Home > Author > Steven H. Strogatz
41 " More generally, when a nonchaotic system is disturbed slightly, the disturbance either doesn’t grow at all or else grows very mildly, increasing in proportion to how much time has passed. One says that the errors grow no faster than linearly in time. "
― Steven H. Strogatz , Sync: The Emerging Science of Spontaneous Order
42 " Whenever the whole is different from the sum of the parts—whenever there’s cooperation or competition going on—the governing equations must be nonlinear. "
43 " pack and a disorganized band of fringe oscillators. When the system was self-synchronizing, Winfree found that no oscillator was indispensable. There was no boss. Any oscillator could be removed and the process would still work. "
44 " I have a friend who gets a tremendous kick out of science, even though he’s an artist. "
― Steven H. Strogatz , The Joy of x: A Guided Tour of Math, from One to Infinity
45 " The amount of time we can successfully predict the state of a chaotic system depends on three things: how much error we’re willing to tolerate in the forecast; how precisely we can measure the initial state of the system; and a time scale that’s beyond our control, called the Lyapunov time, which depends on the inherent dynamics of the system itself. "
46 " In a chaotic system, the required precision in the initial measurement grows exponentially, not linearly. "
47 " Lyapunov time sets a horizon beyond which acceptable prediction becomes impossible. For a chaotic electrical circuit, the horizon is something like a thousandth of a second; for the weather, it’s unknown but seems to be a few days; and for the solar system itself, five million years. "
48 " When subjects go to sleep later in their body temperature cycles, they actually sleep less, even though they have been awake longer. "
49 " Now we see the connection to the Josephson junction. This sine function is the same one that appeared earlier in the direct-current Josephson effect, where the supercurrent is proportional to the sine of the phase across the junction. That’s the analogy: The phase across the junction is like the angle of the pendulum. As it turns out, all the other terms in the equation have counterparts as well. The flow of normal electrons corresponds to the damping of the pendulum caused by friction. The pendulum’s mass is like the junction’s capacitance. And the torque applied to the pendulum is like the external current driving the junction. "
50 " If we make things even more complicated and allow the torque itself to vary in time, like the back and forth agitation of a washing machine, the pendulum’s whirling can become chaotic, rotating this way and that, changing direction haphazardly. The verification of the corresponding electrical spasms in a Josephson junction was one of the early experimental triumphs of chaos theory. Before that, physicists had always seen the pendulum as a symbol of clockwork regularity. Suddenly it was a paradigm of chaos. "
51 " The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. "
― Steven H. Strogatz , Infinite Powers: How Calculus Reveals the Secrets of the Universe
52 " We take it for granted that we can sing and dance together, march in step, clap in unison. Sync is second nature to us. But because it comes so easily, we have poor insight about what it actually demands. It seems to involve at least a low level of intelligence, the ability to time our behavior and anticipate that of others. Which is why the reports of concerted flashing among thousands of fireflies aroused such skepticism for so many years, and why we are impressed by the chorusing of crickets or the seductive tactics of male fiddler crabs, who court a female by waving their gargantuan claws at her in unison. "
53 " Mindless, lifeless things can sync spontaneously. The sympathy of clocks taught us that the capacity for sync does not depend on intelligence, or life, or natural selection. It springs from the deepest source of all: the laws of mathematics and physics. "
54 " No matter how erratically something moves, the area accumulated under its speed curve up to time t always equals the total distance it has traveled up to that time. "
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56 " Superfluid helium is a realization of the hypothetical quantum liquid that we imagined when performing the thought experiment with the buckets on the staircase. Its behavior is almost surreal. "
57 " This weird behavior is a manifestation of quantum sync. All liquids become highly ordered when cooled to very low temperatures. Normally they freeze into a crystal. But the two isotopes of helium, helium-3 and helium-4, never solidify, at least not at ordinary pressures. They remain liquids all the way down to absolute zero. "
58 " All the electrical components on a computer chip are clocked to operate in sync. A microelectronic crystal beats billions of times each second, switching the digital circuitry on and off in concert, which helps the millions of circuits on the chip communicate with one another efficiently. This centralized design, with all components slaved to a tyrannical master clock, has some notable disadvantages: 15 percent of the circuitry is wasted on distributing the clock signal, and the clock itself consumes 20 percent of the power. But engineers still favor this design because of its conceptual simplicity, and because the alternative—a democracy of many local clocks, as in firefly swarms and circadian pacemaker cells—is still not well enough understood to be easily imitated in practice. "
59 " Pecora started the transmitter and receiver in different states, and then asked the computer to predict their behavior far into the future. As the numbers poured out, they bobbled erratically—the aperiodicity expected of chaos—but amazingly, their values converged toward each other. They were synchronizing. By driving the receiver with a chaotic signal transmitted from a duplicate of itself, Pecora had coaxed them to fluctuate in lockstep. "
60 " Over the past 40 years, a number of practical applications have been found for these remarkable manifestations of quantum sync. Josephson’s superconducting sandwiches, now known universally as “Josephson junctions,” have spawned the most sensitive detectors known to science. For instance, a device called a SQUID (for superconducting quantum interference device) takes advantage of the extreme sensitivity of a supercurrent to a magnetic field. A SQUID can measure a displacement a thousand times smaller than an atomic nucleus, or a magnetic field 100 billion times weaker than Earth’s. SQUIDs are used in astronomy, to detect faint radiation from distant galaxies; in nondestructive testing, to spot hidden corrosion beneath the aluminum skin of airplanes; and in geophysics, to help locate sources of oil deep underground. "