第三章Lyapunov指数的非线性控制.ppt

Sensitivity on the initial conditions also happens in linear systems SIC leads to chaos only if the trajectories are bounded (the system cannot blow up to infinity). With linear dynamics either SIC or bounded trajectories. With nonlinearities could be both. xn+1= 2xn But this is “explosion process”, not the deterministic chaos! Why? There is no boundness. There is no folding without nonlinearities! The Lyapunov Exponent A quantitative measure of the sensitive dependence on the initial conditions is the Lyapunov exponent ?. It is the averaged rate of divergence (or convergence) of two neighboring trajectories in the phase space. Actually there is a whole spectrum of Lyapunov exponents. Their number is equal to the dimension of the phase space. If one speaks about the Lyapunov exponent, the largest one is meant. x0 p1(0) t - time flow p2(t) p1(t) p2(0) x(t) Definition of Lyapunov Exponents Given a continuous dynamical system in an n-dimensional phase space, we monitor the long-term evolution of an infinitesimal n-sphere of initial conditions. The sphere will become an n-ellipsoid due to the locally deforming nature of the flow. The i-th one-dimensional Lyapunov exponent is then defined as following: On more formal level The Multiplicative Ergodic Theorem of Oseledec states that this limit exists for almost all points x0 and almost all directions of infinitesimal displacement in the same basin of attraction. Order: λ1 λ2 … λn The linear extent of the ellipsoid grows as 2λ1t The area defined by the first 2 principle axes grows as 2(λ1+λ2)t The volume defined by the first 3 principle axes grows as 2(λ1+λ2+λ3)t and so on… The sum of the first j exponents is defined by the long-term exponential growth rate of a j-volume element. Signs of the Lyapunov exponents Any continuous time-dependent DS without a fixed point will have ?1 zero exponents. The sum of the Lyapunov exponents must be negative in dissipative DS ? ? at least one negative Lyapunov exponent