Oscillatory frequencies in spatiotemporal system with local inhomogeneity

Target waves usually emit concentric circular waves, whereas spiral waves rotate around a central core (topological defect) region, the two forms of waves are closely related due to the similarity of their spatial structures. Spiral waves can be generated spontaneously in a homogeneous system, while target waves usually cannot be self-sustained in the same system. Therefore, spiral waves can be found in diverse natural systems, and target waves can be produced from the spirals with special boundary configurations or central pacemakers. The pacemaker of target wave is an oscillatory source or medium inhomogeneity. To model the inhomogeneity in some realistic situations, we introduce local parameter shifts and simulate the transition from spiral waves to target waves. In this research, the evolution of the spiral waves in the complex Ginzburg-Landau equation is investigated by numerical simulations, and the multi-spiral patterns can be transformed into stable target waves with local inhomogeneous parameter shifts in a two-dimensional (2D) spatiotemporal system. The detailed study shows that the initial multi-spiral waves can be influenced by introducing inhomogeneity in the local area of the system space, and the oscillatory frequency of the system plays an important role in changing the pattern. A successful transition from inwardly propagating spirals to target waves can be observed when the oscillatory frequencies of non-controlled and local inhomogeneous region, which have equal values, are both less than the inherent frequency of system. When we inspect the relationship between oscillatory frequencies and the characteristics of the inhomogeneous region, an intriguing V-shaped line is found in parameter-frequency diagram, and the V-shaped area presents three features. Firstly, the left and right sides of the V-shaped area are symmetrical. Secondly, the propagating directions of target waves from the left and right sides are opposite. An inwardly propagating target wave is formed on the left side of the V-shaped area, and an outwardly propagating target wave stably exists on the right side of the line. Thirdly, as local inhomogeneous parameter 2 increases, the V-shaped area moves towards the local inhomogeneous parameter 2 and decreases simultaneously, and the width of the V-shaped area remains approximately the same. To our knowledge, this V-shaped line is a novel observation, hence the changes of the system frequencies are thought to be provoking. This work presents the numerical experiments and theoretical analyses for the stable conditions of target waves, and therefore provides the ideas in the applications of signal propagation and mode competition.