insight - Dynamical Systems - # Invariant Graphs and Dynamics of Continuous Piecewise Linear Planar Maps

Dynamics of a Family of Continuous Piecewise Linear Planar Maps

Q: 1. Can the results on the generic dynamics for a < 0 be strengthened to show that the property of having at most three ω-limit sets is satisfied by a full Lebesgue measure set of initial conditions in Γ, rather than just an open and dense set?

The results regarding the dynamics of the family of piecewise linear maps ( F_{a,b} ) for ( a < 0 ) suggest that the property of having at most three ω-limit sets could indeed be strengthened to apply to a full Lebesgue measure set of initial conditions in the invariant graph ( \Gamma ). The current findings indicate that for an open and dense set of initial conditions, the dynamics are relatively simple, with at most three possible ω-limit sets. However, the conjecture that this property holds for a full Lebesgue measure set stems from the nature of piecewise linear maps, which often exhibit robust dynamical behaviors across large subsets of their parameter space. To establish this stronger result, one would need to demonstrate that the conditions leading to the existence of at most three ω-limit sets are not just locally valid (in the sense of being true for an open and dense set) but are also globally applicable across the entirety of ( \Gamma ). This could involve analyzing the structure of the invariant graphs and the behavior of orbits under the map ( F ) in a more comprehensive manner, potentially leveraging ergodic theory or measure-theoretic approaches to show that the dynamics are not only generic but also prevalent in the sense of Lebesgue measure.

Q: 2. Are there other families of piecewise linear maps, beyond the one studied here, that exhibit a similar collapse of the dynamics to one-dimensional invariant sets while still retaining the complexity of one-dimensional dynamics?

Yes, there are several families of piecewise linear maps that exhibit a similar phenomenon of collapsing dynamics to one-dimensional invariant sets while retaining the complexity characteristic of one-dimensional dynamics. One notable example is the family of Lozi maps, which are a specific type of piecewise linear map defined on the plane. These maps can exhibit chaotic behavior and have been studied extensively for their rich dynamical properties, including the existence of invariant sets that capture the dynamics in a lower-dimensional space. Another example is the Grove-Ladas family of maps, which generalizes the Lozi maps and includes parameters that can lead to complex dynamics. These maps can also exhibit behaviors where the dynamics are effectively captured by one-dimensional invariant sets, such as intervals or graphs, while still allowing for intricate behaviors like periodic orbits and chaotic attractors. Additionally, max-type maps and other families of piecewise linear maps defined by absolute values or similar constructions can also show this collapsing behavior. The study of these maps often reveals a rich interplay between the geometry of the invariant sets and the dynamics, leading to a variety of behaviors that can be both periodic and chaotic.

Core Concepts

For the family of piecewise linear maps Fa,b(x, y) = (|x| - y + a, x - |y| + b), the dynamics strongly depend on the parameters a and b. When a ≥ 0, all orbits are eventually periodic, while for a < 0, the dynamics is concentrated on one-dimensional invariant graphs that capture the final dynamics of the map.

Abstract

The paper studies the family of piecewise linear maps Fa,b(x, y) = (|x| - y + a, x - |y| + b), where (a, b) ∈ R^2. The dynamics of this family is analyzed in detail, with the results depending on the sign of the parameter a.

For a ≥ 0:

All orbits are eventually periodic.
There is at most one fixed point and up to two 3-periodic orbits, depending on the values of a and b.

For a < 0 (without loss of generality, the case a = -1 is considered):

There exists a compact invariant graph Γ such that all orbits eventually reach Γ.
The structure of Γ is characterized, with Γ being the union of at most 23 compact segments with one of four possible slopes (0, 1, -1, ∞).
The dynamics of F restricted to Γ is studied in detail. It is shown that for an open and dense set of initial conditions, there are at most three possible ω-limit sets, which can be periodic orbits, Cantor sets, or other complicated subsets of Γ.
A full characterization of when the restriction of F to Γ has positive or zero entropy is provided, with the entropy being a discontinuous function of the parameter b/a.

The study of this family of maps is interesting as it provides a natural two-dimensional discrete dynamical system for which the final dynamics is one-dimensional, yet it presents the full richness of one-dimensional dynamics.

