Analytic Continuation and Differential Geometry Views on Slow Manifolds and Separatrices

Analytic continuation and differential geometry views on slow manifolds and separatrices

Dirk Lebiedz, Jorn¨ Dietrich, Marcus Heitel, Johannes Poppe Institute for Numerical Mathematics, Ulm University, Germany

Abstract— We start from a mechano-chemical analogy con- from corresponding mathematical fields, in particular those sidering the time evolution of a homogeneous chemical re- that relate (complex) analysis with differential geometry and action modeled by a nonlinear dynamical system (ordinary topology. Since the phase flow in imaginary time direction differential equation, ODE) as the movement of a phase space point on the solution manifold such as the movement of a is related to complex exponential functions [3], it turns out mass point in curved spacetime. Based on our variational to be natural to study Fourier transforms of imaginary time problem formulation [6] for slow invariant manifold (SIM) trajectories to analyze spectral properties of the dynamical computation and ideas from general relativity theory we system. For linear systems classical complex analysis re- argue for a coordinate free analysis treatment [5] and a veals that the spectrum of the system Jacobian is reflected differential geometry formulation in terms of geodesic flows [8]. In particular, we propose analytic continuation of the by oscillatory modes in imaginary time direction (see Fig. dynamical system to the complex time domain to reveal deeper 1,2 and [3]). We demonstrate for some (linear model and structures and allow the application of the rich toolbox of nonlinear Davis-Skodje) benchmark example systems with Fourier and complex analysis to the SIM problem. time scale separation that presence respectively absence of high frequency modes in the Fourier spectrum distinguish I.ANALYTIC CONTINUATION initial values on or off slow invariant attracting manifolds We introduce analytic continuation of smooth autonomous of given dimension. dynamical systems from real to complex-time domain in Holomorphic curves respectively Riemann surfaces over order to study slow invariant manifolds (SIM) in terms of complex time allow the definition of a symplectic form spectral properties, geometry and topology of holomorphic related to a surface integral within the solution mani- curves, respectively embedded Riemann surfaces. Slow at- fold spanned by dynamical system trajectories. This non- tracting manifolds and, closely related to these, separatrices degenerate bilinear 2-form can be discussed in the context [4] play an important role as backbone structures in phase of our previous studies of variational formulations of the space and allow to distinguish asymptotic behavior of trajec- SIM computation problem in the Hamiltonian framework tories. We propose that a holomorphic complex-time view [7] whose underlying cotangent bundle geometric structure might be useful for identification and analytical or numerical is symplectic. evaluation of characteristic mathematical properties of these Imaginary respectively complex time has been discussed flow-invariant structures. in various contexts of modern physical theories such as We analyze time-holomorphic solution trajectory manifolds special and general relativity, quantum mechanics and quan- of real-analytic ordinary differential equation (ODE) initial tum field theory. The time dependent Schrodinger¨ equation value problems after analytic continuation to the complex is formally of parabolic heat-equation type in imaginary time domain: time. For time-independent potential energy in the quantum n anal. contin. x˙ = f(x), x(t) ∈ , t ∈ , x(0) = x0 −−−−−−→ Hamiltonian operator (with the consequence of separa- arXiv:1904.04613v1 [math.DS] 9 Apr 2019 R R n bility of the time-dependent Schrodinger¨ equation), time- z˙ = F (z), z = x + iy ∈ C , t = σ + iτ ∈ C, z(0) = z0 d differentiation i dt can be interpreted as an energy operator. The real part is

2 2 6

Re(z S 6 4

4 Ts*S 2 s* 2 ) 2

0 i¡ (z 0 s*+e w -2

-4 s*+w -2 -6 5 -4 N S 5 s* 0 0 z -6 1 -5 -5 ) (z ) (z ) 2 1 1 Im(z Fig. 1. 2-D linear ODE system (eq. (1) in [7], γ = 5) in complex time: Projection of complex solution curve (Riemann surface) for given real Fig. 3. Noether’s theorem and continuous symmetry (rotation Lie group) initial value x0 = x(0) off the SIM, red line: real time solution trajectory of trajectories near SIM in complex phase space (taken from [2])

