Physica D 147 (2000) 59–82 Bi-instability and the global role of unstable resonant orbits in a driven laser Thomas W. Carr a,∗ , Lora Billings b,1 , Ira B. Schwartz b , Ioanna Triandaf b a Department of Mathematics, Southern Methodist University, Dallas, TX 75275-0165, USA b Special Project in Nonlinear Science, Code 6700.3, Plasma Physics Division, Naval Research Laboratory, Washington, DC 20375, USA Received 4 June 1999; received in revised form 19 June 2000; accepted 2 August 2000 Communicated by J.D. Meiss Abstract Driven class-B lasers are devices which possess quadratic nonlinearities and are known to exhibit chaotic behavior. We describe the onset of global heteroclinic connections which give rise to chaotic saddles. These form the precursor topology which creates both localized homoclinic chaos, as well as global mixed-mode heteroclinic chaos. To locate the relevant periodic orbits creating the precursor topology, approximate maps are derived using matched asymptotic expansions and subharmonic Melnikov theory. Locating the relevant unstable fixed points of the maps provides an organizing framework to understand the global dynamics and chaos exhibited by the laser. © 2000 Published by Elsevier Science B.V. PACS: 05.45 Keywords: Resonance; Chaos; Heteroclinic; Saddle-bifurcations; Bi-instability 1. Introduction One of the major areas of research and testbeds of nonlinear dynamics has been that of lasers [1–3]. Common to many types of lasers that undergo bifurcations to chaos are optical bistability and hysteresis. Bistability results from the existence of nonlinear interactions between the electromagnetic field and population inversion, or gain. In most cases, bistability, or generalized multi-stability, has been associated indirectly with the onset of chaos. This can be seen in examples such as coupled chaotic class-B lasers [4,5], modulated class-B lasers [6–11], and fiber lasers [12]. Bistability and chaos also appear in other fields, such as subharmonic bifurcation in population biology [13], chemical kinetics [14], and neurophysiology [15], to name just a few. An outstanding problem in many of these fields is the recognition and understanding of the role of unstable orbits in connecting bistability to chaos. Bi-instability, in contrast to bistability, is the precursor to global chaos that has two fundamental frequency components formed by the merging of two basins of attraction. We define chaos that has two distinct spectral structures as a function of time as global mixed-mode chaos. Global mixed-mode chaos is observed experimentally ∗ Corresponding author. Tel.: +1-214-768-3460; fax: +1-214-768-2355 E-mail address: tcarr@mail.smu.edu (T.W. Carr). 1 ONR/ASEE Postdoc Fellowship. 0167-2789/00/$ – see front matter © 2000 Published by Elsevier Science B.V. PII:S0167-2789(00)00164-0