Related Experiment Videos
Finite invariant sets with bridging points in logistic IFS
Hibiki Kato1,2, Tamotsu Onozaki3, Yoshitaka Saiki1
1Graduate School of Business Administration, Hitotsubashi University, Tokyo 186-8601, Japan.
Chaos (Woodbury, N.Y.)
|June 24, 2026
Summary
Switching between simple nonlinear maps can create new invariant structures. These "toss-and-catch" dynamics in regime-switching systems reveal novel mechanisms for complex behavior in random dynamical systems.
Area of Science:
- Dynamical Systems
- Chaos Theory
- Nonlinear Dynamics
Background:
- Regime-switching dynamical systems involve changes in system behavior.
- Iterated function systems (IFS) model complex dynamics through repeated application of functions.
- Finite invariant sets are crucial for understanding long-term system behavior.
Purpose of the Study:
- To model regime-switching dynamical systems using iterated function systems.
- To investigate finite invariant sets with "toss-and-catch" dynamics.
- To identify mechanisms generating invariant structures in randomly switching systems.
Main Methods:
- Utilized one-dimensional maps (logistic, tent) that randomly alternate.
- Focused on systems exhibiting "toss-and-catch" dynamics between fixed and periodic points.
- Derived exact parameter conditions for specific toss-and-catch structures.
Main Results:
- Identified two distinct mechanisms for finite invariant sets: bridging points and shared map intersections.
- Discovered invariant sets with bridging points not periodic in constituent maps.
- Demonstrated that switching generates invariant structures absent in individual maps.
Conclusions:
- Random switching between simple nonlinear maps can create novel invariant structures.
- Bridging points represent a key mechanism for emergent invariant sets.
- Findings suggest a general pathway for invariant structure emergence in random dynamical systems.
Related Concept Videos
01:24BIBO stability of continuous and discrete -time systems
System stability is a fundamental concept in signal processing, often assessed using convolution. For a system to be considered bounded-input bounded-output (BIBO) stable, any bounded input signal must produce a bounded output signal. A bounded input signal is one where the modulus does not exceed a certain constant at any point in time.
To determine the BIBO stability, the convolution integral is utilized when a bounded continuous-time input is applied to a Linear Time-Invariant (LTI) system.
To determine the BIBO stability, the convolution integral is utilized when a bounded continuous-time input is applied to a Linear Time-Invariant (LTI) system.
01:23Linear 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...
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...
01:29Application of Nonlinear Inequalities
A nonlinear inequality describes a comparison involving an expression that curves or behaves more complexly than a straight line. These inequalities often appear in forms that include squares, products, or variables in the denominator.To solve such an inequality, one starts by rewriting it so that zero appears on one side. For example, the inequality: can be factored as: This form makes it easier to identify the values that cause the expression to equal zero. In this case, the key values are 3...
01:29Improper Integrals: Infinite Intervals
An integral is classified as improper due to an infinite interval when at least one of its limits of integration extends to positive or negative infinity. In such cases, the region under the curve is unbounded, and standard techniques for evaluating definite integrals are not directly applicable. Instead, the improper integral is defined through a limiting process that allows one to determine whether the accumulated area remains finite despite the infinite domain.Application to Exponential...
01:28Inequalities
Inequalities express mathematical relationships where two values are not equal and are compared using symbols such as <, >, ≤, or ≥. These expressions define a range of possible solutions rather than a single value. Interval notation provides a concise way to express these solution sets, especially when the variable spans a continuous range. An open interval, written as (a, b), excludes the endpoints, while a closed interval [a, b] includes them. There are also half-open intervals, such...
01:25Introduction to Nonlinear Inequalities
Linear and nonlinear inequalities are fundamental for analyzing variable relationships and identifying ranges satisfying specific conditions. A linear inequality involves variables raised only to the first power, resulting in a straight-line graph. This line partitions the coordinate plane into two distinct regions: one that satisfies the inequality and one that does not. Each region represents a set of solutions where the linear relationship holds true under the specified constraint.Nonlinear...