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Isolation, Identification, And Enzymatic Screening Of Fungi From Utah Dryland Soils For Industrial And Waste Valorization Applications, Jessica Lorena Silva Cuero Aug 2026

Isolation, Identification, And Enzymatic Screening Of Fungi From Utah Dryland Soils For Industrial And Waste Valorization Applications, Jessica Lorena Silva Cuero

All Graduate Reports and Creative Projects, Fall 2023 to Present

Agricultural waste accumulation presents environmental challenges but also opportunities for biological valorization into value-added products. Fungi are particularly well-suited for this purpose due to their capacity to produce extracellular enzymes involved in the degradation of complex organic substrates. Dryland ecosystems represent a promising source for bioprospecting microbes capable of producing such enzymes. Drylands harbor a high diversity of extremophilic microorganisms that can withstand harsh conditions, extreme heat, UV, and low nutrients using unique adaptations. This suggests that these microbes are resilient and capable of surviving under varying conditions and still produce enzymes for biotransformation. The objective of this study was …


A Design Guide For Extreme Heat In Play Environments: Exploring Landscape Architecture Strategies, Amy Elise Guenther Aug 2026

A Design Guide For Extreme Heat In Play Environments: Exploring Landscape Architecture Strategies, Amy Elise Guenther

All Graduate Reports and Creative Projects, Fall 2023 to Present

Urban heat is a rising concern and is affecting vulnerable populations such as our youth. A particular concern is exposure to extremely high temperatures on playgrounds – an environment that is meant to be safe for children even in extreme weather conditions. There has been an increase in many impervious surfaces such as concrete and asphalt in urban areas. These surfaces absorb solar radiation, which contributes to the heat island effect. These same surfaces are in and around playground areas, which are contributing to the extreme heat felt within these play environments. Therefore, requiring intervention to help mitigate the effects …


The Role Of Public Spaces In Facilitating Social Interactions On Summer Weekends, Sierra R. Murray Aug 2026

The Role Of Public Spaces In Facilitating Social Interactions On Summer Weekends, Sierra R. Murray

All Graduate Theses and Dissertations, Fall 2023 to Present

Public spaces such as parks, plazas, and pedestrian areas are important places where people meet, relax, and interact with others. While these spaces are intended to bring communities together, their design does not always support social interaction. This study explored how everyday design features influence how people use public spaces and engage with one another. Features such as seating, trees and plants, shade, water elements, and the way spaces are shaped were examined. The goal was to better understand how design choices affect social life in shared outdoor environments. 

The research involved observing how people naturally behaved in different public …


Modular Verification For Network-On-Chip Designs Using Probabilistic Verification And Assume-Guarantee Reasoning, Nicholas Waddoups Aug 2026

Modular Verification For Network-On-Chip Designs Using Probabilistic Verification And Assume-Guarantee Reasoning, Nicholas Waddoups

All Graduate Theses and Dissertations, Fall 2023 to Present

To satisfy increasing demands for computer chip performance in personal computing, mobile devices, and commercial server computing, a modern computer chip is constructed with tens (or hundreds) of small individual computing modules. Each of these modules must communicate with one other to share information about the running state of a computer. Historically, when chips were a few modules a simple communication method sufficied. However, as the number of modules in a chip grew, a more effecient method was needed in order to maintain performance across the system as a whole. A Network-on-Chip (NoC) design is the de-facto communication method for …


Reciprocal Teaching With Emergent Bilingual Students: Promoting Academic Reading Strategies And Fostering Hope, Lorissa K. Nelson Aug 2026

Reciprocal Teaching With Emergent Bilingual Students: Promoting Academic Reading Strategies And Fostering Hope, Lorissa K. Nelson

All Graduate Theses and Dissertations, Fall 2023 to Present

Many students struggle with reading comprehension, and research shows that low literacy skills in a student’s first language can unfortunately lead to lower rates of university program completion, regardless of the language of instruction. When students feel empowered and confident in their ability to achieve their academic goals, they are much more likely to take ownership of their learning. In fact, students who actively demonstrate hopeful thinking and behaviors are consistently more successful in their academic journeys.

