>> We recently designed an 8-channel complex waveform generator. Each
>> output stage is composed of a DAC, a lowpass filter, an output
>> amplifier, a test relay, and an output connector. It's this one:
>>
>>
http://www.highlandtechnology.com/DSS/V346DS.html
>>
>> You can see the gold output connectors, and the relays are hiding just
>> behind the front panel.
>>
>> The harmonic distortion seemed a bit high, in the -40 dBc range at 32
>> MHz and max level output. We were poking around with a spectrum
>> analyzer and happened to do a 0-3 GHz sweep and lo, a big line at
>> about 1 GHz. Something's oscillating!
>>
>> Cut to the bottom line: the eight output amps, 1.5 GHz current-mode
>> opamps, are individually stable, but oscillate together. Futzing with
>> some amps may affect the outputs of others, several channels away. And
>> the ensemble oscillations have multiple stable modes, including the
>> occasional "off."
>>
>> What's happening is that the front panel is electromagnetically
>> resonating in a fundamental violin-string mode (peak swing in the
>> middle) at about 1 GHz, and couples pretty well into all the output
>> stages; no doubt the relays are helping. A few well-placed capacitors
>> fix the problem. It took a while to figure this out.
>>
>> So the observation is: when something goes wrong, there are a number
>> of likely causes. Here, they were channel-channel trace couplings, Vcc
>> coupling, amplifier loop stability, pad-plane parasitic capacitance,
>> plain rotten opamps, stuff like that. But a complex system has many
>> possible, convoluted causalities other than the obvious ones. Suppose
>> there are a billion possible interactions, not unreasonable for a
>> system with hundreds of themselves-complex parts, all close and
>> well-coupled and interacting at frequencies like this. Suppose most of
>> those failure modes [1] are wildly improbable, like one chance in a
>> billion of ever happening.
>>
>> 1e9 * 1e-9 = 1
>>
>> The final solution was wildly improbable. If suggested as a cause, one
>> would be tempted to say "no, that's just too bizarre." It was probable
>> that the actual problem *was* wildly improbable.
>>
>> This sort of thing happens all the time in our business, in hardware
>> and software. Insanely unlikely insanely complex things happen,
>> because there are potentially so many of them. That makes it fun to
>> track them down.
>>
>> John
>
>Sensitive Dependence on Initial Conditions
>
http://www.schuelers.com/ChaosPsyche/part_1_14.htm
>
http://en.wikipedia.org/wiki/Butterfly_effect
>
http://everything2.com/index.pl?node_id=861246
>
http://www.perkel.com/nerd/butterflyeffect.htm
>
>Network logic is counterintuitive. Say you need to lay a telephone
>cable that will connect a bunch of cities; let's make that three for
>illustration: Kansas City, San Diego, and Seattle. The total length of
>the lines connecting those three cities is 3,000 miles. Common sense
>says that if you add a fourth city to your telephone network, the
>total length of your cable will have to increase. But that's not how
>network logic works. By adding a fourth city as a hub (let's make that
>Salt Lake City) and running the lines from each of the three cities
>through Salt Lake City, we can decrease the total mileage of cable to
>2,850 or 5 percent less than the original 3,000 miles. Therefore the
>total unraveled length of a network can be shortened by adding nodes
>to it! Yet there is a limit to this effect. Frank Hwang and Ding Zhu
>Du, working at Bell Laboratories in 1990, proved that the best savings
>a system might enjoy from introducing new points into a network would
>peak at about 13 percent. More is different.
>
>On the other hand, in 1968 Dietrich Braess, a German operations
>researcher, discovered that adding routes to an already congested
>network will only slow it down. Now called Braess's Paradox,
>scientists have found many examples of how adding capacity to a
>crowded network reduces its overall production. In the late 1960s the
>city planners of Stuttgart tried to ease downtown traffic by adding a
>street. When they did, traffic got worse; then they blocked it off and
>traffic improved. In 1992, New York City closed congested 42nd Street
>on Earth Day, fearing the worst, but traffic actually improved that
>day.
>
>Then again, in 1990, three scientists working on networks of brain
>neurons reported that increasing the gain-the responsivity-of
>individual neurons did not increase their individual signal detection
>performance, but it did increase the performance of the whole network
>to detect signals.
