Nonlinear Dynamics and Chaos: From Concept to Application

Top: Phase portrait of a radio frequency chaotic oscillator developed at Aerospace. Bottom: Example implementation of a cellular nonlinear network—essentially a programmable analog computer made up of a locally coupled array of nonlinear dynamical systems.

Top: Phase portrait of a radio frequency chaotic oscillator developed at Aerospace. Bottom: Example implementation of a cellular nonlinear network—essentially a programmable analog computer made up of a locally coupled array of nonlinear dynamical systems. Courtesy of IEEE.

Traditionally, engineers sought to minimize noise and distortion in system designs. Today, Aerospace scientists are seeking to explicitly harness these effects for useful engineering purposes.

Christopher P. Silva

 

The vast discipline of nonlinear engineering divides into two complementary practices: one that pursues the elimination of undesired nonlinear effects, and one that seeks to harness nonlinear effects for useful engineering purposes. The first practice involves either the characterization and elimination of unwanted nonlinear distortion, or the prevention of unwanted and possibly damaging anomalous behavior in nonlinear circuits and systems. This practice primarily affects current or near-term systems. The second practice seeks to develop whole new design methodologies and technologies, which in the long term may lead to future advanced systems that may be quite different from what now exists.

The most studied nonlinear phenomenon is the complex, random-like behavior called chaos, which is being addressed in fields ranging from astronomy to zoology. The Aerospace Corporation is researching how to harness the effects of chaotic signals and apply them to communication systems for military applications. Some of the most pressing issues involve privacy and security for processing communication signals. There are also potential applications involving radar and sonar, with important implications for addressing such challenges as urban warfare and the remote detection of improvised explosive devices and suicide bombers. The foundation for this new engineering practice is the dynamical system perspective, with its accompanying set of powerful analysis machinery. Overall, this practice is evolving on many fronts and levels, reaching a state of maturity where it can be applied to real-world problems.

Nonlinear Engineering

The practice of electrical engineering has been dominated by a linear paradigm that has well served the needs of communications signal processing functions. The techniques are well established and mature and solve a large class of problems, such as linear filters, for example. These techniques are based on the classical superposition principle, which states that the response of a given system to a sum of stimuli is given by the sum of the responses to each stimulus acting alone. This view provides a first-order approximation of a naturally nonlinear world. Hence, an engineer can create designs that are intentionally linear knowing they will obey these simple principles. Established practice has also dictated that any higher-order nonlinear effects that result from the violation of the superposition principle could be safely ignored. Such effects were called noise and distortion, and have traditionally been treated more like an oddity and nuisance rather than an inherent and possibly useful feature of nature. The practice of working within a linear paradigm, however, does not allow for important, common, and explicitly nonlinear signal functions such as frequency generation, frequency synthesis, and power amplification. It is these required functions that made up the first elements of nonlinear engineering(see sidebar, Dynamical Systems).

An illustration of the principle of superposition that marks the delineation between linear and nonlinear systems.

An illustration of the principle of superposition that marks the delineation between linear and nonlinear systems.

The further development of nonlinear techniques primarily occurred in academia, and new discoveries found little practical application. However, within the last two decades, there has been a revolution of sorts that stems from three fundamental factors. These factors have synergistically acted to radically evolve and change the practice of nonlinear engineering.

The first factor arises from the demand for increased performance in limited-bandwidth channels in communication systems. As a consequence, nonlinear effects can no longer be ignored in the design of these systems, requiring techniques that are not simple extensions of linear theory. A showcase example is wideband communications for advanced military satellites. Here, there is often a “bent-pipe” architecture in which the transponder high-power amplifier is operated in or near its nonlinear saturated region to maximize power efficiency. But this gain in efficiency is offset by increased distortion in the modulated signals that pass through the amplifier. The distortion is exacerbated by the complexity of the modulations needed to attain high bandwidth efficiencies, because they often contain amplitude variations that elicit these added distortions. In this case, an accurate and formal identification of the amplifier must be accomplished before an effective nonlinear compensation strategy can be developed. There is a significant amount of activity in this arena fueled by the demand for personal wireless communications.

Third-order representation of a continuous dynamical system, where the phase-space trajectory is dictated by the vector field F. For a discrete dynamical system, the trajectory becomes a set of distinct points dictated by a state-transition map Φ.

Third-order representation of a continuous dynamical system, where the phase-space trajectory is dictated by the vector field F. For a discrete dynamical system, the trajectory becomes a set of distinct points dictated by a state-transition map Φ.

The second factor involves several seminal discoveries of nonlinear effects that prompted a flurry of research in applying them to communications signal processing, as well as to many other disciplines. In essence, a new modeling and analysis language emerged that captured a much larger portion of the complexity of nature. This was an about-face from the dominant engineering mindset in that these nonlinear effects were now being sought for their application potential. It marked the beginning of the second branch of nonlinear engineering, in which whole new designs would be sought based on these new effects. These discoveries have primarily occurred in the active fields of chaos, fractals, and wavelets(see sidebar, Fractals). Unlike the linear case, this activity is still relatively immature and presents a wide-open frontier for new practitioners. Because the nonlinear methodology provides a higher-order view of nature in which the superposition principle does not hold, it is typically a large leap beyond linear thinking, involving much more complex analysis on small classes of problems. This difficulty is offset by the tremendous application potential and importance of nonlinear effects.

