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Articles 1 - 30 of 36
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Contributions To The Teaching And Learning Of Fluid Mechanics, Ashwin Vaidya
Contributions To The Teaching And Learning Of Fluid Mechanics, Ashwin Vaidya
Department of Mathematics Facuty Scholarship and Creative Works
This issue showcases a compilation of papers on fluid mechanics (FM) education, covering different sub topics of the subject. The success of the first volume [1] prompted us to consider another follow-up special issue on the topic, which has also been very successful in garnering an impressive variety of submissions. As a classical branch of science, the beauty and complexity of fluid dynamics cannot be overemphasized. This is an extremely well-studied subject which has now become a significant component of several major scientific disciplines ranging from aerospace engineering, astrophysics, atmospheric science (including climate modeling), biological and biomedical science …
Numerical Computations Of Vortex Formation Length In Flow Past An Elliptical Cylinder, Matthew Karlson, Bogdan Nita, Ashwin Vaidya
Numerical Computations Of Vortex Formation Length In Flow Past An Elliptical Cylinder, Matthew Karlson, Bogdan Nita, Ashwin Vaidya
Department of Mathematics Facuty Scholarship and Creative Works
We examine two dimensional properties of vortex shedding past elliptical cylinders through numerical simulations. Specifically, we investigate the vortex formation length in the Reynolds number regime 10 to 100 for elliptical bodies of aspect ratio in the range 0.4 to 1.4. Our computations reveal that in the steady flow regime, the change in the vortex length follows a linear profile with respect to the Reynolds number, while in the unsteady regime, the time averaged vortex length decreases in an exponential manner with increasing Reynolds number. The transition in profile is used to identify the critical Reynolds number which marks the …
Teaching And Learning Of Fluid Mechanics, Ashwin Vaidya
Teaching And Learning Of Fluid Mechanics, Ashwin Vaidya
Department of Mathematics Facuty Scholarship and Creative Works
Fluid mechanics occupies a privileged position in the sciences; it is taught in various science departments including physics, mathematics, environmental sciences and mechanical, chemical and civil engineering, with each highlighting a different aspect or interpretation of the foundation and applications of fluids. Doll’s fluid analogy [5] for this idea is especially relevant to this issue: “Emergence of creativity from complex flow of knowledge—example of Benard convection pattern as an analogy—dissipation or dispersal of knowledge (complex knowledge) results in emergent structures, i.e., creativity which in the context of education should be thought of as a unique way to arrange information so …
Fluids In Music: The Mathematics Of Pan’S Flutes, Bogdan Nita, Sajan Ramanathan
Fluids In Music: The Mathematics Of Pan’S Flutes, Bogdan Nita, Sajan Ramanathan
Department of Mathematics Facuty Scholarship and Creative Works
We discuss the mathematics behind the Pan’s flute. We analyze how the sound is created, the relationship between the notes that the pipes produce, their frequencies and the length of the pipes. We find an equation which models the curve that appears at the bottom of any Pan’s flute due to the different pipe lengths.