>
>
http://www.kk.org/outofcontrol/ch2-g.html
>
>The prime variable Kauffman played with was the connectivity of the
>network. In a sparsely connected network, each node would on average
>only connect to one other node, or less. In a richly connected
>network, each node would link to ten or a hundred or a thousand or a
>million other nodes. In theory the limit to the number of connections
>per node is simply the total number of nodes, minus one. A million-
>headed network could have a million-minus-one connections at each
>node; every node is connected to every other node. To continue our
>rough analogy, every employee of GM could be directly linked to all
>749,999 other employees of GM.
>
>As Kauffman varied this connectivity parameter in his generic
>networks, he discovered something that would not surprise the CEO of
>GM. A system where few agents influenced other agents was not very
>adaptable. The soup of connections was too thin to transmit an
>innovation. The system would fail to evolve. As Kauffman increased the
>average number of links between nodes, the system became more
>resilient, "bouncing back" when perturbed. The system could maintain
>stability while the environment changed. It would evolve. The
>completely unexpected finding was that beyond a certain level of
>linking density, continued connectivity would only decrease the
>adaptability of the system as a whole.
>
>Kauffman graphed this effect as a hill. The top of the hill was
>optimal flexibility to change. One low side of the hill was a sparsely
>connected system: flat-footed and stagnant. The other low side was an
>overly connected system: a frozen grid-lock of a thousand mutual
>pulls. So many conflicting influences came to bear on one node that
>whole sections of the system sank into rigid paralysis. Kauffman
>called this second extreme a "complexity catastrophe." Much to
>everyone's surprise, you could have too much connectivity. In the long
>run, an overly linked system was as debilitating as a mob of
>uncoordinated loners.
>
>Somewhere in the middle was a peak of just-right connectivity that
>gave the network its maximal nimbleness. Kauffman found this
>measurable "Goldilocks'" point in his model networks. His colleagues
>had trouble believing his maximal value at first because it seemed
>counterintuitive at the time. The optimal connectivity for the
>distilled systems Kauffman studied was very low, "somewhere in the
>single digits." Large networks with thousands of members adapted best
>with less than ten connections per member. Some nets peaked at less
>than two connections on average per node! A massively parallel system
>did not need to be heavily connected in order to adapt. Minimal
>average connection, done widely, was enough.
>
>Kauffman's second unexpected finding was that this low optimal value
>didn't seem to fluctuate much, no matter how many members comprised a
>specific network. In other words, as more members were added to the
>network, it didn't pay (in terms of systemwide adaptability) to
>increase the number of links to each node. To evolve most rapidly, add
>members but don't increase average link rates. This result confirmed
>what Craig Reynolds had found in his synthetic flocks: you could load
>a flock up with more and more members without having to reconfigure
>its structure.
>
>Kauffman found that at the low end, with less than two connections per
>agent or organism, the whole system wasn't nimble enough to keep up
>with change. If the community of agents lacked sufficient internal
>communication, it could not solve a problem as a group. More exactly,
>they fell into isolated patches of cooperative feedback but didn't
>interact with each other.
>
>At the ideal number of connections, the ideal amount of information
>flowed between agents, and the system as a whole found the optimal
>solutions consistently. If their environment was changing rapidly,
>this meant that the network remained stable-persisting as a whole over
>time.
>
>Kauffman's Law states that above a certain point, increasing the
>richness of connections between agents freezes adaptation. Nothing
>gets done because too many actions hinge on too many other
>contradictory actions. In the landscape metaphor, ultra-connectance
>produces ultra-ruggedness, making any move a likely fall off a peak of
>adaptation into a valley of nonadaptation. Another way of putting it,
>too many agents have a say in each other's work, and bureaucratic
>rigor mortis sets in. Adaptability conks out into grid-lock. For a
>contemporary culture primed to the virtues of connecting up, this low
>ceiling of connectivity comes as unexpected news.
>
>We postmodern communication addicts might want to pay attention to
>this. In our networked society we are pumping up both the total number
>of people connected (in 1993, the global network of networks was
>expanding at the rate of 15 percent additional users per month!), and
>the number of people and places to whom each member is connected.
>Faxes, phones, direct junk mail, and large cross-referenced data bases
>in business and government in effect increase the number of links
>between each person. Neither expansion particularly increases the
>adaptability of our system (society) as a whole.
>
>
http://www.kk.org/outofcontrol/ch20-d.html
>