The third factor has been the rapid development of computational power that is imperative for nonlinear study and application. The nonlinear field is characterized by complex problems, most of which do not have closed-form solutions and must be addressed qualitatively and numerically. Coupled with the qualitative arsenal of tools from the discipline of nonlinear dynamics, the computer offers a means to perform nonlinear experiments on the desktop, thereby providing the insight and knowledge needed to reduce nonlinearity to a beneficial practice.

One-dimensional logistic map example of a discrete dynamical system governed by Xn+1 = f(Xn) = µ Xn(1 − Xn ). Top: Simple asymptotic behavior (convergence to stable fixed point Xe) found with the bifurcation parameter µ set below 3. Bottom: Complex chaotic behavior observed with µ = 4 (where Xe is now unstable).

One-dimensional logistic map example of a discrete dynamical system governed by Xn+1 = f(Xn) = µ Xn(1 − Xn ). Top: Simple asymptotic behavior (convergence to stable fixed point Xe) found with the bifurcation parameter µ set below 3. Bottom: Complex chaotic behavior observed with µ = 4 (where Xe is now unstable).

Introduction to Chaos

Chaos and Bifurcation

One of the most well known and potentially useful nonlinear dynamical effects is the bounded, random-like behavior called chaos—in essence, deterministic noise. Chaos has been found in a myriad of dynamical systems and in frequency ranges from baseband to optical. This phenomenon, along with its closely related cousin called the fractal and the mathematical tool called wavelets, offers a new paradigm for understanding and modeling the world. It stems from the underlying principle of self-similarity at different scales, which appears to be a ubiquitous property of nature.

There are three basic dynamical properties that collectively characterize chaotic behavior. First, it exhibits an essentially continuous and possibly banded frequency spectrum that resembles random noise. Second, it is sensitive to initial conditions—that is, nearby orbits in the phase space (a geometrical perspective in which the dynamical states are plotted against each other so that time becomes implicit) diverge rapidly. Third, it contains an ergodicity and mixing of the dynamic orbits, which in essence implies the wholesale visit of the entire phase space by the chaotic behavior and a loss of information because of the loss of predictability.

Chaotic behavior can only arise in dynamical systems that are nonlinear, although these systems may be continuous or discrete, with or without dissipation. The behavior can also be transient or steady-state in nature and typically arises after a sequence of qualitative changes in behavior as a function of one or more parameters (termed “bifurcations”). A predominant manifestation of chaos occurs in the steady state of dissipative systems and is termed a “strange attractor” because its topological structure is complex and it attracts outside orbits.

Chaotic Synchronization

The classical synchronization or entrainment of periodic oscillators has been known since at least the seventeenth century, when Christiaan Huygens observed the coupled form of this phenomenon in adjacent clocks on a wall. The driven or injection form of synchronization was discovered later with the observation that a small periodic forcing signal could cause the large natural resonance of a system to lock to it. What was unexpected was that a similar phenomenon could be had with chaotic signals, especially given their distinctive bounded instability character. The discovery of the driven form of chaotic synchronization was announced in 1990, marking a turning point in the investigation of chaos for communication systems, for it allowed chaos to be modulated and demodulated like a generalized carrier.

There are five basic chaotic synchronization techniques, all of which relate to communication applications that are generic across national security space programs:

  • Master-slave synchronization. This was the earliest discovered version of chaotic synchronization. It occurs when an autonomous (that is, unforced) system unidirectionally drives a stable subsystem.
  • Nonautonomous synchronization. Here, a nonautonomous (that is, forced) system unidirectionally drives a stable identical nonautonomous system. This form is known to be quite robust against link interference.
  • Inverse system synchronization. In contrast to nonautonomous synchronization, inverse system synchronization occurs when the receiver is a formal dynamical inverse of the transmitter that will reproduce the latter’s forcing function.
  • Adaptive control synchronization. By far the most prolific class of synchronization approaches, this is based on the numerous variants of adaptive control for chaotic systems (also known as “control chaos”). In fact, these techniques have demonstrated some capability (although easily defeated) of extracting information from unknown systems, or even making distinctly different dynamical systems synchronize, thereby possibly weakening the security claims often made for chaos-based communications. These techniques can also make the other forms of chaotic synchronization more suitable for practical implementation—for example, where there are link degradations and parameter mismatches.
  • Coupled synchronization. This consists of bidirectionally coupled identical systems and is a simple generalization of the traditional classical form involving sinusoidal oscillators.

The first four forms of chaotic synchronization are suitable for standard communications purposes, while the fifth is suitable for network communications. It is also preferable that the linking signal between the component systems be of the scalar variety. Because of the newness of these discoveries, many studies are still needed to address important engineering and operational issues, and to compare findings with traditional synchronization approaches.

Implications for Engineering Applications

Until about the mid-1980s, chaos was primarily studied by physicists and mathematicians in a theoretical sense. These studies sought to determine, understand, and report the many unique properties of chaos and to analyze and predict its behavior. During this period, and especially after the pivotal discovery of chaotic synchronization in 1990, researchers began to pursue practical applications. These investigations naturally focused on applications that could exploit the deterministic, yet random-like behavior of chaos, particularly with regard to inherently secure signal processing and transmission.

