Position: A Dynamical Systems Perspective is Needed to Advance Time Series Modeling

Daniel Durstewitz; Christoph Jürgen Hemmer; Florian Hess; Charlotte Ricarda Doll; Lukas Eisenmann

Figure 6: Normal forms and bifurcation diagrams for saddle-node and supercritical pitchfork and Hopf bifurcations. Shown are the stable and unstable equilibria as solid and dashed black lines, respectively, and the limit cycle as shaded region between its minimum–maximum branches (gray lines), as the control parameter

c

in the respective normal form is varied.

$I$	$C$	$g_{L}$	$E_{L}$	$g_{\mathrm{Na}}$	$E_{\mathrm{Na}}$	$V_{h,\mathrm{Na}}$	$k_{\mathrm{Na}}$	$g_{K}$	$E_{K}$	$V_{h,K}$	$k_{K}$	$\tau_{n}$	$g_{M}$	$V_{h,M}$	$k_{M}$	$\tau_{h}$	$g_{\mathrm{NMDA}}$	$E_{\mathrm{NMDA}}$
$0$	$6$	$8$	$-80$	$20$	$60$	$-20$	$15$	$10$	$-90$	$-25$	$5$	$1$	$25$	$-15$	$5$	$200$	$10.2$	$0$

Appendix

Neuron model parameter settings

Figure 7: Illustration of DSR process. “

\approx

” is meant here in the sense of topological conjugacy (cf. Def. A.4 ) and (potentially) geometric & temporal invariants (like the Lyapunov spectrum or fractal dimension).

Figure 8: Illustration of DSR measures: Comparison of geometric (dis)agreement (

D_{\textrm{stsp}}

), power spectral distance (

D_{\textrm{H}}

), Kaplan-Yorke fractal dimension (

D_{\textrm{KY}}

), and max. Lyapunov exponent (

\lambda_{\textrm{max}}

) on reconstructions of the chaotic Lorenz-63 system of different quality from very poor ( a ) to excellent ( d ), obtained by an AL-RNN assessed at different training stages.

Figure 9: N-tipping in state space: Illustration of the basin boundary crossing of the noisy trajectory from Fig. 1 b.

Figure 10: A DSR model trained on just one short trajectory (center) from the chaotic Rössler ( a ) or Lorenz ( b ) system (left) correctly reconstructs the full chaotic attractor (red trajectory), generalizes to novel initial conditions (red dots), and correctly infers the location of fixed points not shown in training (yellow dots). Part (a) based on Suppl. Fig. 4 of Durstewitz et al. (2023) .

Figure 11: Although only trained on a short trajectory snippet (gray curve) from the Selkov system (left), the trained DSR model (shallow PLRNN ( Hess et al., 2023 ) ; right) correctly infers properties of the surrounding vector field (darker shading = higher flow velocity), enabling it to generalize to new initial conditions.

Figure 12: Illustration of a DSR training corpus. Time series (center) from different chaotic attractors (left), generated by the equations on the right. The top time series are from non-autonomous systems where a control parameter (in light blue on the right) was systematically varied, while the bottom time series are from autonomous and stationary system simulations (

135

such systems are collected in Gilpin (2022) , from which through recombination

\approx 10^{4}

systems are created in Lai et al. (2025) ).

Figure 13: a ) Example of generic skew-product form of a dynamical system that implements non-autonomous DS by separating a slow base (driver) system from a fast fiber (forced, dependent) system ( Kloeden and Yang, 2020 ) . b ) Specific example (using a piecewise-linear RNN) of how such a form may be enforced in training through a particular regularization (inductive bias) on parameters which encourages the RNN to segregate its latent space into base and fiber system, inspired by Schmidt et al. (2021) .

Figure 14: Long-term forecasts of ETTh1 dataset for different DSR and TS models.

Figure 15: Long-term forecasts of weather temperature dataset for different DSR and TS models.

