IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 54, NO. 5, MAY 2007 1109 Inducing Chaos in Electronic Circuits by Resonant Perturbations Anil Kandangath, Satish Krishnamoorthy, Ying-Cheng Lai, and John A. Gaudet, Member, IEEE Abstract—We propose a scheme to induce chaotic attractors in electronic circuits. The applications that we are interested in stip- ulate the following three constraints: 1) the circuit operates in a stable periodic regime far away from chaotic behavior; 2) no pa- rameters or state variables of the circuit are directly accessible to adjustment and 3) the circuit equations are unknown and the only available information is a time series (or a signal) measured from the circuit. Under these conditions, a viable approach to chaos in- duction is to use external excitations such as a microwave signal, assuming that a proper coupling mechanism exists which allows the circuit to be perturbed by the excitation. The question we ad- dress in this paper is how to choose the waveform of the excita- tion to ensure that sustained chaos (chaotic attractor) can be gen- erated in the circuit. We show that weak resonant perturbations with time-varying frequency and phase are generally able to drive the circuit into a hierarchy of nonlinear resonant states and even- tually into chaos. We develop a theory to explain this phenom- enon, provide numerical support, and demonstrate the feasibility of the method by laboratory experiments. In particular, our experi- mental system consists of a Duffing-type of nonlinear electronic os- cillator driven by a phase-locked loop (PLL) circuit. The PLL can track the frequency and phase evolution of the target Duffing cir- cuit and deliver resonant perturbations to generate robust chaotic attractors. Index Terms—Duffing oscillator, inducing chaos, phase-locked loop (PLL), resonant perturbations. I. INTRODUCTION M ANY modern devices rely on sophisticated electronic circuits. An important class of these devices is electronic tracking and guidance systems. To accomplish its intended mis- sion, a hostile electronic tracking and guidance system operates in a parameter regime where their performance can be character- ized as stable or regular. If one regards the device as a dynamical system, it is required that the state variables of the system behave in a regular fashion. This is conceivable because the electronic circuits in the system, when in operation, are indeed dynamical systems that evolve state variables such as voltages and currents continuously in time. Inducing chaos in the circuits is likely to cause the system to fail in its intended mission. There have been many previous works on inducing or main- taining chaos in nonlinear systems. These can be categorized Manuscript received June 16, 2005; revised September 30, 2006. This work was supported by AFOSR under Grant. F49620-03-1-0290 and Grant FA9550-06-1-0024. This paper was recommended by Associate Editor Y. Nishio. A. Kandangath, S. Krishnamoorthy, and Y.-C. Lai are with Department of Electrical Engineering, Arizona State University, Tempe, AZ 85287 USA (e-mail: anil.kumar; satishk; Ying-Cheng.Lai@asu.edu). J. A. Gaudet is with Air Force Research Laboratory, AFRL/DEHE, Kirtland AFB, NM 87117 USA. Digital Object Identifier 10.1109/TCSI.2007.893510 into three classes: 1) inducing chaos by random noise [1]–[11]; 2) converting transient chaos into sustained chaos by small per- turbations—the problem of maintaining chaos [12]–[17] and 3) inducing chaos by resonant perturbations [18]–[23]. In the first class, the main question concerns how chaos can arise under the influence of random noise. The pioneering work of Crutch- field et al. [1], [2] established that, in the common route to chaos via period-doubling bifurcations, noise tends to smooth out the transition and induce chaos in parameter regimes where there is no chaos otherwise. The observability and scaling of fractal structures near the transition to chaos in random maps were addressed in [6], [7]. Features of transition to chaos in noisy dynamical systems, such as intermittency and the smooth- ness of the Lyapunov exponents, were also found in the tran- sition from strange nonchaotic to strange chaotic attractors in quasi-periodically driven systems [24] and in the bifurcation to chaos with multiple positive Lyapunov exponents in rela- tively high-dimensional systems [25]–[27]. More recently, the mechanism for transition to chaos in continuous-time dynam- ical systems was investigated [11] where it was found that non- hyperbolicity plays a fundamental role in shaping the transition. The second class of problems deals with systems in parameter regimes where there are nonattracting chaotic sets that physi- cally lead to transient chaos. That is, under its own evolution, from a random initial condition the system behaves chaotically only for a finite amount time before settling into a nonchaotic attractor. Since sustained chaos is believed to be beneficial in circumstances such as biological applications [13], [14], [28], it is desirable that chaos be maintained, which can indeed be achieved by applying small perturbations to an available param- eter or state variable of the system [12], [16], [17]. For the third class, previous works demonstrated that for simple nonlinear oscillators such as the Duffing system, resonant perturbations can be used to drive the system in and out of chaotic motion [18]–[23], provided that the system equations are known so that the external excitation can be designed accordingly. The problem of inducing chaos that we wish to address in this paper is significantly more challenging than those investi- gated previously. In particular, we are interested in applications where the following three constraints naturally arise: 1) The electronic circuit to be defeated operates in stable state that is far away from any chaotic dynamics; 2) the internal structure and parameters of the circuit cannot be modified, i.e., no param- eters or state variables of the circuit are directly accessible to adjustment and 3) the system equations are unknown and only a measured signal (time series) from the system is available. In previous works, however, not all these three requirements were assumed. For instance, in the work of maintaining chaos [14], [17], [18], although system equations were unknown, the 1549-8328/$25.00 © 2007 IEEE