurcation curves for cycles (the pitchfork of cycles and the period doubling ) are reported for the sake of completeness (dotted lines). Around the cusp points of , cycles originating from different bifurcations smoothly change into each other. (b) Corresponding quantitative curves obtained by numerical continuation and analytically .

(Color online) Bifurcation curves for equilibria. (a) Qualitative bifurcation diagram with sketches of the bifurcating invariant sets for low values of (for high values of , refer to Fig. 7 ). Two bifurcation curves for cycles (the pitchfork of cycles and the period doubling ) are reported for the sake of completeness (dotted lines). Around the cusp points of , cycles originating from different bifurcations smoothly change into each other. (b) Corresponding quantitative curves obtained by numerical continuation and analytically .

(Color online) System unfolding around the codimension-2 bifurcation point . (a) Qualitative bifurcation diagram with sketches of the bifurcating invariant sets. (b) Corresponding quantitative curves obtained by numerical continuation.

(Color online) System unfolding around the codimension-2 bifurcation point . (a) Qualitative bifurcation diagram with sketches of the bifurcating invariant sets. (b) Corresponding quantitative curves obtained by numerical continuation.

(Color online) Bifurcation diagram in the neighborhoods of the orbit switch, Belyakov, and Shil’nikov-Hopf degeneracies of the heteroclinic curve. (a) Qualitative bifurcation diagram with sketches of the bifurcating invariant sets. (b) Corresponding quantitative curves obtained by numerical continuation. (c) Typical double heteroclinic loop along the heteroclinic bifurcation curve near (left panel) and enlargements around for parameter pairs above (central panel) and below (right panel) the orbit switch point . The unstable (stable) eigendirections are marked by outward (inward) arrows. The single (double) arrow denotes the leading (nonleading) eigendirection.

(Color online) Bifurcation diagram in the neighborhoods of the orbit switch, Belyakov, and Shil’nikov-Hopf degeneracies of the heteroclinic curve. (a) Qualitative bifurcation diagram with sketches of the bifurcating invariant sets. (b) Corresponding quantitative curves obtained by numerical continuation. (c) Typical double heteroclinic loop along the heteroclinic bifurcation curve near (left panel) and enlargements around for parameter pairs above (central panel) and below (right panel) the orbit switch point . The unstable (stable) eigendirections are marked by outward (inward) arrows. The single (double) arrow denotes the leading (nonleading) eigendirection.

(Color) (a) Detail of one of the right periodic regions of the bifurcation diagram of Fig. 4 . (b) Stable periodic solutions observable for parameter pairs corresponding to the points a to h in (a). The plot scales are the same for all figures (and the same as in Fig. 3 ). (c) Blue-sky scenario, with both period and length of the limit cycles increasing, along the upper and lower part of the white segment in (a). The labels mark the points corresponding to the stable periodic solutions reported in (b).

(Color) (a) Detail of one of the right periodic regions of the bifurcation diagram of Fig. 4 . (b) Stable periodic solutions observable for parameter pairs corresponding to the points a to h in (a). The plot scales are the same for all figures (and the same as in Fig. 3 ). (c) Blue-sky scenario, with both period and length of the limit cycles increasing, along the upper and lower part of the white segment in (a). The labels mark the points corresponding to the stable periodic solutions reported in (b).

(Color online) (a) Qualitative sketch of the structure organizing the periodic windows. All *T* curves (solid, dashed, and dot-dashed) mark fold bifurcations of cycles, whereas *F* curves mark period-doubling bifurcations. The letters k indicate the left and right periodic windows for cycles with oscillations. (b) Real bifurcation curves superimposed upon a detail of the brute-force bifurcation diagram.

(Color online) (a) Qualitative sketch of the structure organizing the periodic windows. All *T* curves (solid, dashed, and dot-dashed) mark fold bifurcations of cycles, whereas *F* curves mark period-doubling bifurcations. The letters k indicate the left and right periodic windows for cycles with oscillations. (b) Real bifurcation curves superimposed upon a detail of the brute-force bifurcation diagram.

## Tables

The parameters , , and expressed as functions of the circuit parameters.

The parameters , , and expressed as functions of the circuit parameters.

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