A self-consistent dynamical system with multiple absolutely continuous invariant measures

In this paper we study a class of \emph{self-consistent dynamical systems}, self-consistent in the sense that the discrete time dynamics is different in each step depending on current statistics. The general framework admits popular examples such as coupled map systems. Motivated by an example of M. Blank, we concentrate on a special case where the dynamics in each step is a $\beta$-map with some $\beta \geq 2$. Included in the definition of $\beta$ is a parameter $\varepsilon > 0$ controlling the strength of self-consistency. We show such a self-consistent system which has a unique absolutely continuous invariant measure (acim) for $\varepsilon=0$, but at least two for any $\varepsilon > 0$. With a slight modification, we transform this system into one which produces a phase transition-like behavior: it has a unique acim for $0< \varepsilon < \varepsilon^*$, and multiple for sufficiently large values of $\varepsilon$. We discuss the stability of the invariant measures by the help of computer simulations employing the numerical representation of the self-consistent transfer operator.Comment: Electronic copy of final peer-reviewed manuscript accepted for publicatio

Synchronization versus stability of the invariant distribution for a class of globally coupled maps

We study a class of globally coupled maps in the continuum limit, where the individual maps are expanding maps of the circle. The circle maps in question are such that the uncoupled system admits a unique absolutely continuous invariant measure (acim), which is furthermore mixing. Interaction arises in the form of diffusive coupling, which involves a function that is discontinuous on the circle. We show that for sufficiently small coupling strength the coupled map system admits a unique absolutely continuous invariant distribution, which depends on the coupling strength $\varepsilon$. Furthermore, the invariant density exponentially attracts all initial distributions considered in our framework. We also show that the dependence of the invariant density on the coupling strength $\varepsilon$ is Lipschitz continuous in the BV norm. When the coupling is sufficiently strong, the limit behavior of the system is more complex. We prove that a wide class of initial measures approach a point mass with support moving chaotically on the circle. This can be interpreted as synchronization in a chaotic state

PENGEMBANGAN LEMBAR KERJA SISWA (LKS) PADA PEMBELAJARAN MEKANIKA TEKNIK DENGAN PENDEKATAN KONTEKSTUAL UNTUK SISWA KELAS X TGB SMK NEGERI 2 SURABAYA

Mata pelajaran mekanika teknik merupakan mata pelajaran yang memerlukan pemahaman dan penjelasan yang lebih detail dan luas pada materinya.Untuk dapat meningkatkan pemahaman siswa dibutuhkan suatu perangkat yang berorientasi pada pemunculan masalah yang berhubungan&nbsp; dengan&nbsp; kehidupan&nbsp; nyata (kontekstual). Lembar Kerja Siswa (LKS) merupakan salah satu perangkat pembelajaran yang dapat dikembangkan dengan pendekatan kontekstual. Tujuan penelitian ini adalah: (1) mendapatkan produk Lembar kerja siswa (LKS) mekanika teknik dengan pendekatan kontekstual yang layak/valid. (2) mengetahui hasil belajar siswa setelah menggunakan Lembar Kerja Siswa (LKS) yang dikembangkan. (3) mengetahui respon siswa terhadap pembelajaran menggunakan Lembar kerja siswa (LKS) yang dikembangkan. Jenis Penelitian ini adalah penelitian pengembangan yang mengacu pada model 4D dengan tahapan define, design, develop, dan disseminate. Namun pada penelitian ini, pengembangan LKS dilakukan hanya sampai tahap develop. Subyek penelitian ini adalah siswa kelas X TGB 1 SMK Negeri 2 Surabaya. Instrumen yang gunakan adalah lembar validasi LKS, tes hasil belajar dan angket respon siswa. Hasil penelitian menunjukkan bahwa: (1) skor kelayakan LKS bedasarkan telaah para ahli sebesar 4,11yang termasuk dalam kategori sangat layak sehingga dapat digunakan untuk pembelajaran. (2) Dari skor hasil belajar didapatkan 9 siswa sangat baik pencapaianya dalam pembelajaran mekanika teknik pada materi momen kopel menggunakan LKS dengan pendekatan kontekstual, 18 siswa mendapatkan kriteria baik dan terdapat 4 siswa yang mendapatkan kriteria kurang &nbsp;(3) respon positif siswa dapat dilihat bahwa sebanyak 74,2% siswa menyatakan senang mengikuti pembelajaran yang menggunakan LKS, 77,4% siswa menyatakan memahami setiap petunjuk yang terdapat di LKS, 77,4% siswa dapat memahami permasalahan pada LKS, 93,5% siswa dapat memahami materi momen dan kopel berdasarkan langkah-langkah kegiatan yang ada pada LKS, 64,5% siswa menyatakan bahwa tampilan (gambar, warna, dll) yang terdapat pada Lembar Kegiatan Siswa&nbsp; menarik, 54,8% siswa menyatakan berminat untuk mengikuti kegiatan belajar dengan menggunakan LKS, 77,4% siswa menyatakan belajar dengan tugas kinerja seperti yang terdapat pada LKS sangat bermanfaat bagi mereka, 74,19% siswa menyatakan bahwa materi lain perlu menggunakan LKS, 67,7% siswa menyatakan bahwa soal yang ada pada bagian latihan sesuai dengan materi, 80,6% siswa menyatakan soal-soal di LKS telah sesuai dengan kehidupan sehari-hari. Kata Kunci: LKS, Pendekatan Kontekstual, Mekanika Teknik, Momen dan Kopel Engineering mechanics subjects are subjects that require understanding and a more detailed explanation and wide on the material. For can enhance students' understanding of a need for a device that is oriented to the appearance of the problems associated with real life (contextual). Student Worksheet is one of the lessons that can be developed with a contextual approach. The aim of this study were: (1) obtain the product Worksheets students engineering mechanics with proper contextual approach / valid. (2) determine student learning outcomes after using the Student Worksheet developed. (3) determine students 'response to students' learning using the worksheet developed This is development research&nbsp; reffering to 4D steps, that are define, design, develop, and disseminate. However, in this research, development is done until the develop only. The subjects of this research are students of sixth grade TGB 1 of SMK Negeri 2 Surabaya. The instrument used was a validation sheet of worksheet, achievement test and students responses sheet. The result showed that: (1) review of the validation of worksheet based on the experts at 4,11 which is included in the category of very decent so it can be used for learning. (2) From the scores obtained 9 student learning outcomes excellent achievement in engineering mechanics teaching at the moment of coupling material using worksheets with contextual approach , 18 students get a good criteria and there are four students who earn less &nbsp;(3) positive responses of students can be seen that as many as 74.2% of students stated enjoy participating in learning to use worksheets, 77.4% of students stated understand any instructions contained in LKS, 77.4% of the students can understand the problems of the LKS, 93, 5% of the students can understand the material moments and coupling based measures existing activities in LKS, 64.5% of students stated that the display (images, colors, etc.) contained in the Student Activity Sheet attractive, 54.8% of students expressed interest to follow the learning activities using worksheets, 77.4% of students stated learning with task performance as contained in the LKS is very beneficial for them, 74.19% of students stated that the other material need to use worksheets, 67.7% of students stated that the problems that exist in part of the exercise in accordance with the material, 80.6% of students stated the questions in LKS are in accordance with everyday life.. Keyword: Student&rsquo;s worksheet, contextual approach, engineering mechanics,&nbsp; Moment and couplin

