Figure 1

Switching diagram: open (blue) boxes describe the vortex dynamics without switching and others correspond to parameters where the polarity reversal is observed [the red circles indicate parameters where the vortices escape from the origin during the first 10 ns, the open (yellow) circles represent parameters where the autocorrelation function (2) rapidly decays, for the (black) diamonds the switching process is periodical, and the green triangles correspond to an intermittent process]. (a) The example of the intermittent process. (b) The examples of the vortex trajectories in case of the complex vortex dynamics. Black points mark the places of the polarity switching. The trajectories $b$${}_1$ and $b$${}_2$ correspond to 12 and 14 GHz, respectively. (c) The example of the chaotic process. (d) The example of the regular process.Reuse & Permissions

Figure 2

Nonlinear resonance curves from micromagnetic simulations. Insets: (a) The temporal evolution of the polarity, (b) the FFT spectrum of the vortex polarity for $B_0=9$ mT, $f=13.2$ GHz during 15 ns. Arrow indicates the pumping frequency.Reuse & Permissions

Figure 3

Autocorrelation function (2) for $B_0=110$ mT and $f=18$ GHz (blue dashed line) and $B_0=70$ mT and $f=14$ GHz (red solid line). The first process demonstrates a periodical process with periodical $C\left(t\right)$ and the second one demonstrates a chaotic behavior with rapidly decaying $C\left(t\right)$.Reuse & Permissions

Figure 4

FFT spectrum of the vortex polarity ($B_0=70$ mT, $f=14$ GHz): solid line corresponds to the numerical data, the dashed line is the fitting to the pink noise.Reuse & Permissions

Figure 5

(a) The pseudophase diagram for the chaotic vortex polarity dynamics (the time step 1 ns, the filled circles mark polarities $p=\pm 1$); (b) the same diagram for the regular dynamics.Reuse & Permissions

Figure 6

Comparison of 2D and 3D simulations. Symbols correspond to 3D simulations: The red circles describe the switching process, the open blue boxes describe the dynamics without switching. The dotted line represents the border of the switching region for 3D simulations, colored domains correspond to 2D simulations (see Fig. 1).Reuse & Permissions

Figure 7

Schematic of the reduced core model: Thick red arrows (numbers $\overline{1,4}$) indicate free magnetic moments and thin black ones (numbers $\overline{5,12}$) indicate fixed magnetic moments with $m_{\mathit{n}}^z=0$. The turning phase $\psi$ describes the deviation of the magnetization angle from the equilibrium (see Appendix for details).Reuse & Permissions

Figure 8

Nonlinear resonance curve: The numerical solution of Eqs. (7) (the dashed line) and the analytical solution (12) (the solid line) for the following parameters: $\Lambda =0.9415$, $h=0.0002$, $\eta =0.01$, $\mu \left(0\right)=0.337$, $\psi \left(0\right)=0$. The effective double-well potential $\mathcal{U}$ [see Eq. (9)] is plotted in the inset.Reuse & Permissions

Figure 9

Map of dynamical regimes in the coordinates $\omega /{\omega }_0-h/\eta$. $\eta =0.002$, other parameters are the same as in Fig. 8. The red region corresponds to the chaotic dynamics. Three scales of gray indicate (in order of growing intensity) the switching process with one stable focus on the Poincaré diagram and the final dynamics near the upward or downward polarity. The blue region corresponds to the dynamics with more than one stable focus on the Poincaré diagram. (a) The phase diagram (projection) for the regular dynamics. The black circles indicate the equilibrium polarities $\pm {\mu }_0$. (b) Example of the regular dynamics, including the $f/3$ peak in FFT spectrum. (c) Example of the chaotic oscillations. ${\tau }_1=186\times {10}^3$. (d) The same as (a) for the chaotic dynamics. (e) The same as (b) for the dynamics with five stable focuses on the Poincaré map. (f) Example of a process with the final state near $-{\mu }_0$.Reuse & Permissions

Figure 10

Evolution of strange attractors with change of pumping frequency. $h/\eta =10.5$, other parameters are the same as in Fig. 9. The number $P$ means number of points on the corresponding Poincaré map.Reuse & Permissions

Figure 11

(a) Number of switchings as function of number of oscillations in the wave train ($N$) and the field intensity ($B_0$). Symbols correspond to values $N$ and $B_0$, where unidirectional switching is observed. (b) Controlled unidirectional switching by sequence of short-wave trains (contained $N=6$ periods on frequency 14 GHz, duration 0.43 ns) with period 3.2 ns. Gray regions show time of applying external magnetic field. (c) The same for the core model. Separate pulse contains $N=3$ periods (duration $58.9$ in arb. units), interval between pulses 589 arb. units.Reuse & Permissions