Conventions, Habits, And U.S. Teachers’ Meanings For Graphs, Kevin C. Moore, Jason Silverman, Teo Paoletti, Dave Liss, Stacy Musgrave
Conventions, Habits, And U.S. Teachers’ Meanings For Graphs, Kevin C. Moore, Jason Silverman, Teo Paoletti, Dave Liss, Stacy Musgrave
Department of Mathematics Facuty Scholarship and Creative Works
In this paper, we use relevant literature and data to motivate a more detailed look into relationships between what we perceive to be conventions common to United States (U.S.) school mathematics and individuals’ meanings for graphs and related topics. Specifically, we draw on data from pre-service (PST) and in-service (IST) teachers to characterize such relationships. We use PSTs’ responses during clinical interviews to illustrate three themes: (a) some PSTs’ responses implied practices we perceive to be conventions of U.S. school mathematics were instead inherent aspects of PSTs’ meanings; (b) some PSTs’ responses implied they understood certain practices in U.S. school …
Pre-Service Teachers’ Figurative And Operative Graphing Actions, Kevin C. Moore, Irma E. Stevens, Teo Paoletti, Natalie L.F. Hobson, Biyao Liang
Pre-Service Teachers’ Figurative And Operative Graphing Actions, Kevin C. Moore, Irma E. Stevens, Teo Paoletti, Natalie L.F. Hobson, Biyao Liang
Department of Mathematics Facuty Scholarship and Creative Works
We report on semi-structured clinical interviews to describe U.S. pre-service secondary mathematics teachers’ graphing meanings. Our primary goal is to draw on Piagetian notions of figurative and operative thought to identify marked differences in the students’ meanings. Namely, we illustrate students’ meanings dominated by fragments of sensorimotor experience and compare those with students’ meanings dominated by the coordination of mental actions in the form of covarying quantities. Our findings suggest students’ meanings that foreground operative aspects of thought are more generative with respect to graphing. Our findings also indicate that students can encounter perturbations due to potential incompatibilities between figurative …
Cause And Consequence Of Aβ – Lipid Interactions In Alzheimer Disease Pathogenesis, Vijayaraghavan Rangachari, Dexter N. Dean, Pratip Rana, Ashuwin Vaidya, Preetam Ghosh
Cause And Consequence Of Aβ – Lipid Interactions In Alzheimer Disease Pathogenesis, Vijayaraghavan Rangachari, Dexter N. Dean, Pratip Rana, Ashuwin Vaidya, Preetam Ghosh
Department of Mathematics Facuty Scholarship and Creative Works
Self-templating propagation of protein aggregate conformations is increasingly becoming a significant factor in many neurological diseases. In Alzheimer disease (AD), intrinsically disordered amyloid-β (Aβ) peptides undergo aggregation that is sensitive to environmental conditions. High-molecular weight aggregates of Aβ that form insoluble fibrils are deposited as senile plaques in AD brains. However, low-molecular weight aggregates called soluble oligomers are known to be the primary toxic agents responsible for neuronal dysfunction. The aggregation process is highly stochastic involving both homotypic (Aβ-Aβ) and heterotypic (Aβ with interacting partners) interactions. Two of the important members of interacting partners are membrane lipids and surfactants, to …
A Greedy Algorithm For Finding A Large 2-Matching On A Random Cubic Graph, Deepak Bal, Patrick Bennett, Tom Bohman, Alan Frieze
A Greedy Algorithm For Finding A Large 2-Matching On A Random Cubic Graph, Deepak Bal, Patrick Bennett, Tom Bohman, Alan Frieze
Department of Mathematics Facuty Scholarship and Creative Works
A 2-matching of a graph G is a spanning subgraph with maximum degree two. The size of a 2-matching U is the number of edges in U and this is at least n-k(U) where n is the number of vertices of G and κ denotes the number of components. In this article, we analyze the performance of a greedy algorithm 2greedy for finding a large 2-matching on a random 3-regular graph. We prove that with high probability, the algorithm outputs a 2-matching U with k(U)=Θ(n1/5).
Teacher Questioning And Invitations To Participate In Advanced Mathematics Lectures, Teo Paoletti, Victoria Krupnik, Dimitrios Papadopoulos, Joseph Olsen, Tim Fukawa-Connelly, Keith Weber
Teacher Questioning And Invitations To Participate In Advanced Mathematics Lectures, Teo Paoletti, Victoria Krupnik, Dimitrios Papadopoulos, Joseph Olsen, Tim Fukawa-Connelly, Keith Weber
Department of Mathematics Facuty Scholarship and Creative Works