Over the years, many other unique effects with their own engineering implications have been discovered and investigated in chaotic systems (e.g., nonlinear amplification with signal enhancing/extracting capabilities). The bifurcation aspects of nonlinear systems provide a profound and critical insight into current circuit and system designs, especially as it applies to their fundamental stability. As a consequence, the investigation of the engineering applications of nonlinear dynamics and chaos has become a vast, rapidly maturing, and multidisciplinary undertaking, especially on an international scale.

Schematic illustrations of chaotic synchronization. Top: Original master-slave form, where the w' subsystem will lock to the w subsystem when driven by the v subsystem. Bottom: Third-order, scalar-driven example of its nonautonomous manifestation, where the receiver forcing frequency ωr and phase Φr must be tuned to achieve lock.

Schematic illustrations of chaotic synchronization. Top: Original master-slave form, where the w’ subsystem will lock to the w subsystem when driven by the v subsystem. Bottom: Third-order, scalar-driven example of its nonautonomous manifestation, where the receiver forcing frequency ωr and phase Φr must be tuned to achieve lock.

Chaos-Based Communications

Modulation Approaches

A whole range of methods have been proposed for implementing and demonstrating analog or digital modulation of information using a chaotic carrier. Such modulations range from simple addition to more complex combinations of information with the carrier that is much more indirect and subtle than the traditional amplitude, phase, or frequency modulation of a classical sinusoidal carrier. Because of the complexity of the carrier and the modulation, these approaches can provide privacy and security for communications without even encrypting the information. In essence, the information is hidden in the “noise” in transmission and can be extracted in the receiver using the inherent determinism of chaos.

Major chaos-based modulation methods being investigated and developed internationally for communication applications include:

  • Additive chaotic masking. This was the earliest form of modulation, wherein the information is added to the carrier as a small perturbation and usually demodulated using a cascaded form of master-slave synchronization.
  • Chaotic switching. In this chaos-based version of traditional digital modulation, an analog signal of finite duration represents a digital symbol consisting of one or more bits. In this case, the digital symbol is uniquely mapped to an analog waveform segment coming from distinct strange attractors (known also as “attractor-shift keying”), or an analog waveform segment from a distinct region of a single strange attractor, thereby forming a chaotic signal constellation.
  • Forcing function modulation. In this approach, a sinusoidal forcing function in a nonautonomous chaotic system is analog or digitally modulated with the information in a classical manner, with the transmitted signal being some other state variable. This modulation typically involves the nonautonomous or inverse synchronization methods and is the basis for the Aerospace development effort addressing high-data-rate, chaos-based communications.
  • Multiplicative chaotic mixing. This can be considered the chaos-based version of the traditional direct-sequence spread-spectrum approach, except in this case, the receiver actually divides by the chaotic carrier to extract the original information.
  • Parametric modulation. In this case, the information directly modulates a circuit parameter value (such as resistance, capacitance, or inductance), and some state variable from the chaotic system is sent that contains the information in a complex manner. As with forcing function modulation, this is an indirect modulation approach that typically offers higher levels of privacy and security and can also provide chaotic multiplexing capabilities, wherein two or more messages can modulate different circuit parameters and be sent and recovered using one transmission signal.
  • Independent source modulation. This is another indirect modulation form where the information becomes an independent voltage/current source that is inserted in the chaotic transmitter circuit.
  • Generalized modulation. This form involves the generalization of additive masking/multiplicative modulation, where the information and chaotic carrier are combined in a more general invertible manner.
Example of chaotic masking modulation, one of several means for chaotic communications. Top: System configuration, where a cascade of master-slave synchronization systems in the receiver regenerates the chaotic carrier x. Bottom: Experimental results for speech.

Example of chaotic masking modulation, one of several means for chaotic communications. Top: System configuration, where a cascade of master-slave synchronization systems in the receiver regenerates the chaotic carrier x. Bottom: Experimental results for speech.

Survey of Baseband to Optical Systems

As chaos-based communication options were being explored in the early 1990s, a whole series of baseband communication links were demonstrated by simulation and experiment. These were primarily proof-of-concept exercises using simple modulation signals ranging from tones to speech, and were based on the various forms of chaotic synchronization and modulation. The baseband nature of these developments was primarily forced by the abundance of chaotic generators in the low-frequency range and the ease by which practical circuits can be implemented in this regime.

Some of the advantages and features of chaos-based communications that were proposed during these early studies included digital and analog implementations that synchronize more rapidly, robustly, and simply because of their natural dynamical properties; unique analog communications capabilities such as privacy, low probability of intercept (LPI), low probability of detection (LPD), and frequency reuse that are of interest to the military; and other unique signaling functions not possible with digital techniques, such as indirect chaotic modulation for enhanced security and multiplexing, chaotic signal constellations allowing for direct high-power transmitters, noise reduction with cascaded receivers, and spatial security using a ring of transmitters. These development efforts are still at an early exploratory stage.

