Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Classification of Systems-I01:26

Classification of Systems-I

Linearity is a system property characterized by a direct input-output relationship, combining homogeneity and additivity.
Homogeneity dictates that if an input x(t) is multiplied by a constant c, the output y(t) is multiplied by the same constant. Mathematically, this is expressed as:
First Order Systems01:21

First Order Systems

First-order systems, such as RC circuits, are foundational in understanding dynamic systems due to their straightforward input-output relationship. Analyzing their responses to different input functions under zero initial conditions reveals significant insights into system behavior.
When a first-order system is subjected to a unit-step input, its response is characterized by its transfer function. By applying the Laplace transform of the unit-step input to the transfer function, expanding the...
Transient and Steady-state Response01:24

Transient and Steady-state Response

In control systems, test signals are essential for evaluating performance under various conditions. The ramp function is effective for systems undergoing gradual changes, while the step function is suitable for assessing systems facing sudden disturbances. For systems subjected to shock inputs, the impulse function is the most appropriate test signal.
These test signals are integral in designing control systems to exhibit two key performance aspects: transient response and steady-state response.
Types of Responses of Series RLC Circuits01:11

Types of Responses of Series RLC Circuits

A second-order differential equation characterizes a source-free series RLC circuit, marking its distinct mathematical representation. The complete solution of this equation is a blend of two unique solutions, each linked to the circuit's roots expressed in terms of the damping factor and resonant frequency.
Classification of Systems-II01:31

Classification of Systems-II

Continuous-time systems have continuous input and output signals, with time measured continuously. These systems are generally defined by differential or algebraic equations. For instance, in an RC circuit, the relationship between input and output voltage is expressed through a differential equation derived from Ohm's law and the capacitor relation,
Multi-input and Multi-variable systems01:22

Multi-input and Multi-variable systems

Cruise control systems in cars are designed as multi-input systems to maintain a driver's desired speed while compensating for external disturbances such as changes in terrain. The block diagram for a cruise control system typically includes two main inputs: the desired speed set by the driver and any external disturbances, such as the incline of the road. By adjusting the engine throttle, the system maintains the vehicle's speed as close to the desired value as possible.
In the absence of...

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same authorSame journal

Global stability of the Atlantic overturning circulation: edge state, long transients and boundary crisis under CO2 forcing.

Philosophical transactions. Series A, Mathematical, physical, and engineering sciences·2026
Same author

Kolmogorov modes and linear response of jump-diffusion models.

Reports on progress in physics. Physical Society (Great Britain)·2025
Same author

Time-lagged recurrence: A data-driven method to estimate the predictability of dynamical systems.

Proceedings of the National Academy of Sciences of the United States of America·2025
Same author

Multistability and intermediate tipping of the Atlantic Ocean circulation.

Science advances·2024
Same author

Analysis of a two-layer energy balance model: Long time behavior and greenhouse effect.

Chaos (Woodbury, N.Y.)·2023
Same author

Scale dependence of fractal dimension in deterministic and stochastic Lorenz-63 systems.

Chaos (Woodbury, N.Y.)·2023
Same journal

Correction to: 'Stokes settling and particle-laden plumes: implications for deep-sea mining and volcanic eruption plumes' (2020), by Mingotti et al.

Philosophical transactions. Series A, Mathematical, physical, and engineering sciences·2026
Same journal

A stable hothouse triggered by a tipping mechanism.

Philosophical transactions. Series A, Mathematical, physical, and engineering sciences·2026
Same journal

Beyond distance: quantifying point cloud dynamics with persistent homology and dynamic optimal transport.

Philosophical transactions. Series A, Mathematical, physical, and engineering sciences·2026
Same journal

Morse index classification and landscape of Kuramoto system for Hebbian-based binary pattern recognition.

Philosophical transactions. Series A, Mathematical, physical, and engineering sciences·2026
Same journal

Detecting the most probable transition phenomenon in a nutrient-phytoplankton-zooplankton system.

Philosophical transactions. Series A, Mathematical, physical, and engineering sciences·2026
See all related articles

Related Experiment Video

Updated: Jun 19, 2026

Fabrication and Validation of an Organ-on-chip System with Integrated Electrodes to Directly Quantify Transendothelial Electrical Resistance
10:51

Fabrication and Validation of an Organ-on-chip System with Integrated Electrodes to Directly Quantify Transendothelial Electrical Resistance

Published on: September 26, 2017

Interpretable and equation-free response theory for complex systems.

Valerio Lucarini1

  • 1School of Mathematical, Physical and Computational Sciences, College of Science and Engineering, University of Leicester , Leicester, UK.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|June 18, 2026
PubMed
Summary
This summary is machine-generated.

We developed a new response theory for Markov chains, enabling predictions of system sensitivity to perturbations. This data-driven approach works even without knowing the underlying equations, offering interpretable insights into complex systems.

Keywords:
KoopmanismMarkov chainsMarkov state modellingProny methodmodel reductionmultiple time scalesresponse theory

More Related Videos

Interactive and Visualized Online Experimentation System for Engineering Education and Research
08:35

Interactive and Visualized Online Experimentation System for Engineering Education and Research

Published on: November 24, 2021

X-ray Beam Induced Current Measurements for Multi-Modal X-ray Microscopy of Solar Cells
10:16

X-ray Beam Induced Current Measurements for Multi-Modal X-ray Microscopy of Solar Cells

Published on: August 20, 2019

Related Experiment Videos

Last Updated: Jun 19, 2026

Fabrication and Validation of an Organ-on-chip System with Integrated Electrodes to Directly Quantify Transendothelial Electrical Resistance
10:51

Fabrication and Validation of an Organ-on-chip System with Integrated Electrodes to Directly Quantify Transendothelial Electrical Resistance

Published on: September 26, 2017

Interactive and Visualized Online Experimentation System for Engineering Education and Research
08:35

Interactive and Visualized Online Experimentation System for Engineering Education and Research

Published on: November 24, 2021

X-ray Beam Induced Current Measurements for Multi-Modal X-ray Microscopy of Solar Cells
10:16

X-ray Beam Induced Current Measurements for Multi-Modal X-ray Microscopy of Solar Cells

Published on: August 20, 2019

Area of Science:

  • Complex systems
  • Statistical mechanics
  • Dynamical systems theory

Background:

  • Response theory analyzes system sensitivity to perturbations.
  • Existing methods struggle with complex, multiscale systems, lacking interpretability and data-driven flexibility.
  • Markov state modelling (MSM) offers a framework for analyzing complex systems.

Purpose of the Study:

  • To develop interpretable, data-driven response theory formulas for Markov chains.
  • To predict responses of observables and higher-order correlations in complex systems.
  • To provide a foundation for equation-agnostic analysis of system dynamics.

Main Methods:

  • Linear and nonlinear response formulas derived for Markov chains.
  • Algebraic expansions inspired by Koopman analysis to identify time scales and modes.
  • Application to a simple metastable system for illustration.

Main Results:

  • Simple, implementable expressions for predicting system responses.
  • Methodology is amenable to purely data-driven implementation, even with unknown dynamics.
  • Explicit and interpretable expressions for Green's functions at all orders were obtained.

Conclusions:

  • The proposed framework offers a powerful, interpretable, and data-driven approach to response theory for complex systems.
  • Provides a dynamical foundation for methods like the Prony method in signal analysis.
  • Enhances understanding of critical transitions and control in complex systems.