System	ALRNN			shPLRNN			RC			AutoARIMA			PatchTST			NBEATS
	$D_{\text{stsp}}$	$D_{H}$	MASE	$D_{\text{stsp}}$	$D_{H}$	MASE	$D_{\text{stsp}}$	$D_{H}$	MASE	$D_{\text{stsp}}$	$D_{H}$	MASE	$D_{\text{stsp}}$	$D_{H}$	MASE	$D_{\text{stsp}}$	$D_{H}$	MASE
ETTh1	$0.46\pm 0.34$	$0.16\pm 0.02$	$3.09\pm 0.07$	$0.21\pm 0.13$	$0.18\pm 0.01$	$1.61\pm 0.08$	$0.35\pm 0.01$	$0.17\pm 0.01$	$1.09\pm 0.00$	$3.92$	$0.22$	$1.46$	$3.85\pm 2.66$	$0.16\pm 0.05$	$4.78\pm 1.82$	$3.14\pm 0.57$	$0.38\pm 0.13$	$3.77\pm 0.42$
Traffic	$1.14\pm 0.57$	$0.14\pm 0.03$	$1.90\pm 0.04$	$0.25\pm 0.08$	$0.09\pm 0.01$	$2.35\pm 0.06$	$0.10\pm 0.02$	$0.13\pm 0.02$	$1.26\pm 0.27$	$8.98$	$0.41$	$2.16$	$1.23\pm 0.68$	$0.24\pm 0.10$	$0.98\pm 0.08$	$1.54\pm 0.12$	$0.13\pm 0.03$	$0.92\pm 0.06$
Cloud Requests	$1.28\pm 0.12$	$0.12\pm 0.00$	$3.02\pm 0.05$	$1.14\pm 0.10$	$0.12\pm 0.00$	$2.88\pm 0.03$	$0.73\pm 0.00$	$0.74\pm 0.01$	$23.12\pm 2.06$	$0.61$	$0.10$	$1.90$	$3.21\pm 2.04$	$0.15\pm 0.03$	$2.40\pm 0.37$	$2.10\pm 0.34$	$0.12\pm 0.02$	$4.05\pm 0.79$
Weather (Temp.)	$0.73\pm 0.02$	$0.11\pm 0.00$	$1.80\pm 0.04$	$0.71\pm 0.28$	$0.12\pm 0.00$	$1.62\pm 0.12$	$0.07\pm 0.02$	$0.12\pm 0.03$	$2.53\pm 0.28$	$0.11$	$0.11$	$2.60$	$1.04\pm 0.87$	$0.12\pm 0.03$	$3.19\pm 1.25$	$1.22\pm 0.83$	$0.11\pm 0.02$	$2.09\pm 0.12$
Weather (Press.)	$0.23\pm 0.09$	$0.29\pm 0.02$	$4.66\pm 0.22$	$0.25\pm 0.03$	$0.30\pm 0.02$	$4.80\pm 0.24$	$1.90\pm 1.05$	$0.89\pm 0.05$	$4.68\pm 0.62$	$13.26$	$1.00$	$4.63$	$5.90\pm 0.38$	$0.45\pm 0.01$	$5.63\pm 0.54$	$5.42\pm 2.45$	$0.49\pm 0.18$	$8.47\pm 3.98$

Appendix

Table 1: Performance comparison of custom trained DSR and TS models on empirical time series in terms of geometrical divergence (

D_{\text{stsp}}

), long-term temporal distance (

D_{H}

), and forecast error (MASE). Best in red , second-best in blue .

System	DynaMix			Chronos-t5-base			Chronos-2			TiRex			Panda
	$D_{\text{stsp}}$	$D_{H}$	MASE	$D_{\text{stsp}}$	$D_{H}$	MASE	$D_{\text{stsp}}$	$D_{H}$	MASE	$D_{\text{stsp}}$	$D_{H}$	MASE	$D_{\text{stsp}}$	$D_{H}$	MASE
ETTh1	$0.60\pm 0.01$	$0.15\pm 0.00$	$1.61\pm 0.41$	$1.58\pm 0.23$	$0.20\pm 0.01$	$1.30\pm 0.03$	$3.44$	$0.16$	$1.71$	$9.20$	$0.29$	$1.46$	$8.18$	$0.69$	$2.86$
Traffic	$0.86\pm 0.02$	$0.18\pm 0.02$	$0.98\pm 0.10$	$0.78\pm 0.24$	$0.11\pm 0.01$	$0.62\pm 0.01$	$0.55$	$0.13$	$0.65$	$3.41$	$0.81$	$0.59$	$4.47$	$0.45$	$0.99$
Cloud Requests	$0.67\pm 0.00$	$0.07\pm 0.01$	$2.62\pm 0.02$	$3.72\pm 1.91$	$0.53\pm 0.08$	$2.43\pm 0.02$	$2.10$	$0.14$	$1.66$	$3.06$	$0.89$	$1.44$	$7.20$	$0.23$	$2.38$
Weather (Temp.)	$0.06\pm 0.00$	$0.07\pm 0.01$	$1.89\pm 0.04$	$8.92\pm 3.75$	$0.44\pm 0.24$	$7.24\pm 0.40$	$2.27$	$0.17$	$1.59$	$3.00$	$0.90$	$1.91$	$1.15$	$0.14$	$1.60$
Weather (Press.)	$0.18\pm 0.01$	$0.26\pm 0.01$	$4.21\pm 0.02$	$3.23\pm 0.67$	$0.54\pm 0.00$	$4.76\pm 0.16$	$11.16$	$0.47$	$4.94$	$7.29$	$0.75$	$4.75$	$10.23$	$0.68$	$4.72$

