At what positions is the speed of a simple harmonic oscillator half its maximum? That is, what values of x / X give v = vmax / 2 , where X is the amplitude of the motion?

THE STRANGE NEW SCIENCE OF CHAOS ● Chaos = errors or deviations from expected results ○ Scientists try to minimize or eliminate it ● Linear system = the output is directly proportional to the input ● Examples of chaotic systems ○ Water bubbling down a stream ○ Smoke rising from a cigarette ○ Water dripping from a tap ○ The weather ○ Our nervous systems ● Chaotic systems are sensitive to initial conditions = a small change in the starting condition leads to a big change in the result ● Weather is hard to predict ○ The turbulent atmosphere cannot be described by simple linear equations ● Attractor = a preferred state towards which a chaotic system may tend ● Turbulence = much of the universe is fluids of one kind or another in motion ● Chaos does not mean randomness, but patterns within what appear to be random ● Our brains depend on chaos, but chaotic behavior is characterized of a heart attack