The first reported example of chaos-based communications used a cascaded form of master-slave synchronization and additive chaotic modulation. The cascading was needed to locally and coherently regenerate the chaotic carrier. This regeneration was found to be quite resilient to noise and interference added to the linking channel, as would be needed for a practical communication system. In this case, the chaotic carrier was modulated by adding a voice message at a much lower level, and was recoverable because of the regenerated chaotic carrier. The message was buried in the “noise” when viewed in the communications channel, indicating how this approach can possibly provide for private transmissions. One must be careful about making such claims, however, because it was later shown that the additive modulation scheme is easily deciphered using so-called de-embedding techniques. These techniques have yet to be applied to traditional digital encryption schemes, which can be thought of as sophisticated mappings of the plain text.

The more sophisticated example of parametric chaotic modulation cannot be imitated by traditional modulation approaches and is much more secure. It occurs when the message modulates a chosen circuit parameter in the system, which in turn influences the state variables of the system in a complex manner. Because the state variables, or combinations thereof, are the signals sent across the communication channel, the manner in which the original message is embedded in this signal is extremely complex and thus provides a first-tier level of security without encrypting the message.

Some design forms of chaos-based communication can be used for baseband communications, whereas for radio-frequency (RF) or microwave communications, these schemes must be combined with traditional carriers and modems. In both cases, the bandwidth of the information is limited to tens of kilohertz; in the latter case, there is an additional loss of LPI capability. Similar to synchronization, there were several important engineering issues that needed to be addressed before operational application could be considered for these new communication approaches.

By the early 2000s, there was a rich set of results for chaos-based communication using more sophisticated techniques (such as adaptive receivers, pulse-position modulation, analog code-division multiple access and spread spectrum), providing more capabilities (such as supporting multiple users and suppressing multipath and jamming interference), and addressing engineering concerns (such as filtering, delay, parameter mismatch, and added channel noise). Despite these advances, the data throughput of such systems remained relatively low because of the bandwidth limitations of their constituent chaotic generators. However, since the early 2000s, the evolution of chaos-based communications has steadily continued, with advances in frequency range, data throughput, and synchronization/modulation techniques.

One example of this evolution involves the progression of chaos-based communications from the RF/microwave arena into the optical range. The motivations for the optical case were similar to those for the RF/microwave regime—namely, bandwidth efficiency, multiuser capabilities, natural large-signal operation, privacy, and security. High-dimensional chaotic behavior is quite easily generated in optical systems using optical injection, opto-electronic feedback, or optical cavities. However, the range of synchronization and modulation methods is more limited than in the RF/microwave case, as was made evident from a 2001–2004 European project called OCCULT (Optical Chaos Communications Using Laser-Diode Terminals). The initial laboratory demonstration sustained a data rate of 3 gigabits per second (Gbps) with a respectable 7 × 10−9 bit-error rate using a high-dimensional chaotic additive masking modulation. The demonstration was later repeated over a large commercial fiber-optic network in Athens, Greece, in 2005, with sustainable data rates of 2.4 Gbps and a similar acceptable bit-error rate.

The second example represents a commercially developed application of chaos-based techniques to the rapidly developing ultrawideband radio services in the 3.1–10.6 gigahertz (GHz) frequency band. In this standard, the minimum or typical communications bandwidth is 500 megahertz (MHz) to 2 GHz, with only a power-spectral-density mask specified and not the carrier or modulator type. The low-data-rate version requires low power consumption, low complexity, low cost, location awareness, high reliability, ad hoc networking capability, and a range of less than 100 meters or so. The dominant implementation for such radios uses a complementary metal-oxide semiconductor (CMOS) system-on-a-chip architecture. The candidate signal sources include impulse, chirp, and chaotic. A 2007 study showed that a direct chaotic approach could be simpler and less expensive than any conventional approach, with comparable bit-error-rate performance. As a consequence, Samsung in South Korea developed a chaotic ultrawideband radio transceiver on a 0.18-micron CMOS integrated circuit. The transceiver used a tunable chaotic signal source, which allowed agile changes in bandwidth and center frequency, and was based on the summation of noncommensurate triangular pulses. The unit was successfully demonstrated at a data rate of up to 15 megabits per second (Mbps).

More recently, in December 2009, the U.S. Army Research Laboratory funded an effort to explore chaos-based communications for satellite applications. Initial work is focused on developing an LPI/LPD chaotic modem system that will be suitable for satellite communications in the X, Ku, and Ka frequency bands.

Aerospace RF/Microwave System

Schematic diagram of the optical chaos modem used in the European Union OCCULT demonstration project based on traditional laser diodes and Mach-Zehnder modulators.

Schematic diagram of the optical chaos modem used in the European Union OCCULT demonstration project based on traditional laser diodes and Mach-Zehnder modulators.

Aerospace has been investigating high-frequency, high-capacity, chaos-based communications systems as alternatives to classical digital systems. The research strategy has consisted of three development phases focusing on oscillation, synchronization, and modulation.

Chaotic ultrawideband radio transceiver developed by Samsung of South Korea. Top: Transceiver architecture. Bottom: Detail on flexible chaotic signal source. Abbreviations: BBA, broadband amplifier; LNA, low-noise amplifier; BPF, bandpass filter.

