Limit Cycle Trajectory Research Articles

Liquidity-profit rate cycles in a capitalist economy

A monetary economy consisting of independent capitalist firms, in which workers spend wages instantly and labor is not scarce, is studied The firms create demand for each other's output through their capital outlays, and create the backing for money through borrowing. When all firms are identical in behavior and initial conditions, and the rate of growth of money is constant, the economy may be unstable around the equilibrium steady state growth path due to strong accelerator effects, and follow a limit cycle trajectory.

A model illustrating amplification of perturbations in an excitable medium

The oscillatory Belousov–Zhabotinskii reaction can be modelled approximately by five irreversible steps: A + Y → X (M1), X + Y → P (M2), B + X → 2X + Z (M3), 2X → Q (M4), Z →ƒY. (M5). These equations are based on the chemical equalities X = HBrO2, Y = Br–, Z = 2Ce(IV), and A = B = BrO–3. If the rate constants kM1 to kM4 are assigned by experimental estimates from oxybromine chemistry, the kinetic behaviour of the model depends critically upon the remaining parameters kM5 and ƒ. When ƒ does not differ too greatly from unity, and when kM5 is not too large, the steady state is unstable to perturbation and the system oscillates by describing a limit cycle trajectory.

Design of Nonlinear Sampled-Data System for Acceleration Measurement

The design of a sampled-data nonlinear system for three-axes acceleration measurement is discussed. The basic specific force equation is given, and the equation for the indicated velocity (integral of indicated acceleration) is derived to show the dependence of the error on the shape of the limit cycle trajectory. Emphasis is on how certain physical parameters, such as damping and dead zone width, may be chosen to fulfill the design requirement of minimum error in the indicated velocity output of the system. Expressions for design purposes relating physical parameters, such as damping, and phase plane switch line spacing to limit cycle period are derived. High limit cycle frequencies, or low periods, are shown to be less objectionable. An expression is given for the approximate dead zone width for minimum error from dynamic average and static offsets of the limit cycle trajectories. A multiscale-normalized phase plane technique is described and used to determine both the transient and final limit cycle trajectories for assumed step acceleration inputs at arbitrary points during an existing limit cycle.