Lyapunov exponents of Eq. (23) at the trivial equilibrium for . Stable exponents are shown in black, while the only unstable exponent is shown as an open circle. Here, .
Comparison of the numerically reconstructed flow defined by Eq. (11) with the original flow of Eq. (23) for I(t) = 0 based on the calculations performed Sec. III B. For a specific initial condition, the solution of Eq. (23) approaches the fixed point . The reconstructed flow is shown with different accuracy levels achieved using higher order terms in the center manifold formulation. To approach the original flow (dashed curve labelled 1), the order parameter equation (Eq. (41)) uses cubic (2) and quadratic (3) order expansions for the center manifold. The trivial ansatz h = 0 (4) is also shown. Parameters are . Initial conditions were chosen to exhibit a monotonic decay towards the equilibrium and were the same for all simulations over the interval , so that where we have . An Euler integration scheme was used.
Comparison of numerical simulations of the reconstructed flow from Eq. (11) with the original flow of Eq. (23) with weak and fast time-periodic driving of the form . The numerical solution of Eq.(23) (dashed curve labelled1) is shown along with the reconstructed flow with cubic order expansion ( in Eqs. (39) and (40)) of the center manifold, and plotted with (2) and without (3) the time-dependent correction (Eq. (49)). The reconstructed flow using the trivial ansatz h = 0 is also shown (4). Further, we add the flow of Eq. (23) without any delay (i.e., for ) (5), showing the difference in the system's response to forcing when no delay is present, and how this difference is accurately captured by the time-dependent center manifold approach. The time-corrected center manifold follows the solution of the DDE with improved accuracy. Parameters are . The initial conditions are chosen the same for each system, i.e., for , where we set .
Deviations between the reconstructed flow governed by Eq. (44) and the original flow of Eq. (23) near a transcritical bifurcation using both standard and time-dependent center manifold reductions with an additive forcing term . As the two flows are in phase, the error is defined as the absolute difference between original and reconstructed flows at input maxima divided by the response amplitude of the original system, taken at the maxima of the oscillation. Panel (a) shows the error when an additive time-dependent component to the center manifold is used, as in Eq. (49). Panel (b) shows the error for the same problem without any time-dependent considerations in the center manifold calculations. Parameters are , , and .
Comparison of the numerical solutions of reconstructed flow according to Eq. (11) with the original flow of Eq. (23) with weak and fast time periodic driving. The input is a superposition of N = 10 oscillations defined by Eq. (51) where the amplitudes follow a Gaussian distribution with mean and standard deviation . Panel (a) shows the original and reconstructed time series and panel (b) is the focus onto the grey-shaded area in (a). The original flow (1, dashed curve) is compared to the reconstructed flow using a center manifold of cubic order expansion, and plotted with (2) and without (3) time-dependent correction. The trivial ansatz h = 0 is also shown (4). In addition, (5) denotes the flow of Eq. (23) for . We observe that the reconstructed flow with time-dependent center manifold follows the original system with great accuracy. Parameters are , , and .
Comparison of the numerical solutions of reconstructed flow according to Eq. (11) with the original flow of Eq. (23) with white noise. The small inset shows a solution path x(t) of the original equation (23), the reconstructed solutions considering the full reduction method, and u(t) alone neglecting the center manifold reduction. The large panel presents a focussed view of the solutions andreveals a very good approximation of the original solution by the reduced model solution. The numerical integration has been performed according to the delayed Euler-Maruyama scheme with time step dt = 0.001. Parameters are . Initial conditions have been chosen as for and u(0) = 0.
Reconstruction error of the method for random inputs and various time delays and noise strengths. The reconstruction error is defined as with the original solution x(t) gained from Eq. (23) and the reconstructed solution obtained from Eqs. (54) and (55). Since the solutions are non-stationary in time, the mean error is computed as the ensemble average over 100 trajectories at an early ((a), ) and a late ((b), ) time instant. The other parameters are identical to the one in Fig. 6.
Additive noise moves the probability density function of the delayed system to higher amplitudes. The probability density functions are given for the original system (all line formats) and order parameter equation (squares) which are in very good agreement. Parameters are (a) , (thick solid line), (dashed thin line), (thin line); (b) (thick solid line), (dashed thin line) and (thin line). The probability has been estimated by trials at t = 5. Other parameters are with the initial conditions (a) , (b) and in both cases for . To ensure the existence of probability density functions, we have introduced absorbing barriers at x = xu and u = xu.
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