Chaotic ultrawideband radio transceiver developed by Samsung of South Korea. Top: Transceiver architecture. Bottom: Detail on flexible chaotic signal source. Abbreviations: BBA, broadband amplifier; LNA, low-noise amplifier; BPF, bandpass filter.

Oscillation

The first research phase sought to develop a high-frequency, broadband chaotic oscillator that would be the building block of the project. This phase was challenging because of the frequency-dependent issues that naturally arise in creating such a broadband oscillator, and because there were relatively few systematic approaches for designing such oscillators at the time.

The first successful realization employed the simple baseband circuit known as Chua’s oscillator. This circuit has become a paradigm for chaos because of its generality and simplicity; its generality stems from its ability to formally realize a whole spectrum of qualitative behaviors, while its simplicity derives from the fact that it is third-order (the minimum for a continuous system) and completely linear except for a nonlinear resistor with a piecewise-linear current-voltage characteristic (the simplest form of nonlinearity). The oscillator consists of a passive portion, which is easy to scale up in frequency, and an active portion, called a negative-resistance generator, which was needed to realize the piecewise-linear resistor. The negative-resistance generator was synthesized in such a way as to allow for the tuning of the breakpoints and slopes of the resistor—an important feature needed for synchronization purposes that would be much more difficult if a general nonlinear characteristic were used instead (for example, as found in a tunnel diode). The attempt to realize the negative-resistance generator at high frequencies brought out frequency-dependent parasitic and delay effects that transformed the intended piecewise-linear resistor into a partially reactive element. This transformation essentially destroyed the strange attractor observed at baseband. Subsequent studies found that this implementation approach could not tolerate even small delays, so that an alternative methodology was required for the desired high-frequency operating range.

Chua’s canonical piecewise-linear (PWL) circuit chosen as the basis for a high-frequency chaotic communications link. Top: Representative PWL resistor current-voltage (I-V) characteristic. Bottom: Circuit diagram.

Chua’s canonical piecewise-linear (PWL) circuit chosen as the basis for a high-frequency chaotic communications link. Top: Representative PWL resistor current-voltage (I-V) characteristic. Bottom: Circuit diagram.

This alternative strategy led to a robust solution, and marked a radical shift from an autonomous to a nonautonomous approach. It also led to a U.S. patent for the so-called Young-Silva chaotic oscillator (YSCO). Two implementation topologies were developed for this oscillator—a series topology based on a controlled voltage source, and a dual parallel topology based on a controlled current source. There are several advantages of this new oscillator implementation. First, the oscillator designs are more forgiving with respect to delays and parasitics, because it is not necessary to realize a negative resistance as in the former, unforced case. Second, the unforced part of the circuit can be second order and hence easy to realize in the microwave regime (using an inductor/capacitor or cavity resonator). Third, the nonautonomous form of synchronization is also quite robust at baseband against interference in the channel—a desirable feature for communications applications. Finally, the system naturally provides for phase modulation of the forcing functions, which again translates into a complicated modulation of the chaotic carrier and hence potentially enhances message security. This implementation also provides several unique and useful features. For example, the shape of the frequency spectrum can be readily controlled by varying the amplitude and frequency of the forcing function. Future development plans include implementing these designs in an integrated circuit to make them more robust and reduce propagation delays, the latter of which leads to higher chaotic bandwidths.

Synchronization

The second research phase examined an inverse system approach to synchronization. This is a less commonly studied approach that extends the well-known concept of inverse linear systems in control and filter theory. In contrast to many of the other synchronization approaches, the receiver here is quite different from the transmitter, with the goal of producing a faithful replica of its forcing function. For communications applications, the forcing function is modulated in a classical style as a subcarrier, with a chosen state variable signal sent across the channel. The reproduced modulated forcing function is then demodulated to arrive at the transmitted message. The synchronization process is again a dynamical one, and general design and circuit implementation methodologies exist for the inverse system (continuous or discrete). This approach is also quite suitable for analog or digital data encryption, corresponding to more conventional self-synchronizing stream ciphering. Inverse chaotic synchronization was demonstrated for the series YSCO (baseband and RF versions) using high-fidelity SPICE (Simulation Program with Integrated Circuit Emphasis) simulations, and both versions of the inverse series YSCO have been constructed, with initial success demonstrated in their hardware synchronization.

duffing-oscillator

Modulation

The third research phase seeks to demonstrate a working communications link using either analog or digital information with a substantial data bandwidth or rate. A phase or frequency modulation of the forcing function has been chosen. Progress in this phase has included a successful MATLAB-based simulation of a complete YSCO-based communications link, which included amplitude, frequency, and phase modulation of the forcing function, as well as the development and field demonstration of a series RF YSCO frequency-modulated transceiver operating at a 1 GHz center frequency that showed the covert nature of the transmitted signal spectrum under simple tonal modulation. The DOD has expressed interest in the latter demonstration because of its suitability for covert battlefield links, such as for remote soldier physiological monitoring and drug delivery and autonomous enemy vehicle tracking. More recently, there have been several additional successful experimental demonstrations of analog and digital modulation using a series RF YSCO.

Circuit board implementation of a series YSCO driven at 100 megahertz.

Circuit board implementation of a series YSCO driven at 100 megahertz.

