On the choice of coarse variables for dynamics

Amit Acharya

Home/Archive/arXiv

arXiv · 0704.1003

On the choice of coarse variables for dynamics

Amit Acharya

View original source ↗Read source document ↗Export metadata ↓

Abstract

Two ideas for the choice of an adequate set of coarse variables allowing approximate autonomous dynamics for practical applications are presented. The coarse variables are meant to represent averaged behavior of a fine-scale autonomous dynamics.

Explore related subjects

nlin.CD math.DS

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Amit Acharya. 2007-04-09. On the choice of coarse variables for dynamics. https://arxiv.org/abs/0704.1003

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Information transfer as a new indicator of approaching local bifurcations: saddle-node, pitchfork and Hopf bifurcations

Early warning signals are key tools for anticipating qualitative changes in the dynamics of complex systems. In this work, we investigate whether an information-theoretic measure designed to characterize directional causal dependencies between observables can provide early warning of the approach to bifurcations. The measure considered is the rate of information transfer, which has been successfully applied to characterize directional dependencies in a broad range of fields, from economics to climate science. Three fundamental local bifurcations are considered: saddle-node, pitchfork, and Hopf bifurcations. We first investigate idealized (quasi-)potential stochastic systems for which analytical expressions for the rate of information transfer can be derived. Distinct signatures are found for the three bifurcation types. In particular, the rate of information transfer displays a strong increase and divergence when approaching the saddle-node bifurcation, whereas smoother but systematic changes are found near the pitchfork and Hopf bifurcations. The approach is then tested using low-order climate models exhibiting the same types of bifurcations. The results confirm that changes in the rate of information transfer can precede the corresponding dynamical transitions, suggesting that directional information transfer provides a useful complementary early-warning indicator for detecting the approach to critical transitions in nonlinear stochastic systems.

nlin.CD↗

arXiv2026

Beyond RMSE and Granger Gain: Multi-Horizon Forecast-Error Entropy for Nonlinear Predictive Coupling

We investigate whether the past of an auxiliary time series provides predictive information about a target beyond that contained in the target's own history. We compare matched out-of-sample target-only and target-plus-source forecasts and introduce the forecast-error distribution entropy (FEDE) predictive-causality framework, where predictive causality is understood in the Granger sense: auxiliary history is informative when it improves prediction beyond the target's own past. FEDE quantifies how the full signed forecast-error distribution changes across prediction horizons. Controlled nonlinear benchmarks hide source-target dependence behind nonlinear transformations not supplied to the predictors. LightGBM recovers these hidden channels, whereas Ridge does not. Across four equal-power nonlinearities and four noise levels, the mean response-sensitivity statistic for FEDE remains 29.00, 27.47, 23.22, and 28.67, compared with 14.09, 10.68, 12.67, and 12.71 for mean log-RMSE gain. Directional and source controls separate the true source-to-target channel from the reverse direction and from independent periodic, quasiperiodic, chaotic, and autoregressive auxiliaries. Chronological Granger-style comparisons show strong model-form dependence: a degree-2 polynomial model is exceptionally sensitive to the matched quadratic channel but becomes weak or oppositely aligned for other nonlinearities, while basic convergent cross mapping is weak or directionally inconsistent in this stochastic, partly non-invertible setting. FEDE remains positively aligned across all 108 tested combinations of window length, history length, and maximum forecast horizon. Together, these results establish FEDE as a sensitive and robust horizon-resolved tool that extracts predictive information beyond conventional error-magnitude measures when the nonlinear interaction form is unknown. Code available in Zenodo.

nlin.CD↗

arXiv2026

Chimera States in Wheel Networks

Chimera states, characterized by the coexistence of coherent and incoherent dynamics, are a hallmark of symmetry-breaking phenomena in networks of coupled oscillators. Here, we investigate the role of higher-order interactions in the emergence of chimera states in wheel networks of Rössler and Lorenz oscillators with pairwise and higher-order interactions under diffusive, conjugate, and mean-field diffusive coupling. Using multiple random initial conditions, we quantify the prevalence of synchronized, desynchronized, chimera, and synchronized cluster states. Chimera states emerge across all three coupling mechanisms as symmetry-broken states among the structurally and dynamically equivalent peripheral oscillators. Higher-order interactions substantially reorganize the dynamical landscape; in particular, they shift chimera states toward weaker pairwise coupling under diffusive coupling, predominantly support them at moderate coupling strengths under conjugate coupling, and yield comparatively less prevalent chimera states under mean-field diffusive coupling. These effects depend strongly on the intrinsic oscillator dynamics, particularly at larger higher-order coupling strengths. Overall, higher-order interactions neither universally promote nor suppress chimera states; rather, their influence is governed by the interplay between network symmetry, coupling mechanism, and intrinsic oscillator dynamics. The observed collective states are quantitatively characterized using two proposed statistical measures, while master-stability-function analysis provides a stability-based explanation for the loss of complete synchronization and the emergence of nonsynchronous dynamical regimes.

nlin.CD↗