Exact analytic solution for a chaotic hybrid dynamical system and its electronic realization.

A novel hybrid dynamical system comprising a continuous and a discrete state is introduced and shown to exhibit chaotic dynamics. The system includes an unstable first-order filter subject to asynchronous switching of a set point according to a feedback rule. Regular samples of the continuous state yield a one-dimensional return map that is a tent function.

Analytic Solution for a Complex Network of Chaotic Oscillators.

Chaotic evolution is generally too irregular to be captured in an analytic solution. Nonetheless, some dynamical systems do have such solutions enabling more rigorous analysis than can be achieved with numerical solutions. Here, we introduce a method of coupling solvable chaotic oscillators that maintains solvability.

Analytic solutions throughout a period doubling route to chaos.

We show examples of dynamical systems that can be solved analytically at any point along a period doubling route to chaos. Each system consists of a linear part oscillating about a set point and a nonlinear rule for regularly updating that set point. Previously it has been shown that such systems can be solved analytically even when the oscillations are chaotic.

Analytically solvable chaotic oscillator based on a first-order filter.

A chaotic hybrid dynamical system is introduced and its analytic solution is derived. The system is described as an unstable first order filter subject to occasional switching of a set point according to a feedback rule. The system qualitatively differs from other recently studied solvable chaotic hybrid systems in that the timing of the switching is regulated by an external clock.

Regularly timed events amid chaos.

We show rigorously that the solutions of a class of chaotic oscillators are characterized by regularly timed events in which the derivative of the solution is instantaneously zero. The perfect regularity of these events is in stark contrast with the well-known unpredictability of chaos. We explore some consequences of these regularly timed events through experiments using chaotic electronic circuits.

Exactly solvable chaos in an electromechanical oscillator.

A novel electromechanical chaotic oscillator is described that admits an exact analytic solution. The oscillator is a hybrid dynamical system with governing equations that include a linear second order ordinary differential equation with negative damping and a discrete switching condition that controls the oscillatory fixed point. The system produces provably chaotic oscillations with a topological structure similar to either the Lorenz butterfly or Rössler's folded-band oscillator depending on the configuration.

Correlation properties of exactly solvable chaotic oscillators.

We derive exact expressions for the autocorrelation and cross-correlation functions of wave forms generated by two exactly solvable chaotic systems. For each system, an analytic expression exists for chaotic solutions, which we use to evaluate correlation integrals explicitly. For some specific parameter values, we calculate the mean and variance of the correlation functions averaged over all possible solutions.

Acoustic detection and ranging using solvable chaos.

Acoustic experiments demonstrate a novel approach to ranging and detection that exploits the properties of a solvable chaotic oscillator. This nonlinear oscillator includes an ordinary differential equation and a discrete switching condition. The chaotic waveform generated by this hybrid system is used as the transmitted waveform.

Exact folded-band chaotic oscillator.

An exactly solvable chaotic oscillator with folded-band dynamics is shown. The oscillator is a hybrid dynamical system containing a linear ordinary differential equation and a nonlinear switching condition. Bounded oscillations are provably chaotic, and successive waveform maxima yield a one-dimensional piecewise-linear return map with segments of both positive and negative slopes.

A matched filter for chaos.

A novel chaotic oscillator is shown to admit an exact analytic solution and a simple matched filter. The oscillator is a hybrid dynamical system including both a differential equation and a discrete switching condition. The analytic solution is written as a linear convolution of a symbol sequence and a fixed basis function, similar to that of conventional communication waveforms.

Time-shifted synchronization of chaotic oscillator chains without explicit coupling delays.

We examine chains of unidirectionally coupled oscillators in which time-shifted synchronization occurs without explicit delays in the coupling. In numerical simulations and in an experimental system of electronic oscillators, we examine the time shift and the degree of distortion (primarily in the form of attenuation) of the waveforms of the oscillators located far from the drive oscillator. Surprisingly, under weak coupling we observe minimal attenuation in spite of a significant total time shift.

Determinism in synthesized chaotic waveforms.

The output of a linear filter driven by a randomly polarized square wave, when viewed backward in time, is shown to exhibit determinism at all times when embedded in a three-dimensional state space. Combined with previous results establishing exponential divergence equivalent to a positive Lyapunov exponent, this result rigorously shows that such reverse-time synthesized waveforms appear equally to have been produced by a deterministic chaotic system.

Time shifts and correlations in synchronized chaos.

