Diophantine conditions and real or complex Brjuno functions

The continued fraction expansion of the real number x=a_0+x_0, a_0\in {\ZZ}, is given by 0\leq x_n<1, x_{n}^{-1}=a_{n+1}+ x_{n+1}, a_{n+1}\in {\NN}, for $n\geq 0.$ The Brjuno function is then $B(x)=\sum_{n=0}^{\infty}x_0x_1... x_{n-1}\ln(x_n^{-1}),$ and the number $x$ satisfies the Brjuno diophantine condition whenever $B(x)$ is bounded. Invariant circles under a complex rotation persist when the map is analytically perturbed, if and only if the rotation number satisfies the Brjuno condition, and the same holds for invariant circles in the semi-standard and standard maps cases. In this lecture, we will review some properties of the Brjuno function, and give some generalisations related to familiar diophantine conditions. The Brjuno function is highly singular and takes value $+\infty$ on a dense set including rationals. We present a regularisation leading to a complex function holomorphic in the upper half plane. Its imaginary part tends to the Brjuno function on the real axis, the real part remaining bounded, and we also indicate its transformation under the modular group.Comment: latex jura.tex, 6 files, 19 pages Proceedings on `Noise, Oscillators and Algebraic Randomness' La Chapelle des Bois, France 1999-04-05 1999-04-10 April 5-10, 1999 [SPhT-T99/116

Linearization of analytic and non--analytic germs of diffeomorphisms of (C,0)({\mathbb C},0)

We study Siegel's center problem on the linearization of germs of diffeomorphisms in one variable. In addition of the classical problems of formal and analytic linearization, we give sufficient conditions for the linearization to belong to some algebras of ultradifferentiable germs closed under composition and derivation, including Gevrey classes. In the analytic case we give a positive answer to a question of J.-C. Yoccoz on the optimality of the estimates obtained by the classical majorant series method. In the ultradifferentiable case we prove that the Brjuno condition is sufficient for the linearization to belong to the same class of the germ. If one allows the linearization to be less regular than the germ one finds new arithmetical conditions, weaker than the Brjuno condition. We briefly discuss the optimality of our results.Comment: AMS-Latex2e, 11 pages, in press Bulletin Societe Mathematique de Franc

Natural boundary for the susceptibility function of generic piecewise expanding unimodal maps

We consider the susceptibility function Psi(z) of a piecewise expanding unimodal interval map f with unique acim mu, a perturbation X, and an observable phi. Combining previous results (deduced from spectral properties of Ruelle transfer operators) with recent work of Breuer-Simon (based on techniques from the spectral theory of Jacobi matrices and a classical paper of Agmon), we show that density of the postcritical orbit (a generic condition) implies that Psi(z) has a strong natural boundary on the unit circle. The Breuer-Simon method provides uncountably many candidates for the outer functions of Psi(z), associated to precritical orbits. If the perturbation X is horizontal, a generic condition (Birkhoff typicality of the postcritical orbit) implies that the nontangential limit of the Psi(z) as z tends to 1 exists and coincides with the derivative of the acim with respect to the map (linear response formula). Applying the Wiener-Wintner theorem, we study the singularity type of nontangential limits as z tends to e^{i\omega}. An additional LIL typicality assumption on the postcritical orbit gives stronger results.Comment: LaTex, 23 pages, to appear ETD

When panic makes you blind: A chaotic route to systemic risk

We present an analytical model to study the role of expectation feedbacks and overlapping portfolios on systemic stability of financial systems. Building on Corsi et al. (2016), we model a set of financial institutions having Value-at-Risk capital requirements and investing in a portfolio of risky assets, whose prices evolve stochastically in time and are endogenously driven by the trading decisions of financial institutions. Assuming that they use adaptive expectations of risk, we show that the evolution of the system is described by a slow-fast random dynamical system, which can be studied analytically in some regimes. The model shows how the risk expectations play a central role in determining the systemic stability of the financial system and how wrong risk expectations may create panic-induced reduction or over-optimistic expansion of balance sheets. Specifically, when investors are myopic in estimating the risk, the fixed point equilibrium of the system breaks into leverage cycles and financial variables display a bifurcation cascade eventually leading to chaos. We discuss the role of financial policy and the effects of some market frictions, as the cost of diversification and financial transaction taxes, in determining the stability of the system in the presence of adaptive expectations of risk