Chaos-Based Radar

Basic Principles and Unique Advantages

The basic principle of radar is to bounce an electromagnetic signal off a target to determine its location, direction of movement, and other properties. The range or distance of the target is determined from the delay between the transmitted and received signals, while the Doppler shift in these signals indicates the velocity of the target relative to the radar. The target’s direction of movement is found using continuous illumination and the angle of arrival of the return signal’s wavefront.

There are two basic modes of radar transmission, continuous and pulsed. Because the power of the received signal is inversely proportional to the fourth power of the range, pulsed transmission is often used so that coherent signal averaging can be done to reduce the effects of noise. Various classes of designed waveforms have been developed, ranging from simple sinusoids to modulated forms to more complex signals. Each type of waveform provides different target information and resolution. For example, only velocity is provided by a sinusoidal waveform, with an accuracy that is set by that of the sinusoidal source.

A schematic illustration of the inverse system form of chaotic synchronization that is natural for forcing function modulation.

A schematic illustration of the inverse system form of chaotic synchronization that is natural for forcing function modulation.

Motivated by many of the unique characteristics offered by chaotic signals, researchers naturally began to consider their use for radar applications. In this case, the use of such a noiselike signal would serve as a deterministic alternative to what is called random-signal or noise radar, which has been under development since the late 1960s. For both deterministic and random noise, the goal is to arrive at stealthy, LPI/LPD radar that also provides range and rate resolutions closely matching those for the ideal, but impractical, white-noise signal. These aspects are clearly advantageous for military applications because enemy targets are being continuously scanned with noise, yet their position and velocity are being determined quite accurately. The range and rate resolution properties of a given radar waveform are often illustrated in ambiguity diagrams for traditional (nonrandom), chaotic, and ideal-noise radar signals.

Why use chaotic signals instead of random noise for radar, given that they both produce desired LPI and resolution features? The answer lies in the inherent determinism of chaos that gives rise to several distinct advantages. First, simple, low-power, lightweight, compact, broadband implementations can be readily developed because such complex waveforms can be easily generated. Second, one can go well beyond the dominant correlation-based receiver signal processing found in traditional radars by exploiting the synchronization capability of chaotic signals. Such processing could very well be improved with respect to its throughput, not to mention the possibility of securely modulating the radar waveform with information that can be used to better probe the nature of the target. Third, through the natural and rapid loss of correlation found in chaotic systems, multiple radars can operate in the same frequency bands and physical locations, yet still have the return signals separable for individual processing (termed “electromagnetic compatibility”). This signal separation could be quite advantageous for processing the returns from an array of radar transmitters, like those proposed for traditional radar signals, and will also ensure multipath immunity for the radar return signals, as has been demonstrated for chaos-based communications. In the same way, some immunity to intentional jamming or other interference could be gained—similar to what a spread-spectrum system does for communications. Finally, there is great flexibility in the design of the radar waveform if it is based on a continuous or discrete chaotic dynamical system. Such flexibility can be used to arrive at very flat spectra for the time-domain waveform and sharp peaks in the ambiguity diagram at the desired target range. For example, it has been demonstrated that with a properly optimized chaotic map, one can achieve better range ambiguity properties than by using filtered Gaussian noise. For national security space systems, this has several implications for improved radar performance.

Survey of Radar/Sonar Developments

The notion of applying chaotic signals to radar applications dates to the mid-1990s, when the generation of such waveforms was already well evolved. Two separate paths developed on how to approach the transmitting side of the system. In one approach, the chaotic signal was generated at low power levels using an analog or digital circuit and then amplified using a traditional high-power amplifier of the solid-state or traveling-wave-tube variety. The other approach pursued the development of direct high-power and high-frequency chaotic sources, up to the millimeter-wave region, using a wide variety of vacuum electronic devices.

Researchers working on these efforts developed a chaotic radar system using a self-mixing or autodyne effect in the chaotic generator that gave rise to a novel return-signal processing method. Beyond these developments, optical chaos coming from nonlinear laser dynamics has been proposed and investigated for developing a chaotic LIDAR (light detection and ranging) system with bandwidths over 10 GHz. In essentially all of these systems, the return signal processing did not exploit the determinism of the chaotic signal via some synchronization approach, but instead used traditional correlation-based processing. In addition to these over-the-air radar systems, there has been some activity since the mid-2000s proposing and investigating the naval application of chaotic sonar.

There has been a rich panorama of applications for these radar systems. Examples include:

  • Vehicular collision avoidance and ranging. This application area seeks to take advantage of the compactness, efficiency, and low cost of chaotic signal generation, as well as its natural electromagnetic compatibility and multipath reduction capabilities. The latter benefit has been shown to be superior to that obtained from conventional direct-sequence spread-spectrum approaches.
  • Imaging radar for security surveillance. This application area harnesses the broadband nature that chaos can readily provide, thereby delivering such features as good penetration into walls, high range resolution, and discrimination of closely spaced targets. Imaging performance for such systems has been proven to surpass more conventional time-modulated ultrawideband radars by reducing false alarms and imaging closely spaced targets obstructed by walls.
  • Other potential applications. Other possibilities include navigation systems where high range or velocity resolutions are required, obstacle approach or intrusion sensor systems, and forward-looking aircraft radar to allow for all weather flying and landing.
Recent results from the Aerospace chaos-based communications development showing a transmitted signal spectrum (blue curve) and recovery of the hidden, modulated forcing function located at 100 megahertz (green curve).