Appendix

Table 2: Performance comparison of DSR and TS foundation models on empirical time series in terms of geometrical divergence (

D_{\text{stsp}}

), long-term temporal distance (

D_{H}

), and forecast error (MASE). Best in red , second-best in blue .

Figure 16: Example forecast of the chaotic Lorenz-63 system using DynaMix and Chronos-t5-base. Chronos’ tendency to parrot its context input leads to a cyclic repetition of a fixed pattern, and Chronos therefore fundamentally fails to capture the truly aperiodic, chaotic behavior. This manifests as cyclic trajectories in the delay-embedded (cf. Appx. A ) state space (left), the close-to-zero maximum Lyapunov exponent (center) and by a sharply peaked (instead of the truly broad) power spectrum (right). Results reproduced from Hemmer and Durstewitz (2025) .

Figure 17: TS forecasts using custom trained AutoARIMA.

Figure 18: TS forecasts using custom trained shallow PLRNN.

Figure 19: TS forecasts using custom trained AL-RNN.

Figure 20: TS forecasts using custom trained Reservoir Computers.

Figure 21: TS forecasts using custom trained PatchTST.

Figure 22: TS forecasts using custom trained NBEATS.

Figure 23: Zero-shot TS forecasts using DynaMix FM.

Figure 24: Zero-shot TS forecasts using Chronos-t5-base FM.

Figure 25: Zero-shot TS forecasts using Chronos-2 FM.

Figure 26: Zero-shot TS forecasts using TiRex FM.

Figure 27: Zero-shot TS forecasts using Panda FM.

Figure 28: Impact of observability, noise, and DSR-specific training methods on short-term prediction. a ) The number of observed variables has only little influence on reconstruction quality (as measured by

D_{\text{stsp}}

), as illustrated here for an ALRNN ( Brenner et al., 2024a ) trained by STF ( Mikhaeil et al., 2022 ) on the

20d

Lorenz-96 system. b ) Short-term prediction error (MASE on 64 step forecasts) increases with noise level (N/S ratio), but does so more strongly for TS models (blue) than for DSR models (red), for stochastic Lorenz-63. c ) Optimal DSR training improves short-term prediction (tested on stochastic Lorenz-63): Training a shPLRNN by GTF ( Hess et al., 2023 ) or STF ( Mikhaeil et al., 2022 ) with dynamically optimal forcing interval (

\tau=16

) yields lower forecast errors than with suboptimal forcing (

\tau=100

) or without GTF; see Mikhaeil et al. (2022) and Hess et al. (2023) who observed the same for DSR performance. Error bars = MAD.

Figure 29: Predictions from a shallow PLRNN ( Hess et al., 2023 ) trained via sparse teacher forcing ( Mikhaeil et al., 2022 ) on sequential MNIST with the digit label provided as an external input. For each digit, a forecast was produced (top panels) and then reshaped to reconstruct the corresponding image (bottom panels).

Figure 30: Proof-of-concept example that other types of architecture than typically used in the DSR literature could also profit from DSR training techniques, in this case a Mamba-2 block ( Dao and Gu, 2024 ) followed by a 1-hidden-layer MLP trained with (red) or without (blue) GTF ( Hess et al., 2023 ) on human EEG data (gray). Both long-term performance measures (

D_{\text{stsp}}

with vs. w/o GTF:

4.86

vs.

11.45

,

D_{\text{H}}

:

0.13

vs.

0.42

) as well as the short-term prediction error (

\text{MASE}(20)

:

10.1

vs.

14.2

) are improved by GTF.

Acronym	Definition
AI	Artificial Intelligence
AL-RNN	Almost-Linear Recurrent Neural Network
AR	Auto-Regressive
ARMA	Auto-Regressive Moving-Average
AutoARIMA	Automatic Auto-Regressive Integrated Moving Average
DS	Dynamical System(s)

Appendix

Table 3: Acronyms used throughout the main text.

Position: A Dynamical Systems Perspective is Needed to Advance Time Series Modeling

Abstract

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