My research explores how we can better support these high school students— specifically concurrent enrollment (CE) students—by examining the impact of Reciprocal Teaching. …


"In Any Other Circumstance I Am Not In Work Mode. I Am Clocked Out": Chappell Roan And The Objectified Persona, Aubry Louise Aug 2026

"In Any Other Circumstance I Am Not In Work Mode. I Am Clocked Out": Chappell Roan And The Objectified Persona, Aubry Louise

All Graduate Theses and Dissertations, Fall 2023 to Present

It is known that celebrities have influence on their audience’s attitudes and behaviors. Within the last few decades, multiple celebrities have created distinctions between their public/stage self and their private self, however, less is known about the ways that they differentiate between the two. This paper looks at celebrity public persona’s and how these persona’s influence their audience. I use Kayleigh Amstutz, otherwise known as Chappell Roan, as my case study. This thesis offers an analysis of how Amstutz talks about Kayleigh differently than Chappell within social media posts, interviews, and commentary from media outlets. I use the concepts of …


Integrated Soil-Water Processes And Irrigation Management For Agricultural Water Optimization In Semi-Arid Regions, Tejinder Singh Aug 2026

Integrated Soil-Water Processes And Irrigation Management For Agricultural Water Optimization In Semi-Arid Regions, Tejinder Singh

All Graduate Theses and Dissertations, Fall 2023 to Present

Water is one of the most important resources for agriculture in the western United States. In semi-arid regions, farmers depend on irrigation to grow crops, but water supplies are becoming less reliable because of drought, declining snowpack, groundwater depletion, and increasing competition for limited water. Improving agricultural water use is therefore not simply about applying less water. It also requires understanding where water goes after it enters a field, how long it remains available to crops, and how farmers can use that information to make better irrigation decisions. This dissertation follows water through irrigated agricultural systems to better understand how …


Does Acceptance Matter? The Moderating Role Of Acceptance Of Pornography Use In Sexual And Relationship Satisfaction, Alexandra L. Gillespie Aug 2026

Does Acceptance Matter? The Moderating Role Of Acceptance Of Pornography Use In Sexual And Relationship Satisfaction, Alexandra L. Gillespie

All Graduate Theses and Dissertations, Fall 2023 to Present

With more or the general population reporting regular pornography use there has been an increase in interest in the consequences of regular use, especially for romantic relationships. Previously completed research has shown that there are likely other explanations for why pornography use is related to both positive and negative outcomes. This study examines how acceptance might change the influence the relationship between frequency of use and relationship and sexual satisfaction for individuals. This relationship may yield greater understanding of why some individuals report benefits and others report negative effects of pornography use. Findings indicated that for men their level of …


Mapler: An R Package For Estimating The Impact Of Climate Change On Maple Syrup Production, Matthew T. White Aug 2026

Mapler: An R Package For Estimating The Impact Of Climate Change On Maple Syrup Production, Matthew T. White

All Graduate Theses and Dissertations, Fall 2023 to Present

Successful maple sap tapping depends on the freeze/thaw cycle (i.e., temperatures fluctuating above/below freezing) during the winter and spring. Climate change threatens to alter the timing and duration of the tapping season. This necessitates research into how maple sap tapping will be impacted by climate change in order to help maple syrup producers prepare for the future. We define a sap day as a day where the freeze/thaw cycle occurred. Using information climate scientists use to predict future temperatures, we calculate how many sap days could occur each year. We develop software to analyze these sap day calculations to determine …


The Role Of Autonomy, Competence, And Relatedness In The Relationship Between Ai Use, Resilience, And Motivation Among Engineering Students, Daniel Kane Aug 2026

The Role Of Autonomy, Competence, And Relatedness In The Relationship Between Ai Use, Resilience, And Motivation Among Engineering Students, Daniel Kane

All Graduate Theses and Dissertations, Fall 2023 to Present

The purpose of this study was to determine how the widespread use of Artificial Intelligence (AI) influences engineering students’ motivation and resilience. Specifically, the role of autonomy, competence, and relatedness as defined in Self-Determination Theory (SDT) was assessed to determine how these basic psychological needs interact with students’ AI use and motivation or resilience. The following research questions guided the study: (1) To what extent do the three basic psychological needs defined in SDT (autonomy, competence, and relatedness) mediate the relationship between the frequency of AI use and students’ levels of motivation (on the self-determination continuum) or resilience in an …


Integrative Taxonomic Analysis Of Cryptic Species Diversity In The Spiny Witch-Hazel Gall Aphid, Hamamelistes Spinosus (Hemiptera: Aphididae: Hormaphidini) In Eastern North America, Gavin Andrew Powell Munson Aug 2026