Recent results from the Aerospace chaos-based communications development showing a transmitted signal spectrum (blue curve) and recovery of the hidden, modulated forcing function located at 100 megahertz (green curve).

Aerospace Wideband Radar System

Encouraged by successful research into chaos-based communications, Aerospace started investigating chaos-based radar. The basic objectives have been to develop a wideband, continuous-wave system with time-domain correlation processing and to determine its LPI and resolution capabilities. The effort has consisted of two successive undertakings: an initial study, which used the series YSCO as the analog chaotic generator, and a more mature study, which used an optimized discrete chaotic map. In both cases, proof-of-concept demonstrations were carried out, including a successful field demonstration that is undergoing external follow-up for product development and operational field testing.

The first effort was a 2005–2006 Harvey Mudd College Engineering Clinic project supported by the Aerospace Corporate University Affiliates Program. It focused on the use of a series YSCO that provided a robust chaotic signal with a bandwidth of around 150 MHz. The chaotic signal was filtered to improve its autocorrelation properties and to remove the forcing function signature in the frequency spectrum. The subsequent signal was frequency modulated onto a sinusoidal carrier in the range of 1–3 GHz, and the transmitter and receiver were switched on and off to pulse the radar signal and prevent damage to the receiver when transmitting. The determination of a known cable delay using the system without frequency modulation was demonstrated, followed by some initial attempts with the frequency modulation turned back on.

The next effort—a three-year independent research project—began with an evaluation of the characteristics of the transmitted frequency-modulated signal for the Harvey Mudd effort, which ranged in bandwidth from 380 to 520 MHz. The study revealed several shortcomings associated with the hardware implementation. As a result, attention next turned to the use of discrete chaotic maps as the signal generator, which offered much more flexibility in the waveform design while significantly easing bandwidth implementation issues. This also took advantage of the ready availability of commercial high-speed arbitrary waveform generators and digital storage oscilloscopes to provide the critical transmitter/receiver functions. A series of iterations led to a mature, bistatic, continuous-wave radar prototype having a bandwidth of more than 5 GHz with a 21 GHz carrier. It used a classical Bernoulli map—whose output was conditioned like that for the series YSCO (albeit much more easily using the software filtering capabilities of the arbitrary waveform generator)—to arrive at desired spectral and autocorrelation properties.

Comparison of radar ambiguity diagrams for selected signals. Top: Classical pulsed, linear frequency-modulated chirp case exhibiting multiple ambiguous resolution cells. Middle: Continuous Lorenz chaos case with only one resolution cell. Bottom: Ideal white noise case, providing an ideal thumbtack resolution cell.

Comparison of radar ambiguity diagrams for selected signals. Top: Classical pulsed, linear frequency-modulated chirp case exhibiting multiple ambiguous resolution cells. Middle: Continuous Lorenz chaos case with only one resolution cell. Bottom: Ideal white noise case, providing an ideal thumbtack resolution cell.

Another area where chaos-based techniques are being explored at Aerospace involves the challenges of countering improvised explosive devices (IEDs), which have been quite successful despite their low-tech nature. IEDs are often placed at the side of roads or embedded in them, with numerous techniques for setting them off, including simple command wires, wireless phones and remotes, and pressure-sensitive detonators. Aerospace research showed that chaos-based techniques could be used against these threats—for example, with the application of chaotic radar to detect IEDs, pressure plates, and their command wires. Under the direct support of a special corporate initiative (administered by Aerospace Intellectual Property Programs), the work has focused on an application-specific demonstration that targets the detection of command wires and suicide bombers. This chaotic radar system was upgraded to include a pan/tilt scanning capability, detection algorithms to reduce the probability of false alarm, and a graphical user interface. Field-testing of this system was conducted twice in special exercises at Camp Roberts in Paso Robles, California. The system was found to be highly effective in the detection of command wires under various conditions such as lying on gravel, hard-baked dirt, and asphalt roads. This detection continued to exhibit a low probability of false alarm even for the relatively high level of clutter or interference found with gravel. Follow-on efforts are now focused to commercially develop an operational-quality demonstration suitable for hosting on unmanned aerial or ground vehicles. Aerospace is also planning to further refine the system to exploit the full benefits of the chaos-based approach, addressing such topics as chaotic source optimization and high-speed circuit implementation, synchronization-based signal processing, the addition of signal modulation, and the use of phased-array or multiple transmitter/receiver configurations. There is also interest in applying this system to 2-D/3-D remote imaging for urban warfare environments.

