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BCB curves and contact bifurcations in piecewise linear discontinuous map arising in a financial market. (English) Zbl 1411.37076

Summary: In this paper, we further study a financial market model established in our earlier paper. The model dynamics is driven by a two-dimensional piecewise linear discontinuous map, which is investigated analytically and numerically for one-sided fixed points being flip saddle and two-sided fixed points being attractors. The existence of chaotic orbit is explained by using the theory of homoclinic intersection between stable and unstable manifolds of the flip saddle invariant set. The structure of chaotic attractor is disclosed. It consists of finite segments rooted on both sides of the \(x\)-axis which are unstable manifolds of flip saddle invariant set. The basins and their structural changes of bounded attractors and coexisting attractors are presented by contact bifurcation theory and numerical simulations. The border collision bifurcation (BCB for short) curves are calculated and coexisting multiattractors are disclosed by overlapping periodicity regions. The results can deepen our understanding of financial markets and dynamical systems.

MSC:

37N40 Dynamical systems in optimization and economics
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
91B24 Microeconomic theory (price theory and economic markets)
37G20 Hyperbolic singular points with homoclinic trajectories in dynamical systems
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37G25 Bifurcations connected with nontransversal intersection in dynamical systems
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