Integrative Taxonomic Analysis Of Cryptic Species Diversity In The Spiny Witch-Hazel Gall Aphid, Hamamelistes Spinosus (Hemiptera: Aphididae: Hormaphidini) In Eastern North America, Gavin Andrew Powell Munson

All Graduate Theses and Dissertations, Fall 2023 to Present

Taxonomy is the practice of identifying, classifying and categorizing species, providing structure for the vast diversity of life. Aphids are an ecologically important group of soft-bodied insects, but they can be difficult to identify because of their tiny anatomical size, reduced or simplified morphological features, and morphological variation within a single species across different generations. In this study, the taxonomy of the spiny witch-hazel gall aphid, Hamamelistes spinosus, is revised. The complex multi-generational life cycle of the aphids, which alternates between witch-hazel and birch trees, has led to confusion regarding past descriptions. Scientists in the late 1800s who were …


Nutrient Leaching Under Traditional And Intercropped Cover Crop Fields In Northern Utah, Katherine Hewitt Aug 2026

Nutrient Leaching Under Traditional And Intercropped Cover Crop Fields In Northern Utah, Katherine Hewitt

All Graduate Theses and Dissertations, Fall 2023 to Present

This study examines whether cover crops can be used in Utah and the Intermountain West to improve water quality and reduce contamination due to nutrient leaching from agricultural soils into the groundwater. Cover crops such as grasses, clover, and other seed mixtures can be planted to improve soil health, increasing the soil’s ability to retain water and nutrients, which reduces the amount of nitrate and phosphate fertilizers lost through leaching and contamination. In Utah, short growing seasons and limited irrigation water availability hinder the ability to establish cover crops. By planting the cover crop between the rows (intercropping) when the …


Flexible Grazing: Evaluating Stocker Cattle As A Drought Management Tool For Utah Livestock Producers?, Ross Israelsen Aug 2026

Flexible Grazing: Evaluating Stocker Cattle As A Drought Management Tool For Utah Livestock Producers?, Ross Israelsen

All Graduate Reports and Creative Projects, Fall 2023 to Present

Drought is a persistent concern in the western United States, often leading to reduced forage production, overgrazing, and economic challenges for livestock producers. This study evaluates the economic feasibility of incorporating stocker cattle as a flexible drought management strategy under varying forage conditions across four Utah grazing locations: Park Valley, Nephi, Vernal, and Beaver. The Rangeland Analysis Platform (RAP) was used to estimate annual forage production and stocking rates from 2001 through 2025. Four stocking-rate scenarios (100%, 75%, 60%, and 50% cow–calf stocking rates) were evaluated using a Net Present Value (NPV) framework to compare the economic performance of traditional …


A Focus On Student Needs In German Instruction, Elisa E. Remund Aug 2026

A Focus On Student Needs In German Instruction, Elisa E. Remund

All Graduate Reports and Creative Projects, Fall 2023 to Present

This portfolio serves as documentation of the author’s professional progress throughout the Master of Science of Second Language Teaching (MSSLT) program at Utah State University, shown through teaching philosophy, hands-on teaching experience and practice, and demonstrates the author’s use of pedagogical strategies and theory, including but not exclusive to comprehensible input, the sociocultural method, dynamic assessment, and the output hypothesis. It details the usage of current research-based strategies to increase classroom engagement and student proficiency, such as the IMAGE Model for teaching culture and language, collaborative learning, and dynamic assessment strategies. Furthermore, the portfolio includes a reflection on the benefits …


Quaking Aspen On Cedar Mountain, Utah: Status And Update, Jaycee Cappaert, Larissa L. Yocom, Paul C. Rogers, Darren J. Mcavoy, R. Justin Derose Aug 2026

Quaking Aspen On Cedar Mountain, Utah: Status And Update, Jaycee Cappaert, Larissa L. Yocom, Paul C. Rogers, Darren J. Mcavoy, R. Justin Derose

Wildland Resources Student Research

Quaking aspen forests are important communities in the western United States where they provide preferential food and habitat for animals, limit wildfires from spreading, and have favorable aesthetics, among other things. Their putative decline is an issue for ecology and management. We remeasured the aspen population on Cedar Mountain, Utah, to assess changes to its condition since it was originally measured in 2008 and laid the groundwork for continued aspen monitoring on Cedar Mountain. Our objectives were to compare current metrics of stand structure and composition, mortality, crown dieback, bole damage, regeneration, and recruitment, to those of the original inventory. …