Coupling the Yoccoz-Birkeland population model with price dynamics: Chaotic livestock commodities market cycles

We propose a new model for the time evolution of livestock commodities prices which exhibits endogenous deterministic stochastic behaviour. The model is based on the Yoccoz\u2013Birkeland integral equation, a model first developed for studying the time-evolution of single species with high average fertility, a relatively short mating season and density-dependent reproduction rates. This equation is then coupled with a differential equation describing the price of a livestock commodity driven by the unbalance between its demand and supply. At its birth the cattle population is split into two parts: reproducing females and cattle for butchery. The relative amount of the two is determined by the spot price of the meat. We prove the existence of an attractor (theorem A) and of a non-trivial periodic solution (theorem B) and we investigate numerically the properties of the attractor: the strange attractor existing for the original Yoccoz\u2013Birkeland model is persistent but its chaotic behaviour depends also on the time evolution of the price in an essential way

Metode Pembelajaran Fisika Berdasarkan Teori Multiple Intelegence pada Materi Perpindahan Kalor

Di Indonesia masih banyak kegiatan pembelajaran menggunakan pembelajaran klasikal padahal tidak semua siswa memiliki kecerdasan yang sama. Sehingga guru perlu mengetahui MI (Multiple Intelegence) yang dominan di kelasnya, supaya pembelajarannya bisa disesuaikan. Pada kenyataannya ada siswa yang memiliki MI cenderung mengarah ke kecerdasan naturalis, maka dibutuhkan RPP (Rencana Pelaksanaan Pembelajaran) yang sesuai dengan kecerdasan siswa sehingga penyerapan materi pada siswa dapat maksimal serta merancang strategi pembelajaran yang sesuai dengan kecenderungan kecerdasan siswa dan bagaimana dampak penggunaan strategi pembelajaran tersebut terhadap pemahaman siswa pada materi perpindahan kalor. Penelitian ini menggunakan metode penelitian tindakan kelas (PTK) dimana guru bertindak sebagai peneliti. Penelitian ini terbagi menjadi empat tahap yaitu tahap perencanaan, pelaksanaan, pengamatan, dan refleksi. Pada tahap perencanaan, kecerdasan majemuk siswa dinilai dengan cara memberikan tes kecerdasan majemuk. Hasil tes kemudian dianalisa untuk mengetahui kecenderungan kecerdasan dan gaya belajar setiap siswa. Selanjutnya guru menyusun instrumen penelitian berupa RPP berdasarkan kecenderungan kecerdasan yang dominan dalam kelas, soal evaluasi, dan pedoman obsevasi. Pada tahap pelaksanaan, RPP diterapkan dalam pembelajaran di kelas dan jalannya pembelajaran direkam dalam lembar observasi. Pada tahap refleksi, hasil evaluasi dianalisis untuk mencari nilai rata-rata dan prosentase keberhasilan belajar siswa, sedangkan data pada lembar observasi dianalisis secara deskriptif-kualitatif. Hasil evaluasi menunjukkan bahwa 22 dari 27 siswa atau sebesar 81% siswa mendapatkan nilai sama atau lebih dari 70. Hasil tersebut menunjukkan bahwa strategi pembelajaran berdasarkan kecenderungan kecerdasan yang dominan dalam kelas dapat membantu siswa memahami materi perpindahan kalor

