A traveler in the desert sees a lake. It shimmers, beckons with coolness, but as you approach, the water recedes. So it is in mathematics: the computer draws chaos, lines dart across the screen, and it seems we are witnessing a real storm. But if you wait long enough, the storm subsides, and the trajectory settles into a simple equilibrium. The question is how long to wait: a minute, a year, or ten million years.

The classic Lorenz system consists of three equations describing convection in a heated fluid. For certain parameters, it behaves like a pendulum that never stops. But at r=24, the computer shows something different: chaos persists, and persists, then suddenly vanishes. For one initial point, the fuss lasts more than $10^7$ time units; for another, only 18,000. Both trajectories eventually freeze at the same point, but along the way they behave as if chaos were eternal.

To avoid being fooled, scientists devised a trap: clamp an unstable periodic orbit, like a beetle with tweezers, and check whether the chaos indicators of the stabilized and free trajectories match. In the Rössler system, they match to the fifth decimal only on the interval [0, 30.4], and then diverge. So the numerical chaos was an illusion, born of too short an observation time.

There is another path: skip the computer altogether. For the Vallis system, which describes El Niño, researchers derived an analytical estimate of the attractor's dimension without running a single numerical experiment. A formula with square roots and parameters B and C gives an upper bound: for B=102, C=3, p=0.83, the dimension does not exceed 2.65903. It is like measuring the depth of a lake without diving in, just by looking at a map.

A mirage remains a mirage, even if you walk toward it all your life. But now we have a compass: analytical methods show where chaos is real and where it is merely a long ripple on the water. A traveler armed with such knowledge will not waste years chasing a ghost.