Dual-mode spiral vortices

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Dual-mode spiral vortices

Yair Mau, Aric Hagberg, and Ehud Meron

Phys. Rev. E 80, 065203(R) – Published 8 December 2009

Abstract

We show that spiral vortices in oscillatory systems can lose stability to secondary modes to form dual-mode spiral vortices. The secondary modes grow at the vortex core where the oscillation amplitude vanishes but are nonlinearly damped by the oscillatory mode away from the core. Gradients of the oscillation phase, induced by the hosted secondary mode, can lead to additional hosting events that culminate in periodic core oscillations or in a novel form of spatiotemporal chaos. The results of this study apply to physical, chemical, and biological systems that go through cusp-Hopf, fold-Hopf, and Hopf-Turing bifurcations.

Received 19 April 2009

DOI:https://doi.org/10.1103/PhysRevE.80.065203

Authors & Affiliations

Yair Mau¹, Aric Hagberg², and Ehud Meron^1,3

¹Physics Department, Ben-Gurion University, Beer-Sheva 84105, Israel
²Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
³Department of Solar Energy and Environmental Physics, BIDR, Ben-Gurion University, Sede Boqer, 84990, Israel

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Issue

Vol. 80, Iss. 6 — December 2009

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Images

Figure 1
(Color online) Single- and dual-mode spiral vortices in the FHN system (1) beyond the Hopf and Turing bifurcations. (a) Close to the Turing bifurcation the Turing mode is damped everywhere, $ζ = 4.5$ . (b) Farther from the Turing bifurcation the Turing mode is hosted at the spiral core, $ζ = 5.75$ . (c) The difference between (a) and (b) showing the Turing mode at the core. The frames show the $u$ field variable of Eqs. (1) with ranges of $[- 0.5, 0.5]$ in (a) and (b) and $[- 0.1, 0.1]$ in (c). Lighter colors are lower values and darker colors are higher values. The initial condition is a spiral wave. Parameters: $ε = 1.5$ , $a_{1} = 0.5$ , $a_{0} = 0.05$ , and $x = y = [0, 256]$ , with no-flux boundary conditions.Reuse & Permissions
Figure 2
(Color online) Bifurcation diagram for $ρ = | A |$ and $v$ showing the nonzero uniform solutions of Eqs. (2): single-mode Hopf (thick black line), single-mode pitchfork (thin blue line), and dual-mode pitchfork-Hopf (vertical-ticked green line). Solid (dashed) lines represent stable (unstable) solutions. The unshaded range denotes the stability range of the single-mode Hopf solution. Parameters: $η = 0.8$ , $κ = 0$ , $μ = 0$ , $δ = 0.8$ , $γ = - i$ , and $β = 0$ .Reuse & Permissions
Figure 3
(Color online) Vortex solutions of the Hopf-Pitchfork system. (a) A single-mode vortex with $v = 0$ at the core, $λ = 0.62$ . (b) and (c) Dual-mode vortices with (b) $v > 0$ and (c) $v < 0$ at the core, $λ = 0.635$ . The top frames show the phase $φ = arg A$ in the $x − y$ plane (colored or shaded from $[0, 2 π]$ in a circle around the origin) and the bottom frames show the amplitude $ρ = | A |$ and $v$ at cross section at $y = 0$ . The initial condition is $A = exp (i φ)$ , $φ = arctan (y / x)$ , and $v = \pm 0.1 exp [- (x^{2} + y^{2}) / 5]$ . Parameters: $η = 0.8$ , $κ = 0$ , $μ = 0$ , $δ = 0.8$ , $γ = - i$ , $α = 0$ , $β = 0$ , $d = 1$ , and $x = y = [- 32, 32]$ , with no-flux boundary conditions.Reuse & Permissions
Figure 4
(Color online) Vortex oscillations induced by spontaneous hosting around the core. Shown are the phase $φ$ (top) and cross sections of $ρ = | A |$ and $v$ (center) at three different times. (a) Phase gradients induced by the hosted $v$ field reduce the oscillation amplitude near the core. (b) The oscillation amplitude decreases and leads to spontaneous hosting. (c) The low amplitude regions diminish as the phase gradients travel outward setting the stage for a new cycle of hosting. (d) The resulting core oscillations produce a time-periodic change of $v$ at the core center. The initial conditions and parameters are the same as in Fig. 3 with $λ = 0.638$ .Reuse & Permissions
Figure 5
(Color online) STC induced by spontaneous hosting far from the pitchfork bifurcation, shown in snapshots of the phase (a) and (c) $φ$ and (b) and (d) $v$ . (e) Strong hosting events induce zeros of the oscillation amplitude and spontaneous vortex-pair nucleation with a fluctuating number of vortices. (c) and (d) Bistability of STC states with $v$ positive (blue, right half) and negative (red, left half) allows for spatial mixture of the two chaotic states and for fronts separating them (c) and (d). The parameters are the same as Fig. 3 with $λ = 0.75$ and $x = y = [- 64, 64]$ .Reuse & Permissions

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