Linear Response for a Family of Self-Consistent Transfer Operators

We study a system of all-to-all weakly coupled uniformly expanding circle maps in the thermodynamic limit. The state of the system is described by a probability measure and its evolution is given by the action of a nonlinear operator, also called a self-consistent transfer operator. We prove that when the coupling is sufficiently small, the system has a unique stable state that satisfies a linear response formula when varying the coupling strength.Comment: Electronic copy of final peer-reviewed manuscript accepted for publicatio

Conditioning problems for invariant sets of expanding piecewise affine mappings: Application to loss of ergodicity in globally coupled maps

We propose a systematic approach to the construction of invariant union of polytopes (IUP) in expanding piecewise affine mappings. The goal is to characterize ergodic components in these systems. The approach relies on using empirical information embedded in trajectories in order to infer, and then to solve, a so-called conditioning problem for some generating collection of polytopes. A conditioning problem consists of a series of requirements on the polytopes' localisation and on the dynamical transitions between these elements. The core element of the approach is a reformulation of the problem as a set of piecewise linear inequalities for some matrices which encapsulate geometric constraints. In that way, the original topological puzzle is converted into a standard problem in computational geometry. This transformation involves an optimization procedure that ensures that both problems are equivalent, ie. no information is dropped when passing to the analytic formulation. As a proof of concept, the approach is applied to the construction of asymmetric IUP in piecewise expanding globally coupled maps, so that multiple ergodic components result. The resulting mathematical statements explain, complete and extend previous results in the literature, and in particular, they address the dynamics of cluster configurations. Comparison with the numerics reveals that, in all examples, our approach provides sharp existence conditions and accurate fits of the empirical ergodic components

Globally coupled Anosov diffeomorphisms: Statistical properties

We study infinite systems of globally coupled Anosov diffeomorphisms with weak coupling strength. Using transfer operators acting on anisotropic Banach spaces, we prove that the coupled system admits a unique physical invariant state, $h_\varepsilon$. Moreover, we prove exponential convergence to equilibrium for a suitable class of distributions and show that the map $\varepsilon\mapsto h_\varepsilon$ is Lipschitz continuous

Through the Looking Glass: An Autoethnographic View of the Perceptions of Race and Institutional Support in the Tenure Process

This autoethnography study reflects on the experiences of three assistant professors of different races of the tenure process at a large public research university. The study was framed by social cognitive career theory (SCCT), which is often used to describe career interest and career choice in a variety of professional domains, considering cognitive - personal, environmental, and experiential factors (Lent, 2005; Tang, Pan, & Newmeyer, 2008). Despite the commonality of the journey through the tenure process, everyone’s path looks different. Isolation, collegial relationships, and for some, unusually high service commitments and family responsibilities, whether because of gender or race, have provided a back drop for the lives of women in academe and have, in large part, defined their experiences on the tenure track. The results of the reflections of our experiences identify that environmental variables are important to the success of tenure seeking faculty

Studi Perubahan Nilai Tanah Dan Penggunaan Lahan Pada Daerah Rawan Genangan Banjir Rob Di Kecamatan Semarang Utara

North Semarang sub-district has a topography with a slope of 0-2%. The lower area of sea surface will be flooded if the sea surface keep increase with landsubsidence, and also caused by optimallyzation landuse around the coastal area.This research using some secondary data. From the process of merging spotheight with the landsubsidence value every year and then classified by the highest tides, the average of sea level and the lowest tides so obtained the form of tidal inudation area. The fix tidal inudation area model will be overlay with the map of landuse and the land value of North Semarang sub-district. The method used is analysis data based on Geographic Information System with seeing the real condition of the field.The research result shows that the safe area has decrease by 35,5 acres within a period of 7 years. It caused by land use changes such as increase of industry area by 45,3 acres within a period of 5 years. The selling price of land in subcript tidal inudation area such as Tanjung Mas village has increased by Rp. 28.000,00 with AD ground code within a period of 5 years whereas the safe area with AD ground code such as Plombokan village has increased by Rp. 559.000,00 within a period of 5 years