By Poisson, Eric
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Additional info for An advanced course in general relativity
11) A specific case that has strong chaotic behavior is the Duffing oscillator F = -(ax + bx3 + cy) (see Chapters 2 and 4). 3 MAPSANDFLOWS 41 Figure 1-27 Trajectory of a chaotic solution to the Lorenz equations for thermofluid convection [Eqs. 9)] (numerical integration). It is worth noting that for a two-dimensional phase space, solutions to autonomous systems cannot exhibit chaos because the solution curves of the “flow” cannot cross one another. However, in the forced oscillator or the three-dimensional phase space, these curves can become “tangled” and chaotic motions are possible.
In the case of the Van der Pol oscillator, the source of energy is a dc voltage. 5) For low amplitudes, energy can flow into the system, but at higher amplitudes the nonlinear damping limits the amplitude. , see Minorsky, 1962, Chap. 28), the constant rotation of the motor provides an energy input. 2 CLASSICAL NONLINEAR VIBRATION THEORY: A BRIEF REVIEW 15 tv (C) Figure 1-8 Example of self-excited oscillations: ( a ) dry friction between a mass and moving belt; ( b ) aeroelastic forces on a vibrating airfoil; and (c) negative resistance in an active circuit element.
Readers desiring more detailed discussion in classical nonlinear vibration should consult books such as Stoker (1950), Minorsky (1962), Nayfeh and Mook (1979), or Hagedorn (1988). We begin with a brief review of linear vibration concepts. Linear Vibration Theory The classic paradigm of linear vibrations is the spring-mass system shown in Figure 1-5 along with its electric circuit analog. The addition of damping c' # 0, R # 0) introduces decay in the free vibrations so that the amplitude of the mass (or charge in the circuit) exhibits the following time dependence: fbl Figure 1-5 ( a ) The classic, mechanical spring-mass-dashpot oscillator.