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The time-averaged Lyapunov exponents support a mechanistic description of thechaos generated in and by nonlinear dynamical systems. The exponents areordered from largest to smallest with the largest one describing theexponential growth rate of the (small) distance between two neighboringphase-space trajectories. Two exponents describe the rate for areas defined bythree nearby trajectories. The sum of the first three exponents is the rate forvolumes defined by four nearby trajectories, and so on. Lyapunov exponents forHamiltonian systems are symmetric. The time-reversibility of the motionequations links the growth and decay rates together in pairs. This pairingprovides a more detailed explanation than Liouville's for the conservation ofphase volume in Hamiltonian mechanics. Although correct for long-time averages,the dependence of trajectories on their past is responsible for the observedlack of detailed pairing for the instantaneous "local" exponents, $\{\\lambda_i(t) \ \}$ . The 2017 Ian Snook Prizes will be awarded to the author(s)of an accessible and pedagogical discussion of local Lyapunov instability insmall systems. We desire that this discussion build on the two nonlinear modelsdescribed here, a double pendulum with Hooke's-Law links and a periodic chainof Hooke's-Law particles tethered to their lattice sites. The latter system isthe $\phi^4$ model popularized by Aoki and Kusnezov. A four-particle version issmall enough for comprehensive numerical work and large enough to illustrateideas of general validity.

Instantaneous Pairing of Lyapunov Exponents in Chaotic Hamiltonian Dynamics and the 2017 Ian Snook Prize