Figure 5

Non-linear system response around the instability. (a) Time evolution of the dominant mode amplitude \(|u_{0,1}|\) as a function of time for different \(\Omega _H\) as indicated and modulation amplitudes (1) \(s_{z,M}=0.1\) and (2) \(s_{z,M}=0.2\), respectively (cf. trajectory III and IV in Fig. 1). Either of these modulation starts at \(t=0\), before only a static field \(s_{z,S}=0.4\) (\(s_{z,M}=0.0\)) is present. For clarity/visibility mode amplitudes \(|u_{0,1}|\) are only shown until \(t=25\) in case of \(\Omega _H\geqslant 0.5\) (1a). (b) Temporal oscillations of the control function \(s_z(t) = s_{z,S} + s_{z,M}\sin {(\Omega _h t)}\). The dotted black and dashed red lines mark the stationary (\(s_{z,S}=0.4\), \(s_{z,M}=0.0\)) and high frequency limit oscillatory (\(s_{z,S}=0.4\), \(s_{z,M}=0.1\)) bifurcation threshold, respectively. (c) as (a) but as a function of the reduced time \(t/T_H\). The red squares show the response to stationary magnetic field with magnetic field strength \(s_z\) given by the actual value of \(s_z(t)\). The red dashed line in (1c) indicates the (time averaged) mode amplitudes for modulated driving (\(\Omega _H=50,\, 100\) almost falls on top of it). Note, that in (1) for modulated driving with \(\Omega _H \gtrsim 0.27\) the system remains supercritical. Other for (2) at which for modulated driving with \(\Omega _H \gtrsim 0.27\) the system remains subcritical. Further control parameter \(Re=80\).