PHYSICAL REVIEW E 84, 047201 (2011) Mixed-mode oscillations via canard explosions in light-emitting diodes with optoelectronic feedback F. Marino, 1 M. Ciszak, 2 S. F. Abdalah, 2,3 K. Al-Naimee, 2,4 R. Meucci, 2 and F. T. Arecchi 1,2 1 Dipartimento di Fisica, Universit` a di Firenze, Via Sansone 1, I-50019 Sesto Fiorentino, Firenze, Italy 2 Consiglio Nazionale delle Ricerche, Istituto Nazionale di Ottica, Largo E. Fermi 6, I-50125 Firenze, Italy 3 High Institute of Telecommunications and Post, Al Salihiya, Baghdad, Iraq 4 Department of Physics, College of Science,University of Baghdad, Al Jadiriah, Baghdad, Iraq (Received 18 February 2011; revised manuscript received 19 September 2011; published 24 October 2011) Chaotically spiking attractors in semiconductor lasers with optoelectronic feedback have been recently observed to be the result of canard phenomena in three-dimensional phase space (incomplete homoclinic scenarios). Since light-emitting diodes display the same dynamics and are much more easily controllable, we use one of these systems to complete the attractor analysis demonstrating experimentally and theoretically the occurrence of complex sequences of periodic mixed-mode oscillations. In particular, we investigate the transition between periodic and chaotic mixed-mode states and analyze the effects of the unavoidable experimental noise on these transitions. DOI: 10.1103/PhysRevE.84.047201 PACS number(s): 05.45.−a, 42.65.Sf Oscillatory dynamics in chemical, biological, and physical systems often takes the form of complex temporal sequences known as mixed-mode oscillations (MMOs) [1]. Typical time traces are characterized by a mixture of L large- amplitude relaxation spikes followed by S small-amplitude quasiharmonic oscillations, while oscillations of intermediate amplitude do not occur. Sequences of this type are ubiquitous in nature and were originally observed in chemical systems more than 100 years ago [2], with the Belouzov-Zhabotinsky reaction being the most thoroughly studied example [3–6]. More recent studies involved surface chemical reactions [7–9], electrochemical systems [10,11], neural and cardiac cells [12,13], calcium dynamics [14], and plasma physics [15], to name just a few. As some bifurcation parameter is varied, MMOs can be ordered in periodic-chaotic sequences, in which intervals of periodic states are separated by chaotic states resembling random mixtures of the adjacent periodic patterns. In other cases, these mixtures can form periodic concatenations following the Farey arithmetic and plotting of a suitably defined winding number against the bifurcation parameter leads to a devil’s staircase. Several mechanisms can be at the origin of these phe- nomena [1], for instance, the quasiperiodic route to chaos on an invariant 2-torus [16] and the loss of stability of a Shilnikov homoclinic orbit [17,18]. However, periodic-chaotic sequences and Farey sequences of MMOs do not necessarily involve a torus or a homoclinic orbit, but can occur also through the canard phenomenon [19]. Here a limit cycle born in a supercritical Hopf bifurcation experiences the abrupt transition from a small-amplitude quasiharmonic cycle to large relaxation oscillations in a narrow parameter range (canard explosions) [20]. Although this sudden transition can be easily misinterpreted as a homoclinic bifurcation, here an exact homoclinic connection to a saddle focus does not occur and therefore application of the Shilnikov theorem is not allowed. Such behavior is typical in three-dimensional (3D) multiple-time-scale dynamical systems, which can be described in terms of a fast 2D oscillatory subsystem, coupled to a slowly evolving variable acting as a quasistatic bifurcation parameter. The strong separation of time scales may induce the switch between periods of small amplitude and relaxation oscillations and makes the flow to pass very closely to the saddle-focus stationary state, thus simulating trajectories close to the Shilnikov condition. For this reason, canard phenomena in 3D systems are often referred to as incomplete homoclinic scenarios [21]. Although most of the studies of this dynamics have been carried out in chemical systems, incomplete homoclinic scenarios have been recently predicted and observed also in semiconductor lasers with optoelectronic feedback [22,23] and optical cavities with movable mirrors [24,25]. In these works, attention has been focused on the chaotically spiking regime, a special kind of MMO where large pulses are separated by an irregular number of quasiharmonic oscillations. In this Brief Report we complete the dynamical picture by extending the analysis also to regular types of MMOs. In particular, we investigate experimentally the transition between periodic and chaotic mixed-mode states and analyze the effects of the unavoidable experimental noise on these transitions. The system here considered is a GaAs light-emitting diode (LED) (with a peak wavelength of 870 nm and spectral width of ∼50 nm) with ac-coupled nonlinear optoelectronic feedback. The LED is driven by a constant positive voltage via a current-limiting resistor (3 k) in series. The output light is sent to a photodetector producing a signal directly proportional to the optical intensity in the whole operation range. The corresponding voltage signal is high-pass filtered (with a cutoff frequency of γ f ∼ 1 kHz) and amplified by means of a variable-gain amplifier characterized by a nonlinear transfer function of the form f F (w) = Aw/(1 + s ′ w), where A is the amplifier gain and s ′ is a saturation coefficient. The feedback voltage signal is then added to the dc voltage driving the LED by means of a mixer. As will be clarified later, a key element to observe MMOs in optoelectronic devices is the existence of a threshold for light emission. In semiconductor lasers this is the current value at which gain (stimulated emission) overcomes the cavity losses. In LEDs the main recombination mechanism is spontaneous emission and the emitted light is simply proportional to the current passing through the device. However, as in electronic diodes, the current-voltage characteristics of a LED are highly nonlinear. With no external applied voltage, an equilibrium 047201-1 1539-3755/2011/84(4)/047201(5) ©2011 American Physical Society