In this study, we were interested in exploring the extent to which advanced mathematics lecturers provide students opportunities to play a role in considering or generating course content. To do this, we examined the questioning practices of 11 lecturers who taught advanced mathematics courses at the university level. Because we are unaware of other studies examining advanced mathematics lecturers’ questioning, we first analyzed the data using an open coding scheme to categorize the types of content lecturers solicited and the opportunities they provided students to participate in generating course content. In a second round of analysis, we examined the extent …
Simplicity And Sustainability: Pointers From Ethics And Science, Mehrdad Massoudi, Ashuwin Vaidya
Simplicity And Sustainability: Pointers From Ethics And Science, Mehrdad Massoudi, Ashuwin Vaidya
Department of Mathematics Facuty Scholarship and Creative Works
In this paper, we explore the notion of simplicity. We use definitions of simplicity proposed by philosophers, scientists, and economists. In an age when the rapidly growing human population faces an equally rapidly declining energy/material resources, there is an urgent need to consider various notions of simplicity, collective and individual, which we believe to be a sensible path to restore our planet to a reasonable state of health. Following the logic of mathematicians and physicists, we suggest that simplicity can be related to sustainability. Our efforts must therefore not be spent so much in pursuit of growth but in achieving …
Simplicity And Sustainability: Pointers From Ethics And Science, Mehrdad Massoudi, Ashwin Vaidya
Simplicity And Sustainability: Pointers From Ethics And Science, Mehrdad Massoudi, Ashwin Vaidya
Department of Mathematics Facuty Scholarship and Creative Works
In this paper, we explore the notion of simplicity. We use definitions of simplicity proposed by philosophers, scientists, and economists. In an age when the rapidly growing human population faces an equally rapidly declining energy/material resources, there is an urgent need to consider various notions of simplicity, collective and individual, which we believe to be a sensible path to restore our planet to a reasonable state of health. Following the logic of mathematicians and physicists, we suggest that simplicity can be related to sustainability. Our efforts must therefore not be spent so much in pursuit of growth but in achieving …
Minimizing The Number Of Independent Sets In Triangle-Free Regular Graphs, Jonathan Cutler, A. J. Radcliffe
Minimizing The Number Of Independent Sets In Triangle-Free Regular Graphs, Jonathan Cutler, A. J. Radcliffe
Department of Mathematics Facuty Scholarship and Creative Works
Recently, Davies, Jenssen, Perkins, and Roberts gave a very nice proof of the result (due, in various parts, to Kahn, Galvin–Tetali, and Zhao) that the independence polynomial of a d-regular graph is maximized by disjoint copies of Kd,d. Their proof uses linear programming bounds on the distribution of a cleverly chosen random variable. In this paper, we use this method to give lower bounds on the independence polynomial of regular graphs. We also give a new bound on the number of independent sets in triangle-free cubic graphs.
Interlace Polynomials Of Friendship Graphs, Christina Eubanks-Turner, Aihua Li
Interlace Polynomials Of Friendship Graphs, Christina Eubanks-Turner, Aihua Li
Department of Mathematics Facuty Scholarship and Creative Works
In this paper, we study the interlace polynomials of friendship graphs, that is, graphs that satisfy the Friendship Theorem given by Erdös, Rényi and Sos. Explicit formulas, special values, and behaviour of coefficients of these polynomials are provided. We also give the interlace polynomials of other similar graphs, such as, the butterfly graph.
Rethinking The Teaching And Learning Of Area Measurement, Nicole Panorkou
Rethinking The Teaching And Learning Of Area Measurement, Nicole Panorkou
Department of Mathematics Facuty Scholarship and Creative Works
This study focused on exploring an innovative way of teaching and learning measurement, what we refer to as Dynamic Measurement or DYME. Without relying on the common approach of counting square units, our goal was to engage students in contextually rich digital dynamic tasks to visualize area as a continuous quantity and evaluate the area of a rectangular region as a multiplicative relationship between the two lengths of the sides. In this paper, we briefly describe the iterative process of designing, testing and refining the tasks for DYME pointing to the significance of the design for developing students’ thinking of …
Inverse Function: Pre-Service Teachers’ Techniques And Meanings, Teo Paoletti, Irma E. Stevens, Natalie L.F. Hobson, Kevin C. Moore, Kevin R. Laforest