Desain Pembelajaran IPA Terpadu Pada Siswa SMP dengan Topik Pemanasan Global

Dengan kurikulum yang berlaku saat ini, dalam KTSP pada jenjang SMP/MTs menuntut pembelajaran IPA (Fisika, Biologi dan, kimia) secara terintegrasi yang dikenal dengan nama IPA Terpadu. Namun penerapan pembelajaran IPA terpadu di SMP saat ini masih mengalami beberapa kendala seperti guru–guru IPA di SMP yang masih berlatar belakang pendidikan berbeda-beda yaitu Fisika, Biologi dan Kimia, sehingga masih banyak guru yang merasa kesulitan dalam melaksanakan pembelajaran terpadu. Karena itu penelitian ini dibuat untuk melengkapi desain pembelajaran IPA terpadu yang telah ada, yang bertujuan untuk memberi contoh bagi para guru dalam pembuatan rencana pelaksanaan pembelajaran (RPP) yang detail agar mempermudah dalam melaksanakan pembelajaran khususnya pada materi pemanasan global. Jenis penelitian yang dikembangkan adalah penelitian tindakan kelas (PTK). Hasil penelitian menunjukkan bahwa desain pembelajaran IPA terpadu dengan materi pemanasan global ini dapat membantu guru dalam memberikan pembelajaran yang lebih baik, serta dapat membantu siswa untuk memahami materi IPA secara menyeluruh. Desain pembelajaran IPA terpadu dengan materi pemanasan global ini dapat meningkatkan keterampilan kerja ilmiah siswa dan hasil belajar siswa secara efektif. Hal tersebut dapat dilihat dari pencapaian hasil belajar siswa yaitu sebanyak 94% siswa dapat memperoleh nilai tes 70. Dengan pembelajaran yang menyenangkan dan kreatif mampu menciptakan pembaharuan pendidikan ke arah yang lebih baik, meski hanya pembaharuan tingkat kelas atau sekolah

Analysis of Bank Leverage via Dynamical Systems and Deep Neural Networks

We consider a model of a simple financial system consisting of a leveraged investor that invests in a risky asset and manages risk by using value-at-risk (VaR). The VaR is estimated by using past data via an adaptive expectation scheme. We show that the leverage dynamics can be described by a dynamical system of slow-fast type associated with a unimodal map on [0,1] with an addi-tive heteroscedastic noise whose variance is related to the portfolio rebalancing frequency to target leverage. In absence of noise the model is purely deterministic and the parameter space splits into two regions: (i) a region with a globally attracting fixed point or a 2-cycle; (ii) a dynamical core region, where the map could exhibit chaotic behavior. Whenever the model is randomly perturbed, we prove the existence of a unique stationary density with bounded variation, the stochastic stability of the process, and the almost certain existence and continuity of the Lyapunov exponent for the stationary measure. We then use deep neural networks to estimate map parameters from a short time series. Using this method, we estimate the model in a large dataset of US commercial banks over the period 2001--2014. We find that the parameters of a substantial fraction of banks lie in the dynamical core, and their leverage time series are consistent with a chaotic behavior. We also present evidence that the time series of the leverage of large banks tend to exhibit chaoticity more frequently than those of small banks

Metode Pembelajaran Kooperatif Tipe Numbered Snowball Throwing Pada Materi Gelombang Transversal dan Gelombang Longitudinal

Penelitian ini diharapkan dapat menjadi referensi guru dalam membuat RPP dengan model Cooperative Learning sehingga dapat digunakan sebagai alat evaluasi dalam meningkatkan efektifitas dan efisiensi belajar yang akan meningkatkan motivasi belajar siswa, serta keterlibatan siswa dalam pembelajaran yang berpengaruh terhadap aktivitas dan hasil belajar siswa. Penelitian ini merupakan penelitian tindakan kelas (PTK) dengan subjek penelitian kelas VIII yang berjumlah 20 orang. Intrumen penelitian yang digunakan adalah lembar observasi siswa, lembar quisioner dan tes tertulis, yang kemudian dianalisis. Hasil penelitian menunjukkan 81% siswa aktif dalam kegiatan diskusi dan 85% siswa sudah mencapai tingkat ketuntasan belajar. Dengan demikian model pembelajaran kooperatif tipe Numbered Snowball Throwing dapat diimplementasikan sebagai strategi pembelajaran dikelas