The Augmented Matching Law, Matias A. Avellaneda Aug 2026

The Augmented Matching Law, Matias A. Avellaneda

All Graduate Theses and Dissertations, Fall 2023 to Present

Every living organism, including humans, has a finite amount of time available: 24 hours in a day, 365 days in a year, and a certain number of years in its life. At every moment, such an organism will be engaging in one of a multitude of possible activities, such as eating, sleeping, or reading this dissertation. We use the term choice to denote the way in which the organism distributes its time among all these activities, and some of these choice patterns are more beneficial to the organism than others, showcasing the importance of understanding the mechanisms underlying choice. A …


Applications Of Machine Learning To Gas Plume Analysis In Longwave Infrared Hyperspectral Images, Scout C. Jarman Aug 2026

Applications Of Machine Learning To Gas Plume Analysis In Longwave Infrared Hyperspectral Images, Scout C. Jarman

All Graduate Theses and Dissertations, Fall 2023 to Present

Each pixel from a hyperspectral camera measures the intensity of light over a continuous range of wavelengths, which is in contrast to traditional color cameras, which just measure the intensity of red, green, and blue wavelengths of light. Longwave infrared hyperspectral images can be used to detect gases from a distance by measuring how different materials emit and absorb heat. This makes them useful for applications such as monitoring industrial emissions or locating hazardous gas leaks. In practice, however, gas signatures in these hyperspectral images are often weak and easily obscured by variations in the background scene, making reliable identification …


A Comparison Of The Effectiveness Of Teacher-Led And Student-Supported Variations Of The Good Behavior Game, Patrick Wells Aug 2026

A Comparison Of The Effectiveness Of Teacher-Led And Student-Supported Variations Of The Good Behavior Game, Patrick Wells

All Graduate Theses and Dissertations, Fall 2023 to Present

This project examined the effectiveness of two variations of the Good Behavior Game (GBG) intervention: the traditional teacher-led and a student-supported variation. The traditional intervention has been shown to be an effective intervention for reducing disruptive classroom behaviors. Despite its effectiveness, proper implementation does require additional effort from the teacher—effort that may better be directed toward improving instruction. A student-supported variation of the intervention has the potential to reduce the effort required of teachers who wish to implement the intervention in their classroom. We utilized a nonconcurrent, multiple baseline design with alternating treatment to compare the effectiveness of both versions …


The Influence Of Habitat And Individual Specialization On Mountain Lion (Puma Concolor) Hunting Strategy In A Multi-Prey System, Hannah B. Klugman Aug 2026

The Influence Of Habitat And Individual Specialization On Mountain Lion (Puma Concolor) Hunting Strategy In A Multi-Prey System, Hannah B. Klugman

All Graduate Theses and Dissertations, Fall 2023 to Present

There is a growing recognition that animal species considered generalists (having a broad and varied diet), are actually specialists, a collection of individuals with different preferences and strategies known as specializations. The diet of a specialist depends on the variety and quantity of available food options, and their own ability to take what they need. Mountain lions (Puma concolor) are known for their ability and tendency to hunt large prey, such as elk (Cervus canadensis) and mule deer (Odocoileus hemionus), but despite the fact they overlap with other potential large prey, like feral horses …


Agronomic And Economic Evaluation Of Alternative Forage Crops Under Irrigation Technologies And Deficit Irrigation, Shikha Sharma Aug 2026

Agronomic And Economic Evaluation Of Alternative Forage Crops Under Irrigation Technologies And Deficit Irrigation, Shikha Sharma

All Graduate Theses and Dissertations, Fall 2023 to Present

Water is becoming more limited in the western United States, including Utah, so farmers need forage crops that can produce well while using less irrigation water. This study examined how different forage crops respond to reduced irrigation and different irrigation application methods, and whether some crops can provide better economic returns under limited water conditions.