Inverse Function: Pre-Service Teachers’ Techniques And Meanings, Teo Paoletti, Irma E. Stevens, Natalie L.F. Hobson, Kevin C. Moore, Kevin R. Laforest
Department of Mathematics Facuty Scholarship and Creative Works
Researchers have argued teachers and students are not developing connected meanings for function inverse, thus calling for a closer examination of teachers’ and students’ inverse function meanings. Responding to this call, we characterize 25 pre-service teachers’ inverse function meanings as inferred from our analysis of clinical interviews. After summarizing relevant research, we describe the methodology and theoretical framework we used to interpret the pre-service teachers’ activities. We then present data highlighting the techniques the pre-service teachers used when responding to tasks that involved analytical and graphical representations of functions and inverse functions in both decontextualized and contextualized situations and discuss …
A Covariational Understanding Of Function: Putting A Horse Before The Cart, Teo Paoletti, Kevin C. Moore
A Covariational Understanding Of Function: Putting A Horse Before The Cart, Teo Paoletti, Kevin C. Moore
Department of Mathematics Facuty Scholarship and Creative Works
Supporting students developing understandings of function has been a notoriously elusive task in mathematics education. We present Thompson and Carlson’s (2017) description of a covariational meaning of function and provide an example of a student who maintains meanings compatible with this description. We use this student’s activity to illustrate nuances in a covariational meaning of function and to highlight how such meanings can be powerful for students. In doing so, we argue that a student who has develop meanings compatible with the covariational meaning of function presented by Thompson and Carlson has the foundational meanings needed to understand a formal …
Entropy Production In A Fluid-Solid System Far From Thermodynamic Equilibrium, Bong Jae Chung, Blas Ortega, Ashuwin Vaidya
Entropy Production In A Fluid-Solid System Far From Thermodynamic Equilibrium, Bong Jae Chung, Blas Ortega, Ashuwin Vaidya
Department of Mathematics Facuty Scholarship and Creative Works
Abstract.: The terminal orientation of a rigid body in a moving fluid is an example of a dissipative system, out of thermodynamic equilibrium and therefore a perfect testing ground for the validity of the maximum entropy production principle (MaxEP). Thus far, dynamical equations alone have been employed in studying the equilibrium states in fluid-solid interactions, but these are far too complex and become analytically intractable when inertial effects come into play. At that stage, our only recourse is to rely on numerical techniques which can be computationally expensive. In our past work, we have shown that the MaxEP is a …
Connecting Advanced And Secondary Mathematics, Eileen Murray, Erin Baldinger, Nicholas Wasserman, Shawn Broderick, Diana White
Connecting Advanced And Secondary Mathematics, Eileen Murray, Erin Baldinger, Nicholas Wasserman, Shawn Broderick, Diana White
Department of Mathematics Facuty Scholarship and Creative Works
There is an ongoing debate among scholars in understanding what mathematical knowledge secondary teachers should have in order to provide effective instruction. We explore connections between advanced and secondary mathematics as an entry point into this debate. In many cases, advanced mathematics is considered relevant for secondary teachers simply because the content is inherently related. In this paper, we instead argue that there are connections between advanced mathematics and secondary mathematics that directly influence teaching. These are not discussions of the mathematical connections, per se, but rather discussions of specific ways in which knowing mathematical connections might influence secondary teachers’ …
Rainbow Perfect Matchings And Hamilton Cycles In The Random Geometric Graph, Deepak C. Bal, Patrick Bennett, Xavier Pérez‐Giménez, Paweł Prałat
Rainbow Perfect Matchings And Hamilton Cycles In The Random Geometric Graph, Deepak C. Bal, Patrick Bennett, Xavier Pérez‐Giménez, Paweł Prałat
Department of Mathematics Facuty Scholarship and Creative Works
Given a graph on n vertices and an assignment of colours to the edges, a rainbow Hamilton cycle is a cycle of length n visiting each vertex once and with pairwise different colours on the edges. Similarly (for even n) a rainbow perfect matching is a collection of independent edges with pairwise different colours. In this note we show that if we randomly colour the edges of a random geometric graph with sufficiently many colours, then a.a.s. the graph contains a rainbow perfect matching (rainbow Hamilton cycle) if and only if the minimum degree is at least 1 (respectively, …
