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Related Concept Videos

Introduction to Differential Equations01:20

Introduction to Differential Equations

A differential equation is a mathematical expression that establishes a relationship between a function and its derivatives. These equations are fundamental in modeling dynamic systems across various fields of science and engineering. The order of a differential equation is defined by the highest order derivative present in the equation. A first-order differential equation includes only the first derivative, while a second-order differential equation includes up to the second derivative of the...
Introduction to Limits01:30

Introduction to Limits

A limit describes the value a function approaches as its input moves closer to a particular point. Even when a function is undefined at a specific value, limits allow us to analyze its behavior near that point. This concept is fundamental in calculus and essential for understanding continuity, derivatives, and integrals.Mathematically, a function f(x) has a limit L at x = a if its values L approach x as x gets arbitrarily close to a. This is written as:This notation expresses that the function...
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...
Linear time-invariant Systems01:23

Linear time-invariant Systems

A system is linear if it displays the characteristics of homogeneity and additivity, together termed the superposition property. This principle is fundamental in all linear systems. Linear time-invariant (LTI) systems include systems with linear elements and constant parameters.
The input-output behavior of an LTI system can be fully defined by its response to an impulsive excitation at its input. Once this impulse response is known, the system's reaction to any other input can be calculated...
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:
An Introduction to Mechanics01:28

An Introduction to Mechanics

Humans have been making ships, shelters, pyramids, weapons, agricultural equipment, and many more items without recording the process or theory behind them for centuries. It would be challenging to document the evolution of mechanics from its origin to the present.
According to records, the history of mechanics starts with Aristotle (384–322 BC). He related mechanics to physical theory, aiming for a universal synthesis.
Newton defined mechanics as the branch of physical science that studies the...

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Updated: May 29, 2026

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
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Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis

Published on: September 23, 2025

An introduction to dynamical systems.

Eric A Sobie1

  • 1Department of Pharmacology and Systems Therapeutics and Systems Biology Center New York, Mount Sinai School of Medicine, New York, NY 10029, USA. eric.sobie@mssm.edu

Science Signaling
|September 22, 2011
PubMed
Summary
This summary is machine-generated.

This resource teaches dynamical systems tools for analyzing ordinary differential equation (ODE) models in biology. It covers ODE solutions, stability analysis, and bifurcations for graduate students.

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An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
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Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
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Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis

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An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
11:03

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

Published on: December 4, 2017

Area of Science:

  • Mathematical Biology
  • Computational Biology
  • Systems Biology

Background:

  • Introduces dynamical systems tools for analyzing ordinary differential equation (ODE)-based models.
  • Applies concepts to biological problems, suitable for graduate students or advanced undergraduates.
  • Explains deriving biochemical signaling network equations from reaction diagrams.

Discussion:

  • Covers principles of numerical solution for systems of ODEs.
  • Illustrates methods for determining stability of steady-state solutions in 1D and 2D ODE systems using graphical techniques.
  • Introduces the concept of bifurcation and its significance in qualitative system behavior changes.

Key Insights:

  • Provides a comprehensive teaching resource including lecture notes, slides, and a problem set.
  • Enables students to implement ODE models of biochemical reactions using MATLAB.
  • Facilitates exploration of dynamical systems concepts through practical application.

Outlook:

  • Aims to enhance understanding of mathematical modeling in biological sciences.
  • Encourages the application of computational tools for analyzing complex biological systems.
  • Prepares students for advanced research in computational and systems biology.