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# Parameterization method for unstable manifolds of delay differential equations

The second author was partially supported by NSF grant DMS - 1318172, NSF grant DMS- 1700154, and by the Alfred P. Sloan Foundation grant G-2016-7320.
• This work is concerned with efficient numerical methods for computing high order Taylor and Fourier-Taylor approximations of unstable manifolds attached to equilibrium and periodic solutions of delay differential equations. In our approach we first reformulate the delay differential equation as an ordinary differential equation on an appropriate Banach space. Then we extend the Parameterization Method for ordinary differential equations so that we can define operator equations whose solutions are charts or covering maps for the desired invariant manifolds of the delay system. Finally we develop formal series solutions of the operator equations. Order-by-order calculations lead to linear recurrence equations for the coefficients of the formal series solutions. These recurrence equations are solved numerically to any desired degree.

The method lends itself to a-posteriori error analysis, and recovers the dynamics on the manifold in addition to the embedding. Moreover, the manifold is not required to be a graph, hence the method is able to follow folds in the embedding. In order to demonstrate the utility of our approach we numerically implement the method for some 1, 2, 3 and 4 dimensional unstable manifolds in problems with constant, and (briefly) state dependent delays.

Mathematics Subject Classification: Primary: 34K28, 34K19; Secondary: 34C45, 37L25, 37M99, 37C27.

 Citation: • • Figure 1.  Parameterization of a 2D local unstable manifold attached to the origin for Wright's equation with $\alpha = 2.2$. The system has an attracting periodic orbit (yellow). We integrate an orbit (red) on the local unstable manifold until it converges to the periodic orbit. We compute the parameterization to Taylor order $K = 130$ by solving the recurrence equation (4). The coordinates used in the figure are discussed in Section A.1.3

Figure 2.  Boundary torus of a parameterized local unstable manifold attached to a periodic orbit for Wright's equation with $\alpha = 9$. We compute to Taylor order $K=42$ and Fourier order $M = 22$. The surface plotted in the frames is obtained by evaluating and plotting the image of the resulting Fourier-Taylor polynomial and exploits no numerical integration procedures. The torus is embedded in an infinite dimensional phase space so has no inside and outside. The coordinates used in the figure are discussed in Section A.1.

Figure 3.  The attractor (yellow) and the unstable manifold (red) of the cubic Ikeda equation at $u=0$. The attractor is computed simply by integrating arbitrary initial conditions using a standard DDE integrator. The unstable manifold is obtained by plotting the parameterization computed to 120 terms. Note that the parameterization captures several turns in the manifold, i.e. is far from the linear approximation (is not for example the graph of any function). The parameterization turns several times quite sharply near the ends

Figure 4.  Left frame: the first 9 eigenvalues of the linearized cubic Ikeda equation at the origin. Right frame: the decay of the coefficients of the parametrization of the unstable manifold at $u=0$

Figure 5.  Left frame: the first 8 eigenvalues of the linearized cubic Ikeda equation at the points $u=\pm1$. Right frame: the decay of the coefficients of the parametrization of the unstable manifolds at $u=\pm1$

Figure 6.  The equilibria (black dots), attractor (yellow) and two-dimensional manifolds at $u=\pm1$ (green) as seen from two different angles. Note how these two manifolds wrap closely around each other near the origin

Figure 7.  The decay of the coefficients of the parametrization of the unstable manifold of the minimal periodic solutions of the cubic Ikeda equation

Figure 8.  The attractor (yellow) and two-dimensional manifolds (blue) of the periodic orbits (magenta). The attractor is computed simply by integrating arbitrary initial conditions using a standard DDE integrator. The periodic orbit is computed using 140 Fourier modes. The unstable manifold is parameterized to Taylor order 51, where again each Taylor coefficient is computed with 140 Fourier modes

Figure 9.  Left frame: the first 8 eigenvalues of the linearized Wright's equation at the point $u=0$. Right frame: the decay of the coefficients of the parametrization of the unstable manifold at $u=0$

Figure 10.  The two-dimensional manifold at $u=0$ (green) as seen from two different angles

Figure 11.  Left frame: the first 10 eigenvalues of the linearized Wright's equation at the point $u=0$. Right frame: the decay of the coefficients of the parametrization of the unstable manifold at $u=0$

Figure 12.  The two-dimensional fast manifold (red) and the two-dimensional slow manifold (green) manifold at $u=0$ as seen from two different angles. The unstable periodic orbit of period $T \approx 0.805$ is shown in yellow. The manifolds seem to intersect one another (and the periodic orbit) because of the projection to 3 dimensions

Figure 13.  Several tori along the boundary of the 4D unstable manifold of the $u=0$ equilibrium of Wright's equation for $\alpha = 9.0$. Depicted are the tori corresponding to $c \in \{0.99, 0.61, 0.21, 0.08 \}$

Figure 14.  The decay of the coefficients computed for the parametrization of the unstable manifold, corresponding to the unstable periodic solution of Wright's equation

Figure 15.  The periodic orbit (yellow) and two-dimensional submanifold (blue) of the unstable manifold of the periodic orbit corresponding to $|z| = 0.25$

Figure 16.  Top frame: decay of the Taylor coefficients. Left frame: the scaling used for each $\epsilon$. Right frame: the defect in the conjugacy for each value of $\epsilon$

Figure 17.  The unstable manifold of the state-dependent perturbation of Wright's equation, for several values of $\epsilon$. The green manifold corresponds $\epsilon = 0$, while the dark blue one at the centre corresponds to $\epsilon = 1$. Note that while these manifolds do lie closely together, they do not exactly overlap

Figure 18.  Left frame: solution of a delay equation with initial history segment colored in red. Right frame: solution of same delay equation with the evolution of the initial segment by time $t = a$ colored red

Figure 19.  The phase space of a delay differential equation. The intuition is that we change to coordinates which "move with" the window of length $\tau$

Figure 20.  Left frame: the evolution of the delay embedding coordinates (green) corresponding to the solution of a delay equation with a given initial history (red). Right frame: the same delay embedding coordinates, but embedded in $\mathbb{R}^3$

Figure 21.  the dynamical meaning of the conjugacy described in equation (39). The desired chart map conjugates the dynamics on the unstable manifold to the linear flow generated by the unstable eigenvalues

Figure 22.  Unstable manifold of a periodic orbit $\gamma$

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