Field studies were conducted at three Utah locations, Cedar City, Logan, and Vernal, from 2020 to 2024. The alternative forage crops included teff, sorghum-sudangrass (sudex), and forage rye. For economic comparison, conventional forage crops included alfalfa, corn, and three-way small grains. Four to five irrigation …


Bees In The Trap: Leveraging Bumble Bee (Hymenoptera: Apidae: Bombus) Bycatch For Ecological And Genetic Insights, Tori L. Strausser Aug 2026

Bees In The Trap: Leveraging Bumble Bee (Hymenoptera: Apidae: Bombus) Bycatch For Ecological And Genetic Insights, Tori L. Strausser

All Graduate Theses and Dissertations, Fall 2023 to Present

Managers rely on insect monitoring traps to detect insect pests that harm crops and protect food and other agricultural production. However, these traps also capture non-target insects, including bumble bees, raising concerns about whether repeated trapping could harm pollinator populations. This thesis explores whether bees caught as non-targets, also called bycatch, in routine pest monitoring traps can be used to better understand bumble bee populations, and whether traps pose risks to their long-term persistence. Using bumble bees collected from pest surveillance traps placed in crop fields across Utah, this research examined seasonal patterns in body size, timing of capture, and …


Bridging Animal Production And Human Nutrition: How Production Ecology Shapes The Nutritional Quality Of Terrestrial Animal Foods And Human Omega-3 Status, Joseph Vinod Varre Aug 2026

Bridging Animal Production And Human Nutrition: How Production Ecology Shapes The Nutritional Quality Of Terrestrial Animal Foods And Human Omega-3 Status, Joseph Vinod Varre

All Graduate Theses and Dissertations, Fall 2023 to Present

The question at the center of this dissertation is simple: does the way food is produced change the quality of the food itself in ways that matter for human health? Across a series of studies on bison meat, eggs, and human dietary intervention, the answer was yes. This work began with a broad look at the modern food system and the nutritional consequences of industrialized diets. In particular, it focused on the growing imbalance between omega-6 and omega-3 fatty acids in the Western diet. This balance matters because omega-3 fats support heart, brain, and immune health, and diets high in …


The Impact Of Early Exposure To Engineering Concepts In The Form Of Family, School Classes, And Stem Extracurriculars On The Long-Term Sense Of Belonging In Undergraduate Women In Engineering, Hannah Lloyd Aug 2026

The Impact Of Early Exposure To Engineering Concepts In The Form Of Family, School Classes, And Stem Extracurriculars On The Long-Term Sense Of Belonging In Undergraduate Women In Engineering, Hannah Lloyd

All Graduate Theses and Dissertations, Fall 2023 to Present

Engineering continues to struggle with the underrepresentation of women, and many women who begin engineering programs report feeling isolated or uncertain about whether they belong. A strong sense of belonging has been linked to greater confidence, persistence, and success in engineering. This study examined how early experiences with engineering before college may influence students' sense of belonging later in their undergraduate engineering programs.

Using survey responses from undergraduate engineering students across the United States, this study explored several types of early exposure to engineering, including family influence, school-based engineering and STEM courses, extracurricular programs such as robotics clubs and engineering …


Investigating Factors That Affect Students' Help-Seeking Strategies In A University-Level Foundational Mathematics Course, Randall I. Haws Aug 2026

Investigating Factors That Affect Students' Help-Seeking Strategies In A University-Level Foundational Mathematics Course, Randall I. Haws

All Graduate Theses and Dissertations, Fall 2023 to Present

Quantitative literacy, or the ability to understand and use mathematical information, is an important skill developed in college. However, required math courses can be a major obstacle for many students, and those who struggle to complete them are more likely to leave school without earning a degree. Understanding why some students seek help while others do not is key to improving student success.

This study explores how students in a university-level foundational mathematics course (college algebra) seek help and what factors influence their use of available support services. A total of 118 students completed a survey about their help-seeking behaviors, …


Learning Latent Structure In High-Dimensional Data Via Geometry And Graphs, Haozhe Chen Aug 2026

Learning Latent Structure In High-Dimensional Data Via Geometry And Graphs, Haozhe Chen

All Graduate Theses and Dissertations, Fall 2023 to Present

Modern datasets often contain many measured variables for each observation, such as gene-expression levels, brain activity signals, or features in tabular data. These data are also often noisy, meaning that useful patterns are mixed with measurement error or irrelevant variation. Although such datasets can appear complex, they are frequently represented by simpler hidden structures, such as trajectories, clusters, or relationships between observations. This dissertation develops methods for uncovering these hidden structures by learning geometric and graph-based representations directly from data. The first part introduces Functional Information Geometry, which represents local patterns in high-dimensional data using functional features and constructs a …


