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Periodic environments may either enhance or suppress a population via resonant or attenuant cycles. We derive signature functions for predicting the responses of two competing populations to 2-periodic oscillations in six model parameters. Two of these parameters provide a non-trivial equilibrium and two provide the carrying capacities of each species in the absence of the other, but the remaining two are arbitrary and could be intrinsic growth rates. Each signature function is the sign of a weighted sum of the relative strengths of the oscillations of the perturbed parameters. Periodic environments are favourable for populations when the signature function is positive and are deleterious if the signature function is negative. We compute the signature functions of four classical, discrete-time two-species populations and determine regions in parameter space which are either favourable or detrimental to the populations. The six-parameter models include the Logistic, Ricker, Beverton–Holt, and Hassell models.

We present an efficient control scheme that stabilizes the unstable periodic orbits of a chaotic system. The resulting orbits are known as cupolets and collectively provide an important skeleton for the dynamical system. Cupolets exhibit the interesting property that a given sequence of controls will uniquely identify a cupolet, regardless of the system's initial state. This makes it possible to transition between cupolets, and thus unstable periodic orbits, simply by switching control sequences. We demonstrate that although these transitions require minimal controls, they may also involve significant chaotic transients unless carefully controlled. As a result, we present an effective technique that relies on Dijkstra's shortest path algorithm from algebraic graph theory to minimize the transients and also to induce certainty into the control of nonlinear systems, effectively providing an efficient algorithm for the steering and targeting of chaotic systems.

We report on the tendency of chaotic systems to be controlled onto their unstable periodic orbits in such a way that these orbits are stabilized. The resulting orbits are known as cupolets and collectively provide a rich source of qualitative information on the associated chaotic dynamical system. We show that pairs of interacting cupolets may be induced into a state of mutually sustained stabilization that requires no external intervention in order to be maintained and is thus considered bound or entangled. A number of properties of this sort of entanglement are discussed. For instance, should the interaction be disturbed, then the chaotic entanglement would be broken. Based on certain properties of chaotic systems and on examples which we present, there is further potential for chaotic entanglement to be naturally occurring. A discussion of this and of the implications of chaotic entanglement in future research investigations is also presented.

Recent experimental and theoretical work has detected signatures of chaotic behavior in nearly every physical science, including quantum entanglement. In some instances, chaos either plays a significant role or, as an underlying presence, explains perplexing observations. There are certain properties of chaotic systems which are consistently encountered and become focal points of the investigations. For instance, chaotic systems typically admit a dense set of unstable periodic orbits around an attractor. These orbits collectively provide a rich source of qualitative information about the associated system and their abundance has been utilized in a variety of applications.

We begin this thesis by describing a control scheme that stabilizes the unstable periodic orbits of chaotic systems and we go on to discuss several properties of these orbits. This technique allows for the creation of thousands of periodic orbits, known as cupolets (Chaotic Unstable Periodic Orbit-lets). We then present several applications of cupolets for investigating chaotic systems. First, we demonstrate an effective technique that combines cupolets with algebraic graph theory in order to transition between their orbits. This also induces certainty into the control of nonlinear systems and effectively provides an efficient algorithm for the steering and targeting of chaotic systems.

Next, we establish that many higher-order cupolets are amalgamations of simpler cupolets, possibly through bifurcations. From a sufficiently large set of cupolets, we obtain a hierarchal subset of fundamental cupolets from which other cupolets may be assembled and dynamical invariants approximated. We then construct an independent coordinate system aligned to the local dynamical geometry and that reveals the local stretching and folding dynamics which characterize chaotic behavior. This partitions the dynamical landscape into regions of high or low chaoticity, thereby supporting prediction capabilities.

Finally, we demonstrate how interacting chaotic systems may be controlled onto cupolets whose periodic behavior is maintained by their continued interaction. This is known as chaotic entanglement and it evokes a classical analog to quantum entanglement. Fundamental cupolets are believed to play important roles in chaotic entanglement. Based on certain properties of chaotic systems and on examples which we present, there is potential for chaotic entanglement to be naturally occurring.

We describe the use of student-generated video content to assess students' engagement with, and understanding of, problem-solving skills. In this framework, students are tasked with using technology to create videos that show them working through, and explaining solutions to, challenging calculus exercises. The videos are then posted online, accessible only to the students and instructors in the class. Such video assignments align with what recent studies have identified as effective homework practices. Indeed, results from student surveys suggest that a significantly higher level of self-regulated learning takes place in creating these videos than in completing traditional written or online homework.