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Invariant graphs and dynamics of a family of continuous piecewise linear planar maps

by Anna... at arxiv.org 10-03-2024

https://arxiv.org/pdf/2410.01052.pdf

Invariant graphs and dynamics of a family of continuous piecewise linear planar maps

Deeper Inquiries

1. Can the results on the generic dynamics for a < 0 be strengthened to show that the property of having at most three ω-limit sets is satisfied by a full Lebesgue measure set of initial conditions in Γ, rather than just an open and dense set?

The results regarding the dynamics of the family of piecewise linear maps ( F_{a,b} ) for ( a < 0 ) suggest that the property of having at most three ω-limit sets could indeed be strengthened to apply to a full Lebesgue measure set of initial conditions in the invariant graph ( \Gamma ). The current findings indicate that for an open and dense set of initial conditions, the dynamics are relatively simple, with at most three possible ω-limit sets. However, the conjecture that this property holds for a full Lebesgue measure set stems from the nature of piecewise linear maps, which often exhibit robust dynamical behaviors across large subsets of their parameter space.
To establish this stronger result, one would need to demonstrate that the conditions leading to the existence of at most three ω-limit sets are not just locally valid (in the sense of being true for an open and dense set) but are also globally applicable across the entirety of ( \Gamma ). This could involve analyzing the structure of the invariant graphs and the behavior of orbits under the map ( F ) in a more comprehensive manner, potentially leveraging ergodic theory or measure-theoretic approaches to show that the dynamics are not only generic but also prevalent in the sense of Lebesgue measure.

2. Are there other families of piecewise linear maps, beyond the one studied here, that exhibit a similar collapse of the dynamics to one-dimensional invariant sets while still retaining the complexity of one-dimensional dynamics?

Yes, there are several families of piecewise linear maps that exhibit a similar phenomenon of collapsing dynamics to one-dimensional invariant sets while retaining the complexity characteristic of one-dimensional dynamics. One notable example is the family of Lozi maps, which are a specific type of piecewise linear map defined on the plane. These maps can exhibit chaotic behavior and have been studied extensively for their rich dynamical properties, including the existence of invariant sets that capture the dynamics in a lower-dimensional space.
Another example is the Grove-Ladas family of maps, which generalizes the Lozi maps and includes parameters that can lead to complex dynamics. These maps can also exhibit behaviors where the dynamics are effectively captured by one-dimensional invariant sets, such as intervals or graphs, while still allowing for intricate behaviors like periodic orbits and chaotic attractors.
Additionally, max-type maps and other families of piecewise linear maps defined by absolute values or similar constructions can also show this collapsing behavior. The study of these maps often reveals a rich interplay between the geometry of the invariant sets and the dynamics, leading to a variety of behaviors that can be both periodic and chaotic.

3. What are the potential applications of the dynamics of the studied family of maps, particularly in the context of power electronics, neural networks, or other areas where piecewise linear maps arise?

The dynamics of the family of piecewise linear maps ( F_{a,b} ) have several potential applications across various fields, particularly in power electronics, neural networks, and other areas where such maps are relevant.

Power Electronics: In power electronics, piecewise linear models are often used to describe the behavior of converters and inverters. The dynamics of these maps can help in understanding the stability and performance of power conversion systems, particularly in scenarios involving switching behaviors. The periodic and chaotic dynamics observed in these maps can inform the design of control strategies to mitigate undesirable oscillations and improve efficiency.

Neural Networks: In the context of neural networks, particularly in models that incorporate piecewise linear activation functions, the dynamics of the studied maps can provide insights into the behavior of network outputs in response to varying inputs. Understanding the invariant graphs and periodic behaviors can aid in the design of networks that exhibit desired learning dynamics, such as convergence to stable states or periodic patterns that mimic biological neural activity.

Mechanical Systems with Friction: The dynamics of piecewise linear maps are also applicable in modeling mechanical systems where friction plays a significant role. The study of orbits and periodic behaviors can help in predicting the motion of systems under various loading conditions, leading to better designs that account for stability and performance.

Economic Models: In economics, piecewise linear maps can model various phenomena, such as market dynamics and consumer behavior. The insights gained from the dynamics of these maps can inform economic theories and help in the development of predictive models that account for non-linearities in market responses.

Overall, the rich dynamical behavior of the family of maps ( F_{a,b} ) provides a valuable framework for analyzing and designing systems in these diverse fields, highlighting the importance of understanding piecewise linear dynamics in practical applications.