Unifying And Expanding Global And Local Variable Importance Methods For Explainable Machine Learning, Kelvyn K. Bladen Aug 2026

Unifying And Expanding Global And Local Variable Importance Methods For Explainable Machine Learning, Kelvyn K. Bladen

All Graduate Theses and Dissertations, Fall 2023 to Present

Machine learning methods are powerful analytical tools used across all scientific disciplines and many other fields of investigation for prediction and inference from diverse data sources. Despite their broad applicability, machine learning methods are often highly complex and difficult to interpret. Developing a greater understanding of which variables most influence a response is essential for increasing the interpretability of these models and supporting informed decision-making. This research focuses on improving how we evaluate the importance of these variables.

One common approach is to shuffle the values of a variable and see how much the model accuracy gets worse. Another approach …


Three Essays In Discrete Choice Analysis: Valuation, Attention Level, And Spatial Heterogeneity, Marcelo Pignatari Aug 2026

Three Essays In Discrete Choice Analysis: Valuation, Attention Level, And Spatial Heterogeneity, Marcelo Pignatari

All Graduate Theses and Dissertations, Fall 2023 to Present

Understanding how people value food labels and environmental goods is essential for designing policies, incentives, and market mechanisms that improve the well-being of the population. In this study, we found that a local label indicating that fruit was produced in Utah makes Utahns value strawberries, gala apples, and dried tart cherries more. An organic label has a similar effect, which is especially strong for dried tart cherries. In contrast, a freshness label only increases value for fruits with a shorter shelf life, such as strawberries. We also found that people who chose to access clarifying information about what these labels …


Advancing Management Strategies For Two Promising Mason Bee Species (Osmia Ribifloris And Osmia Bruneri) For Berry Pollination, Miranda L. Jones Aug 2026

Advancing Management Strategies For Two Promising Mason Bee Species (Osmia Ribifloris And Osmia Bruneri) For Berry Pollination, Miranda L. Jones

All Graduate Theses and Dissertations, Fall 2023 to Present

Bee management can broadly refer to human manipulation of one or multiple parts of a bee’s lifecycle. Managing honey bees, commonly referred to as beekeeping, is responsible for providing most pollination for global food crops. However, using more than one bee species for crop pollination can have many benefits. Benefits include better crop pollination and insurance in case one bee species underperforms or its population collapses. Some bees within the genus Osmia are already widely managed for crop pollination because their developmental lifecycle allows bee managers to readily deploy pollinating bees in sync with varying crop blooms, and for certain …


Sensitivity Analysis Of A Low-Power Wireless Capacitive Charging System To Inform Thermal Limitations And Battery Size Reduction Potential For Electric Passenger Vehicles, Karmen Teuscher Aug 2026

Sensitivity Analysis Of A Low-Power Wireless Capacitive Charging System To Inform Thermal Limitations And Battery Size Reduction Potential For Electric Passenger Vehicles, Karmen Teuscher

All Graduate Theses and Dissertations, Fall 2023 to Present

The purpose of using electric vehicles instead of internal combustion engine vehicles that burn gas is to reduce pollutants in the air and potential environmental impacts such as climate change. EVs produce much less air pollutants than ICEVs but manufacturing the EV batteries and the repeated recharging still contributes to the overall air pollution.

Capacitive power transfer systems can wirelessly charge vehicles through the road as they drive along highways or stop at traffic lights. If vehicles charge as they drive, an opportunity opens to reduce the size of the battery while maintaining the same driving range. Reducing the size …


Visualizing The Phase Space Of Two Interacting Spherical Magnets Via Lagrangian Descriptors, Matthew Pontius Aug 2026

Visualizing The Phase Space Of Two Interacting Spherical Magnets Via Lagrangian Descriptors, Matthew Pontius

All Graduate Theses and Dissertations, Fall 2023 to Present

This thesis studies the motion of one spherical magnet sliding on another fixed spherical magnet. Although the setup is simple, the resulting motion can range from regular and predictable to chaotic as the energy increases. To understand this behavior, mathematical tools are used to visualize how all possible motions are organized in phase space. These methods reveal patterns such as stable regions, repeating motions, and chaotic trajectories. The results show how order and chaos coexist in this system and provide insight into how small changes in the initial conditions can lead to very different outcomes.