Partitioning Random Graphs Into Monochromatic Components, Deepak Bal, Louis Debiasio
Partitioning Random Graphs Into Monochromatic Components, Deepak Bal, Louis Debiasio
Department of Mathematics Facuty Scholarship and Creative Works
Erdős, Gyàrfàs, and Pyber (1991) conjectured that every r-colored complete graph can be partitioned into at most r – 1 monochromatic components; this is a strengthening of a conjecture of Lovàsz (1975) and Ryser (1970) in which the components are only required to form a cover. An important partial result of Haxell and Kohayakawa (1995) shows that a partition into r monochromatic components is possible for sufficiently large r-colored complete graphs. We start by extending Haxell and Kohayakawa’s result to graphs with large minimum degree, then we provide some partial analogs of their result for random graphs. In particular, we …
Metastable States In Terminal Orientation Of Hinged Symmetric Bodies In A Flow, Doralia Castillo, Bong Jae Chung, Klaus Schnitzer, Karina Soriano, Haiyan Su, Ashuwin Vaidya
Metastable States In Terminal Orientation Of Hinged Symmetric Bodies In A Flow, Doralia Castillo, Bong Jae Chung, Klaus Schnitzer, Karina Soriano, Haiyan Su, Ashuwin Vaidya
Department of Mathematics Facuty Scholarship and Creative Works
Symmetric bodies such as cylinders and spheroidal bodies, in their terminal stable states, are long known to have their long axis align themselves perpendicular to the direction of the flow. This property has been confirmed in primarily sedimentation based theoretical, experimental and numerical techniques and the transition to a terminal stable state is believed to coincide with the onset of significant inertial effects in the flow. However, the threshold at which this transition occurs is yet unknown. We conduct modified experiments with hinged bodies and a CFD study to examine the nature of the transition of prolate spheroids and cylinders …
The Maximum Number Of Complete Subgraphs Of Fixed Size In A Graph With Given Maximum Degree, Jonathan Cutler, A. J. Radcliffe
The Maximum Number Of Complete Subgraphs Of Fixed Size In A Graph With Given Maximum Degree, Jonathan Cutler, A. J. Radcliffe
Department of Mathematics Facuty Scholarship and Creative Works
In this article, we make progress on a question related to one of Galvin that has attracted substantial attention recently. The question is that of determining among all graphs G with n vertices and Δ(G) ≤ r, which has the most complete subgraphs of size t, for t≥3. The conjectured extremal graph is aKr+1 ∪ Kb, where n = a(r + 1) + b with 0 ≤ b ≤ r. Gan et al. (Combin Probab Comput 24(3) (2015), 521–527) proved the conjecture when a ≤ 1, and also reduced the general conjecture to the case t = 3. We prove …
On The Three Dimensional Interaction Between Flexible Fibers And Fluid Flow, Bogdan Nita, Ryan Allaire
On The Three Dimensional Interaction Between Flexible Fibers And Fluid Flow, Bogdan Nita, Ryan Allaire
Department of Mathematics Facuty Scholarship and Creative Works
In this paper we discuss the deformation of a flexible fiber clamped to a spherical body and immersed in a flow of fluid moving with a speed ranging between 0 and 50 cm/s by means of three dimensional numerical simulation developed in COMSOL . The effects of flow speed and initial configuration angle of the fiber relative to the flow are analyzed. A rigorous analysis of the numerical procedure is performed and our code is benchmarked against well established cases. The flow velocity and pressure are used to compute drag forces upon the fiber. Of particular interest is the behavior …
Freedericksz Transition In Nematic Liquid Crystal Couette Flow, Arup Mukherjee
Freedericksz Transition In Nematic Liquid Crystal Couette Flow, Arup Mukherjee
Department of Mathematics Facuty Scholarship and Creative Works
This article investigates the effects of an external magnetic field on the Freedericksz transition for an elastically anisotropic nematic liquid crystal sample occupying the annular region between two concentric cylinders in relative (slow) rotation. Assuming both azimuthal and radial magnetic fields and strong anchoring conditions for the liquid crystal director perpendicular to the surface of the cylinders, we investigate and characterize the differences in the director distortions and the critical field value for the onset of the transition.
