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Border collision bifurcation of a resonant closed invariant curve.

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Memory effects on binary choices with impulsive agents: Bistability and a new BCB structure.

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This study models binary choices with memory, extending one-dimensional models to two dimensions. The research reveals persistent period-adding structures and the novel possibility of bistability in agent behavior dynamics.

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Area of Science:

  • Agent-based modeling
  • Dynamical systems theory
  • Behavioral economics

Background:

  • Seminal works by Schelling established models for binary choices among agents.
  • The role of memory in agent-based binary choice models remains an active research area.
  • Recent literature models impulsive agent dynamics using one-dimensional piecewise smooth maps.

Purpose of the Study:

  • To investigate the dynamics of binary choices incorporating a memory effect.
  • To extend existing one-dimensional models to a two-dimensional framework.
  • To analyze the resulting piecewise smooth map, its bifurcations, and parameter space structure.

Main Methods:

  • Developed a two-dimensional piecewise smooth map incorporating a memory effect (dependence on present and last state).
  • Analyzed a limiting case: a piecewise linear discontinuous map.
  • Investigated system dynamics, bifurcations, parameter space organization, and basins of attraction.

Main Results:

  • The two-dimensional map exhibits trajectories confined to a pair of straight lines.
  • The period-adding structure, characteristic of one-dimensional models, is preserved.
  • Coexistence of two attracting cycles (bistability) becomes possible, unlike in the one-dimensional case.

Conclusions:

  • Memory effects in binary choice models lead to complex two-dimensional dynamics.
  • The period-adding structure and bistability offer new insights into agent behavior organization.
  • Codimension-two bifurcation points play a crucial role in shaping parameter and phase space structures.