We examine the quantum-classical correspondence from a classical perspective by discussing the potential for chaotic systems to support behaviors normally associated with quantum mechanical systems. Our main analytical tool is a chaotic system’s set of cupolets, which are highly-accurate stabilizations of its unstable periodic orbits. Our discussion is motivated by the bound or entangled states that we have recently detected between interacting chaotic systems, wherein pairs of cupolets are induced into a state of mutually-sustaining stabilization that can be maintained without external controls. This state is known as chaotic entanglement as it has been shown to exhibit several properties consistent with quantum entanglement. For instance, should the interaction be disturbed, the chaotic entanglement would then be broken. In this paper, we further describe chaotic entanglement and go on to address the capacity for chaotic systems to exhibit other characteristics that are conventionally associated with quantum mechanics, namely analogs to wave function collapse, various entropy definitions, the superposition of states, and the measurement problem. In doing so, we argue that these characteristics need not be regarded exclusively as quantum mechanical. We also discuss several characteristics of quantum systems that are not fully compatible with chaotic entanglement and that make quantum entanglement unique.

Cupolets are a relatively new class of waveforms that represent highly accurate approximations to the unstable periodic orbits of chaotic systems, and large numbers can be efficiently generated via a control method where small kicks are applied along intersections with a control plane. Cupolets exhibit the interesting property that a given set of controls, periodically repeated, will drive the associated chaotic system onto a uniquely defined cupolet regardless of the system’s initial state. We have previously demonstrated a method for efficiently steering from one cupolet to another using a graph-theoretic analysis of the connections between these orbits. In this paper, we discuss how connections between cupolets can be analyzed to show that complicated cupolets are often composed of combinations of simpler cupolets. Hence, it is possible to distinguish cupolets according to their reducibility: a cupolet is classified either as composite, if its orbit can be decomposed into the orbits of other cupolets or as fundamental, if no such decomposition is possible. In doing so, we demonstrate an algorithm that not only classifies each member of a large collection of cupolets as fundamental or composite, but that also determines a minimal set of fundamental cupolets that can exactly reconstruct the orbit of a given composite cupolet. Furthermore, this work introduces a new way to generate higher-order cupolets simply by adjoining fundamental cupolets via sequences of controlled transitions. This allows for large collections of cupolets to be collapsed onto subsets of fundamental cupolets without losing any dynamical information. We conclude by discussing potential future applications.

In chaotic entanglement, pairs of interacting classically-chaotic systems are induced into a state of mutual stabilization that can be maintained without external controls and that exhibits several properties consistent with quantum entanglement. In such a state, the chaotic behavior of each system is stabilized onto one of the system’s many unstable periodic orbits (generally located densely on the associated attractor), and the ensuing periodicity of each system is sustained by the symbolic dynamics of its partner system, and vice versa. Notably, chaotic entanglement is an entropy-reversing event: the entropy of each member of an entangled pair decreases to zero when each system collapses onto a given period orbit. In this paper, we discuss the role that entropy plays in chaotic entanglement. We also describe the geometry that arises when pairs of entangled chaotic systems organize into coherent structures that range in complexity from simple tripartite lattices to more involved patterns. We conclude with a discussion of future research directions.

The Euclidean Discus Toss is an active and tactile learning activity that models the extended Euclidean algorithm with a frisbee relay. The extended Euclidean algorithm involves both iterative and recursive programming and is regularly taught throughout the mathematics and computer science curricula. The Euclidean Discus Toss invites students to toss and catch frisbees in a collaborative and hands-on effort designed to sharpen modular arithmetic skills, enhance familiarity with iterative and recursive algorithms, and strengthen classroom community. The activity is fun, low-stakes, and can be customized to meet a variety of pedagogical objectives.

In chaos control, one usually seeks to stabilize the unstable periodic orbits (UPOs) that densely inhabit the attractors of many chaotic dynamical systems. These orbits collectively play a significant role in determining the dynamics and properties of chaotic systems and are said to form the skeleton of the associated attractors. While UPOs are insightful tools for analysis, they are naturally unstable and, as such, are difficult to find and computationally expensive to stabilize. An alternative to using UPOs is to approximate them using cupolets. Cupolets, a name derived from chaotic, unstable, periodic, orbit-lets, are a relatively new class of waveforms that represent highly accurate approximations to the UPOs of chaotic systems, but which are generated via a particular control scheme that applies tiny perturbations along Poincaré sections. Originally discovered in an application of secure chaotic communications, cupolets have since gone on to play pivotal roles in a number of theoretical and practical applications. These developments include using cupolets as wavelets for image compression, targeting in dynamical systems, a chaotic analog to quantum entanglement, an abstract reducibility classification, a basis for audio and video compression, and, most recently, their detection in a chaotic neuron model. This review will detail the historical development of cupolets, how they are generated, and their successful integration into theoretical and computational science and will also identify some unanswered questions and future directions for this work.