Exploring Connections Between Advanced And Secondary Mathematics, Erin E. Baldinger, Eileen Murray, Diana White, Shawn Broderick, Nicholas Wasserman
Exploring Connections Between Advanced And Secondary Mathematics, Erin E. Baldinger, Eileen Murray, Diana White, Shawn Broderick, Nicholas Wasserman
Department of Mathematics Facuty Scholarship and Creative Works
The second meeting of this Working Group continues to explore questions about the connections between abstract algebra and school mathematics. Our goal is to focus in on questions around the way in which teachers’ practice might be influenced based on their understanding of such connections. In particular, we will gather interested individuals in an effort to deepen our understanding of existing connections between abstract algebra and secondary mathematics and which of these connections are important for secondary teachers to know and understand. Moreover, we aim to further research in this area by first considering connections between abstract algebra and school …
Computing The Optimal Path In Stochastic Dynamical Systems, Martha Bauver, Eric Forgoston, Lora Billings
Computing The Optimal Path In Stochastic Dynamical Systems, Martha Bauver, Eric Forgoston, Lora Billings
Department of Mathematics Facuty Scholarship and Creative Works
In stochastic systems, one is often interested in finding the optimal path that maximizes the probability of escape from a metastable state or of switching between metastable states. Even for simple systems, it may be impossible to find an analytic form of the optimal path, and in high- dimensional systems, this is almost always the case. In this article, we formulate a constructive methodology that is used to compute the optimal path numerically. The method utilizes finite-time Lyapunov exponents, statistical selection criteria, and a Newton-based iterative minimizing scheme. The method is applied to four examples. The first example is a …
Determination Of Critical Nucleation Number For A Single Nucleation Amyloid-Β Aggregation Model, Preetam Ghosh, Ashuwin Vaidya, Amit Kumar, Vijayaraghavan Rangachari
Determination Of Critical Nucleation Number For A Single Nucleation Amyloid-Β Aggregation Model, Preetam Ghosh, Ashuwin Vaidya, Amit Kumar, Vijayaraghavan Rangachari
Department of Mathematics Facuty Scholarship and Creative Works
Aggregates of amyloid-β (Aβ) peptide are known to be the key pathological agents in Alzheimer disease (AD). Aβ aggregates to form large, insoluble fibrils that deposit as senile plaques in AD brains. The process of aggregation is nucleation-dependent in which the formation of a nucleus is the rate-limiting step, and controls the physiochemical fate of the aggregates formed. Therefore, understanding the properties of nucleus and pre-nucleation events will be significant in reducing the existing knowledge-gap in AD pathogenesis. In this report, we have determined the plausible range of critical nucleation number (n*, the number of monomers associated within the nucleus …
Improving Teaching Through Collaborative Reflective Teaching Cycles, Eileen Murray
Improving Teaching Through Collaborative Reflective Teaching Cycles, Eileen Murray
Department of Mathematics Facuty Scholarship and Creative Works
Reflection and collaboration are two activities teachers can use to change and improve their practice. However, finding the time and space to do so can be challenging. The collaborative reflective teaching cycle is a structured activity teachers can use to engage in reflection and collaboration. This article describes how a seventh grade teaching team implemented a series of cycles and in what ways the cycles impacted their practice. Implications for instruction and suggestions for use of the cycles in practice are discussed.
When To Spray: A Time-Scale Calculus Approach To Controlling The Impact Of West Nile Virus, Diana Thomas, Marion Weedermann, Lora Billings, Joan Hoffacker, Robert Washington-Allen
When To Spray: A Time-Scale Calculus Approach To Controlling The Impact Of West Nile Virus, Diana Thomas, Marion Weedermann, Lora Billings, Joan Hoffacker, Robert Washington-Allen
Department of Mathematics Facuty Scholarship and Creative Works
West Nile Virus (WNV) made its initial appearance in the New York City (NYC) metropolitan area in 1999 and was implicated in cases of human encephalitis and the extensive mortality in crows (Corvus sp.) and other avian species. Mosquitoes were found to be the primary vectors and NYC’s current policy on control strategies involved an eradication program that depends on the synchronicity of the summer mosquito population’s increases with the occurrence of cases in humans. The purpose of this paper is to investigate whether this is the most effective control strategy because past mathematical models assumed discrete behavior that …
Multi-Scale Continuum Mechanics: From Global Bifurcations To Noise Induced High-Dimensional Chaos, Ira B. Schwartz, David S. Morgan, Lora Billings, Ying-Cheng Lai
Multi-Scale Continuum Mechanics: From Global Bifurcations To Noise Induced High-Dimensional Chaos, Ira B. Schwartz, David S. Morgan, Lora Billings, Ying-Cheng Lai
Department of Mathematics Facuty Scholarship and Creative Works
Many mechanical systems consist of continuum mechanical structures, having either linear or nonlinear elasticity or geometry, coupled to nonlinear oscillators. In this paper, we consider the class of linear continua coupled to mechanical pendula. In such mechanical systems, there often exist several natural time scales determined by the physics of the problem. Using a time scale splitting, we analyze a prototypical structural–mechanical system consisting of a planar nonlinear pendulum coupled to a flexible rod made of linear viscoelastic material. In this system both low-dimensional and high-dimensional chaos is observed. The low-dimensional chaos appears in the limit of small coupling between …