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Period doubling bifurcations in a two-box model of the brusselator. (English) Zbl 0531.34030

Let us consider the system of ordinary differential equations (1) \(\dot x=v(x,\mu)\), where \(x=[x_ 1,y_ 1,x_ 2,y_ 2]\in {\mathbb{R}}^ 4\), \(\mu \in {\mathbb{R}}^ 1\) is a parameter and vector field \(v(\cdot,\mu)\) is invariant under the diffeomorphism g for every \(\mu\in {\mathbb{R}}\). The diffeomorphism g is defined by the following relation \(g(x_ 1,y_ 1,x_ 2,y_ 2)=[x_ 2,y_ 2,x_ 1,y_ 1].\)
Such systems often arise when two identical oscillators are coupled and the coupling between them is symmetrical. A typical result of this paper is following: Let us denote \(\Delta =\{[x_ 1,y_ 1,x_ 2,y_ 2]\in {\mathbb{R}}^ 4,\quad x_ 1=x_ 2\wedge y_ 1=y_ 2\}\) the diagonal in \({\mathbb{R}}^ 4\). The periodic solution \(x_{{\hat \mu}}(t)\) of (1) will be called a \(\Delta\)-symmetric solution iff its trajectory \(\gamma_{{\hat \mu}}\) is an invariant set of the mapping g, i.e. \(g(\gamma_{{\hat \mu}})=\gamma_{{\hat \mu}}\) and \(\Delta \cap \gamma_{{\hat \mu}}=\emptyset\). Theorem: The \(\Delta\)-symmetric solution cannot bifurcate by the period doubling bifurcation. Comment: Period doubling bifurcation in nonautonomous systems with certain symmetry is investigated in the paper of J. W. Swift and K. Wisenfeld, Physical Review Letters 52, 705-708 (1984).

MSC:

34C25 Periodic solutions to ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
37G99 Local and nonlocal bifurcation theory for dynamical systems
37N99 Applications of dynamical systems
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