Related Experiment Video
Updated: May 30, 2026
06:44
Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
Published on: September 23, 2025
Nondeterminism in the limit of nonsmooth dynamics
1Department of Engineering Mathematics, University of Bristol, Queen's Building, Bristol BS8 1TR, United Kingdom.
Physical Review Letters
|July 21, 2011
Summary
Discontinuous derivatives model threshold switching. This study proves well-defined solutions exist for multivalued flows, quantifying dynamics via loss of determinism in sensitive systems.
Area of Science:
- Physics
- Dynamical Systems
- Nonlinear Dynamics
Background:
- Discontinuous time derivatives model threshold-dependent switching across various fields, including friction, electronics, and biology.
- In continuous flows, such derivatives can lead to multiple outcomes from identical initial conditions, posing challenges for deterministic analysis.
Purpose of the Study:
- To demonstrate the existence of well-defined solution sets for systems exhibiting multivalued dynamics due to grazing discontinuities.
- To develop a method for quantifying dynamics in the limit of infinite sensitivity to initial conditions.
Main Methods:
- Analysis of systems with discontinuous time derivatives and grazing bifurcations.
- Application of the concept of 'loss of determinism' to quantify dynamics.
- Modeling and analysis of a superconducting resonator and a negatively damped oscillator.
Main Results:
- Established the existence of well-defined solution sets for flows that become multivalued when grazing a discontinuity.
- Quantified the dynamics of sensitive systems by analyzing the loss of determinism.
- Successfully applied the framework to specific physical systems like superconducting resonators.
Conclusions:
- The study provides a rigorous mathematical framework for understanding systems with discontinuous derivatives and multivalued dynamics.
- The developed methods offer new ways to analyze and predict the behavior of highly sensitive dynamical systems.
- Findings have implications for understanding complex phenomena in physics and engineering.
Related Concept Videos
01:19Limits with Oscillating Discontinuities
An oscillating discontinuity is a type of discontinuity in which a function’s values fluctuate infinitely often as the input approaches a particular point. Unlike jump discontinuities, where the function suddenly shifts between two values, or infinite discontinuities, where the function diverges without bound, an oscillating discontinuity arises from rapid back-and-forth variation. Because the function never stabilizes toward a single value, no finite limit exists at that point.One of the most...
01:29Evaluating Limits by Direct Substitution
In the analysis of functions that represent continuous physical phenomena, it is often necessary to determine the output value as the input approaches a specific point. When a combination of algebraic terms defines the function and exhibits no discontinuities or abrupt changes near the point of interest, the limit of the function can be evaluated directly. This process, known as direct substitution, involves replacing the variable in the expression with the value it approaches.Direct...
01:30Introduction 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...
01:23Types of Limits I
Limits are a key mathematical concept for understanding how functions behave as their input approaches specific values, particularly when the function is undefined. They help reveal trends and discontinuities by examining the values a function approaches rather than its actual value.One-sided limits focus on the direction from which a value is approached. When a function behaves differently depending on whether the input approaches from the left or the right, the two one-sided limits may not...
01:27The Precise Definition of a Limit
Understanding the formal definition of a limit is essential for precise mathematical analysis. This concept allows us to rigorously determine how a function behaves near a particular point without relying on ambiguous notions such as "getting close." The ε-δ definition plays a foundational role in calculus, ensuring analytical clarity and logical consistency in limit evaluation.The formal definition states that the limit of a function f(x) as x approaches a is L, written asif for every ε >...
01:30The Squeeze Theorem
Certain mathematical functions exhibit unpredictable or highly variable behavior near specific input values, making direct evaluation of their limits challenging. This complexity may arise from rapid oscillations or irregular patterns that obscure the function’s trend. In such cases, the Squeeze Theorem offers a reliable method for determining limits.According to the Squeeze Theorem, if a function is confined between two other functions near a particular point, and both outer functions approach...