Methods Inf Med 2000; 39(02): 114-117
DOI: 10.1055/s-0038-1634284
Original Article
Schattauer GmbH

Time Series and the Dynamics of Demand Pacing

D. T. Kaplan
1   Department of Mathematics & Computer Science Macalester College, St. Paul, Minnesota, USA
› Author Affiliations
Further Information

Publication History

Publication Date:
07 February 2018 (online)

Abstract:

Motivated by a common practice in cardiology, we analyze the dynamics of a demand paced system where one seeks to create a stable periodic response. By using techniques originally developed for controlling chaotic systems, one can enhance the information contained in time series regarding hidden, unstable periodic orbits. This makes it possible, for example, to track drifts in a system‘s dynamics.

 
  • REFERENCES

  • 1 Garfinkel A, Spano ML, Ditto WL, Weiss JN. Controlling cardiac chaos. Science 1992; 257: 1230-5.
  • 2 Kaplan DT. Applying blind chaos control to find periodic orbits. 1999 unpublished.
  • 3 Christini DJ, Kaplan DT. Adaptive estimation and control of unstable periodic dynamics in excitable biological systems. 1999 preprint.
  • 4 Courtemanche M, Glass L, Keener JP. Instabilities of a propagating pulse in a ring of excitable media. Physical Review Letters 1993; 70: 2182-5.
  • 5 This claim depends on one’s point of view. During successful periodic pacing the detailed dynamics might still be very complex and irregular, causing fluctuations that are small compared to the pacing period. But, seen from a coarse measurement, the dynamics might appear quite periodic.
  • 6 Sauer T. In: Nonlinear Dynamics and Time Series: Building a Bridge Between the Natural and Statistical Science. Cutler C, Kaplan DT. eds. Fields Inst Publications v 1997; 11: 63-75.
  • 7 Witkowski FX, Kavanagh KM, Penkoske PA, Plonsey R, Spano ML, Ditto WL, Kaplan DT. Evidence for determinism in ventricular fibrillation. Physical Review Letters 1995; 75 (Suppl. 06) 1230-33.
  • 8 Kaplan DT. Finding and characterizing unstable fixed points Proceedings of Chaos in Brain?. In press.
  • 9 Note that if the dynamics involved more than two previous values of t, this condition would be largely the same. For example, if the dynamics were three-dimensional τ l+1 = at + b t-1 + c t-2 + d the condition for stable pacing would be (a + b + c) C + d > C. Thus, two-dimensional dynamics can correctly model the stability of pacing when the autonomous dynamics involve more than two dimensions.