We introduce a new method for predicting characteristics of the synchronized state achieved by a wide class of unidirectional coupling schemes. Specifically, we derive a transfer function from the coupling model that provides estimates of the correlation between the drive and response waveforms, and of the time shift (i.e.

Deconstructing spatiotemporal chaos using local symbolic dynamics.

We find that the global symbolic dynamics of a diffusively coupled map lattice is well approximated by a very small local model for weak to moderate coupling strengths. A local symbolic model is a truncation of the full symbolic model to one that considers only a single element and a few neighbors. Using interval analysis, we give rigorous results for a range of coupling strengths and different local model widths.

A simple Lorenz circuit and its radio frequency implementation.

A remarkably simple electronic circuit design based on the chaotic Lorenz system is described. The circuit consists of just two active nonlinear elements (high-speed analog multipliers) and a few passive linear elements. Experimental implementations of the circuit exhibit the classic butterfly attractor and the hysteretic transition from steady state to chaos observed in the Lorenz equations.

Synthesizing folded band chaos.

A randomly driven linear filter that synthesizes Lorenz-like, reverse-time chaos is shown also to produce Rössler-like folded band wave forms when driven using a different encoding of the random source. The relationship between the topological entropy of the random source, dissipation in the linear filter, and the positive Lyapunov exponent for the reverse-time wave form is exposed. The two drive encodings are viewed as grammar restrictions on a more general encoding that produces a chaotic superset encompassing both the Lorenz butterfly and Rössler folded band paradigms of nonlinear dynamics.

Chaos without nonlinear dynamics.

A linear, second-order filter driven by randomly polarized pulses is shown to generate a waveform that is chaotic under time reversal. That is, the filter output exhibits determinism and a positive Lyapunov exponent when viewed backward in time. The filter is demonstrated experimentally using a passive electronic circuit, and the resulting waveform exhibits a Lorenz-like butterfly structure.

Symbolic dynamics of coupled map lattices.

We present a method to reduce the [FORMULA: SEE TEXT] dynamics of coupled map lattices (CMLs) of N invertibly coupled unimodal maps to a sequence of N-bit symbols. We claim that the symbolic description is complete and provides for the identification of all fixed points, periodic orbits, and dense orbits as well as an efficient representation for studying pattern formation in CMLs. We give our results for CMLs in terms of symbolic dynamical concepts well known for one-dimensional chaotic maps, including generating partitions, Gray orderings, and kneading sequences.

Lag and anticipating synchronization without time-delay coupling.

We describe a new method for achieving approximate lag and anticipating synchronization in unidirectionally coupled chaotic oscillators. The method uses a specific parameter mismatch between the drive and response that is a first-order approximation to true time-delay coupling. As a result, an adjustable lag or anticipation effect can be achieved without the need for a variable delay line, making the method simpler and more economical to implement in many physical systems.

Experimental observation of delay-induced radio frequency chaos in a transmission line oscillator.

We report an experimental study of fast chaotic dynamics in a delay dynamical system. The system is an electronic device consisting of a length of coaxial cable terminated on one end with a diode and on the other with a negative resistor. When the negative resistance is large, the system evolves to a steady state.

Information flow in chaos synchronization: fundamental tradeoffs in precision, delay, and anticipation.

We use symbolic dynamics to examine the flow of information in unidirectionally coupled chaotic oscillators exhibiting synchronization. The theory of symbolic dynamics reduces chaos to a shift map that acts on a discrete set of symbols, each of which contains information about the system state. Using this transformation we explore so-called achronal synchronization, in which the response lags or leads the drive by a fixed amount of time.

Control of long-period orbits and arbitrary trajectories in chaotic systems using dynamic limiting.

We demonstrate experimental control of long-period orbits and arbitrary chaotic trajectories using a new chaos control technique called dynamic limiting. Based on limiter control, dynamic limiting uses a predetermined sequence of limiter levels applied to the chaotic system to stabilize natural states of the system. The limiter sequence is clocked by the natural return time of the chaotic system such that the oscillator sees a new limiter level for each peak return.

Synchronizing the information content of a chaotic map and flow via symbolic dynamics.

In this paper we report an extension to the concept of generalized synchronization for coupling different types of chaotic systems, including maps and flows. This broader viewpoint takes disparate systems to be synchronized if their information content is equivalent. We use symbolic dynamics to quantize the information produced by each system and compare the symbol sequences to establish synchronization.