Other Application Areas

Beside the two basic application areas of communications and radar reviewed here, there exists a whole range of other important nonlinear-effect-based applications that have been, and continue to be, investigated. Many of these have critical implications for national security space, with some representing disruptive capability or performance improvements, and they are all being actively pursued on an international level. Examples include pseudorandom-sequence generation for traditional digital cryptography systems, spread-spectrum systems, and multiuser communication systems; analog and digital encryption for 1-D signals, images, and real-time video; stochastic resonance for signal-in-noise enhancement (noise can be beneficial); cellular nonlinear networks for analog signal/image processing (essentially an analog computer with the equivalent of terabits-per-second processing capability); bifurcation engineering involving analysis, control, and exploitation (which encompasses the nonlinear stability issue and whole new signal processing designs); control and anticontrol chaos to eliminate or create chaos in systems (e.g., improved chemical mixing via anticontrol chaos); and chaos-based electronic measures for jamming wireless transmissions and damaging enemy circuits. A technology survey, status, and implications evaluation for these other application areas may be presented in a follow-up article in Crosslink.

Camp Roberts chaos-based-radar field demonstration details and results. Top: Overall gravel road scene and magnified view, the latter indicating the difficulty in visually seeing the command wire. Bottom: Radar return correlation diagram that clearly indicates the command wire location at more than 150 feet.

Camp Roberts chaos-based-radar field demonstration details and results. Top: Overall gravel road scene and magnified view, the latter indicating the difficulty in visually seeing the command wire. Bottom: Radar return correlation diagram that clearly indicates the command wire location at more than 150 feet.

Conclusion

The field of nonlinear dynamics provides an important new framework and perspective for the design and analysis of circuits and systems, offering a vast set of powerful qualitative and quantitative techniques. The many unique features and capabilities offered by nonlinear effects such as chaos provide numerous opportunities for the development of whole new technology areas. The exploitation of nonlinear effects has evolved from a state of mere knowledge advancement to product and commercial development and insertion, yielding mature products competitive with, or better than, current technologies. However, recent years have seen a troubling trend toward a dominant international role in nonlinear engineering, despite the many important U.S. national security implications that these approaches provide. This trend cuts across the whole engineering cycle, from research to product development and insertion. Motivated by these trends, Aerospace has begun research and development in the main areas of communications and radar. The importance of maintaining and continuing such investments cannot be overemphasized.

Acknowledgments

The author would like to acknowledge corporate support of this work through the Aerospace Sponsored Research program and the Mission Oriented Investigation and Experimentation program. Thanks also go to Herbert Wintroub (in whose memory this article is dedicated), Rich Haas, Al Geiger, André Montoya, Joe Straus, Andy Quintero, and Grant Aufderhaar for their advocacy and support. The author also thanks his colleagues, especially Albert Young (co-inventor of the Young-Silva chaotic oscillator) and Samuel Osofsky (project leader on chaos-based radar), who made it possible to carry out the basic development efforts described herein. The author would also like to thank his management for support, especially Jerry Michaelson, Allyson Yarbrough, Keith Soo Hoo, Robert Frueholz, Diana Johnson, Samuel Osofsky, and Yat Chan.

Further Reading

  • Centre for Chaos Control and Synchronization, http://www.ee.cityu.edu.hk/~cccn (as of Nov. 9, 2010).
  • R. Devaney, An Introduction to Chaotic Dynamical Systems (Benjamin/Cummings, Menlo Park, CA, 1986).
  • W. Ditto and L. Pecora, “Mastering Chaos,” Scientific American, Vol. 269, pp. 78–84 (Aug. 1993).
  • U. Feldmann, M. Hassler, and W. Schwarz, “Communication by Chaotic Signals: The Inverse System Approach,” International Journal of Circuit Theory and Applications, Vol. 24, No. 5, pp. 551–579 (1996).
  • J. Gleick, Chaos (Viking Press, New York, 1987).
  • E. Hunt and G. Johnson, “Keeping Chaos at Bay,” IEEE Spectrum, Vol. 30, pp. 32–36 (Nov. 1993).
  • G. Kolumbán, M. Kennedy, and L. Chua, “The Role of Synchronization in Digital Communications Using Chaos—Part I: Fundamentals of Digital Communications,” IEEE Transactions on Circuits and Systems, Vol. 44, No. 10, pp. 927–936 (Oct. 1997).
  • L. Larson, J.-M. Liu, and L. Tsimring, Digital Communications Using Chaos and Nonlinear Dynamics (Springer Science + Business Media, New York, 2006).
  • F. Lau and C. Tse, Chaos-Based Digital Communication Systems (Springer-Verlag, Berlin, 2004).
  • F.-Y. Lin, “Chaotic Radar Using Nonlinear Laser Dynamics,” IEEE Journal of Quantum Electronics, Vol. 40, No. 6, pp. 815–820 (Jun. 2004).
  • Z. Liu, X. Zhu, W. Hu, and F. Jiang, “Principles of Chaotic Signal Radar,” International Journal of Bifurcation and Chaos, Vol. 17, No. 5, pp. 1735–1739 (May 2007).
  • L. Pecora and T. Carroll, “Synchronization in Chaotic Systems,” Physical Review Letters, Vol. 64, pp. 821–824 (Feb. 19, 1990).
  • H.-O. Peitgen, H. Jurgens, and D. Saupe, Chaos and Fractals: New Frontiers of Science (Springer-Verlag, New York, 1992).
  • S. Strogatz, Chaos, DVD No. 1333 (The Great Courses, Chantilly, VA).
  • W. Tam, F. Lau, and C. Tse, Digital Communications with Chaos: Multiple Access Techniques and Performance (Elsevier Science, Oxford, 2007).

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