Analog computational circuits for chaotic dynamics
Development Boards


NL09D1XX Datasheet (7/1/09)
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Design principles and operation

The NL09D1XX development boards implement analogues of topological models of chaotic dynamical systems. Such models are called branched manifolds; they have the structure of multiply-connected ribbons of finite width and zero thickness. If we restrict the bends, turns, and twists in the ribbons to 90o, we can define a set of canonical pieces sufficient to assemble any branched manifold corresponding to an arbitrary dynamical system. We can then define a simple circuit fragment that is the analogue of each canonical piece, and by connecting the circuit fragments with the same topology as the model branched manifold, we obtain a complete circuit that is an electronic analogue of the chaotic dynamical system. We call this a circuit manifold. The circuit behavior is guaranteed to correspond to the behavior of the physical system, because every part of the circuit is the analogue of a corresponding part of the physical system. The correspondence between the branched manifold and the circuit manifold is shown here for a fragment of a chaotic system:

The following analog circuit fragments are sufficient to assemble any circuit manifold: split, inversion, shift, scale, and merge. Circuit fragments implementing these functions are shown here:

Analysis shows that combining these circuit fragments in the order shown can represent an arbitrary branched manifold, hence an arbitrary chaotic system. For example, the following figure shows the five fragments connected to provide three independent signal paths which are processed differently and then combined at the output.

The circuit manifold provides an input/output path for an analog signal; the output can be routed to the input for iteration (corresponding to the chaotic system traversing many cycles of its quasi-periodic motion), or it can be routed to the input of a different (but identical) module. In the latter case, implementation of a large number of such modules in series (or with more complicated mesh connectivity) provides the equivalent of a parallel-architecture processor.

Particularly interesting is the arrangement of N modules connected into a ring. This circuit will automatically generate the N values of the dynamical variable in N passes through the processor, thus directly revealing periodicity (up to N). Because the motion of chaotic systems is organized by unstable periodic orbits, this circuit will directly reveal the topological structure of the branched manifold, hence the physical system. The following figure shows an 18-module ring processor. The 18 analog signals are brought into an external multiplexer for analysis and display.

The NL09D1XX product series provides several implementations of circuit manifolds, using the above modules and design procedure. Signals are provided to pins that enable jumper connections within the modules and to other modules. Overlay boards are available that provide default connections for some common connectivities, e.g., rings, eliminating the need for extensive manual jumper wiring. The circuits are 100% analog, including the switches used in the split modules.

NL09D101 - Single-path processor

The NL09D101 development board consists of the five canonical analog modules described above, providing a single analog signal path. Multiple boards can be connected in series, or in arbitrary mesh connectivities. Using multiple boards converts the time-dependent chaotic signal into a spatially-dependent signal (the values at the mesh nodes).

NL09D104 - Quad-path processor

The NL09D104 development board consists of four sets of the five canonical analog modules described above, providing four independent analog signal paths. The output from one path can be routed to the input of another path, providing up to four modules in series, including a ring circuit. Multiple boards can be used to generate larger numbers of independent paths.

These products are implemented according to the principle of TOPOLOGICTM, in which the circuit has structure and function that are analogues of the physical system. The modules are generally constructed to be data transfer units, enabling direct coupling of modules in exact correspondence with a pseudo-physical model such as a topological model of the phase space behavior of a chaotic system. .NanoLogic is the inventor of this technology, which is protected by U. S. patents (pending).