(z ) 2 5 II.DIFFERENTIAL GEOMETRY AND GEODESIC FLOW 4 5 3 Invariant manifolds in dynamical systems are intrinsic math- 2 ematical objects whose characteristics do not depend on a 1 )

2 0

(z chosen coordinate system (e.g. reaction progress variables 0

-1 for parameterization). We start from the ideas to treat the

-2 -5 slow manifold problem in the fully coordinate independent 5 -3 5 setting of differential geometry and the result that neces- 0 -4 0 sary (and in special cases also sufficient) conditions for a -5 (z ) -5 -5 (z ) 1 1 SIM can be formulated in terms of tensor analysis. The invariance equation, e.g., is reflected by vanishing time- Fig. 2. 2-D linear ODE system (eq. (1) in [7], γ = 5) in complex time: Projection of complex solution curve (Riemann surface) for given real sectional curvature in extended phase space including time initial value x0 = x(0) on the SIM, red line: real time solution trajectory as a coordinate axis [5]. In a second step we derive here a Riemannian metric which makes the solution trajectories geodesic flow lines with respect to the Levi-Civita connec- particular, we show that an extended complex analysis view tion induced by the metric [8]. In this setting the stretching- on flows of dynamical systems sheds new light on the based analysis of Adrover et al. [1] can be recast in the problem of slow invariant manifolds. language of manifold curvature concepts, see [8].

ACKNOWLEDGMENT A. Analysis of Riemann surfaces The Klaus-Tschira foundation is gratefully acknowledged We demonstrate the value of complex time dynamical for financial funding of the project. systems (analytic ODE) by analyzing a diagonalizable 2- REFERENCES D linear dynamical system with two different eigenvalues (see eq. (1) in [7] with γ = 5). Figs. 1 and 2 visualize [1] A. Adrover, F. Creta, M. Giona and M. Valorani. Stretching-based diagnostics and reduction of chemical kinetic models with diffusion. projections of the Riemann surfaces in complex phase space J. Comput. Phys. 225, 1442–1471 (2007) C2 generated by solution trajectories with initial values off [2] J. Dietrich. Symmetries of slow invariant manifolds. Master Thesis (Fig. 1) and on (Fig. 2) the SIM which is here the slow University of Ulm, Germany (2018) [3] J. Dietrich, D. Lebiedz. Slow invariant manifolds of analytic dynam- eigenspace. The oscillatory modulation of the surface can ical system. 7th IWMRRF Trondheim, Norway (2019) be correlated with an active or relaxed fast mode [3]. [4] M. Heitel, D. Lebiedz. Characterization of separatrices in holomorphic dynamical systems in the light of complex time. 7th IWMRRF Trondheim, Norway (2019) B. Symmetry considerations [5] P. Heiter and D. Lebiedz. Towards differential geometric characterization of slow invariant manifolds in extended phase space: Sectional Based on the conception that 1-D SIMs seem to ’balance’ curvature and flow invariance. SIAM J. Appl. Dyn. Syst. 17, 732–753 contraction rates of trajectories from different phase space (2018) [6] D. Lebiedz, J. Siehr, and J. Unger. A variational principle for directions in [2] an analytic continuation of real to complex computing slow invariant manifolds in dissipative dynamical systems. phase space is applied to investigate the significance of SIM SIAM J. Sci. Comput. 33, 703–720 (2011) symmetry properties (Fig. 3) exploiting Noether’s theorem. [7] D. Lebiedz, J. Unger. On unifying concepts for trajectory-based slow invariant attracting manifold computation in kinetic multiscale This might help to restrict the potential choice of a suitable models. Math. Comp. Model. Dyn. 22, 87–112 (2016) Lagrangian function in a variational setting [7] formulated [8] J. Poppe, D. Lebiedz. Stretching-based diagnostics in a differential as an inverse problem for SIM identification. geometry setting. 7th IWMRRF Trondheim, Norway (2019)