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高频地波雷达生成海洋表面矢量流图
周浩;文必洋
【期刊名称】《海洋与湖沼》
【年(卷),期】2002(033)001
【摘要】武汉大学研制的双站高频地波雷达系统OSMAR2000利用测得的两幅单站径向海流图生成矢量海流图.经典矢量流图生成方法不能直接应用到
OSMAR2000系统中.本文提出一种先在极坐标系下用自然三次样条函数将径向流插值到公共网格上然后直接进行矢量合成的矢量海流图生成方法.OSMAR2000在东海的表面矢量流实测结果与作对比验证的传统海流计测量结果十分吻合.对比数据表明,该方法是可行的,且优于先进行径向流线性插值后矢量合成的矢量流图生成方法.这也是国内首次利用高频地波雷达实现海洋表面矢量流的实时监测.
【总页数】7页(P1-7)
【作者】周浩;文必洋
【作者单位】武汉大学电子信息学院,武汉,430072;武汉大学电子信息学院,武汉,430072
【正文语种】中文
【中图分类】TN958.93
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子杰;文必洋;石振华;田建生;侯杰昌;柯亨玉;高火涛
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4.一种海洋表面矢量流图生成算法的研究 [J], 吴世才;刘海文;文必洋
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江西省宜春市2024高三冲刺（高考物理）苏教版测试(培优卷)完整试卷一、单项选择题（本题包含8小题，每小题4分，共32分。
在每小题给出的四个选项中，只有一项是符合题目要求的）(共8题)第(1)题如图所示，直线MN是一匀强磁场的边界，三个相同的带正电粒子分别沿图示1、2、3三个方向以相同的速率从O点射入磁场，沿箭头1、3两个方向的粒子分别经t1、t3时间均从p点离开磁场，沿箭头2方向(垂直于MN)的粒子经t2时间从q点离开磁场，p是Oq的中点，则t1、t2、t3之比为A．1：2：3B．2：3：4C．1：3：5D．2：3：10第(2)题科学家在宜居带“金发姑娘区”发现一颗类地行星TRAPPIST—le，若将该行星视为半径为R、质量分布均匀的球体，一物体从离行星表面（）处由静止释放，经时间，该物体落回到行星表面，引力常量为G，则该行星的平均密度约为（）A．B．C．D．第(3)题下列关于物理学研究方法的叙述正确的是（ ）A．在探究加速度与力、质量的关系时，采用了“理想实验法”B．用速度—时间图像推导匀变速直线运动的位移公式时，采用了“理想模型法”C．伽利略对自由落体运动规律的研究，采用了类比的思想方法D．在无需考虑物体的大小和形状时，用质点来代替物体的方法是“理想模型法”第(4)题图甲描绘了在时刻，由两波源S1、S2在均匀介质中形成的波阵面平面图（实线和虚线分别代表波峰和波谷）。
已知这两个波源的振动频率相等，振幅相同。
P和Q是该平面上的两个固定点，S1、S2间的距离是0.2m。
图乙是P点的振动图像（位移-时间关系）。
下列说法中正确的是（ ）A． S1S2的连线上共有4个干涉减弱点B． S1S2连线的延长线上没有干涉增强点C．t=0.2s时，Q点的位移是A D．波源 S1发出的机械波波速为0.2m/s第(5)题中国空间站轨道高度为400~450千米，地球半径约为6370千米。
当航天员出舱在空间站舱外作业时，其所受地球的引力大约是他在地面所受地球引力的（ ）A．0.9倍B．0.25倍C．0.1倍D．0.01倍第(6)题《读卖新闻》报道，日本预计于2023年春夏前后把处理后的福岛核废水排放入海，国际舆论哗然。
singular point数学Singular Point: Exploring the Mathematics Behind ItIntroductionIn the vast field of mathematics, singular points play a pivotal role in various branches such as calculus, differential equations, and complex analysis. These unique points have fascinated mathematicians for centuries, capturing their attention and driving their exploration. In this article, we will delve into the concept of singular points, examining their characteristics, applications, and significance in the realm of mathematics.Defining Singular PointsA singular point can be defined as a point within a mathematical function or equation where certain properties exhibit exceptional behavior, distinct from the surrounding points. It is a point of interest that usually stands out due to its peculiar properties, making it a key component in understanding the overall behavior of a mathematical system.Characteristics of Singular PointsSingular points possess several distinct characteristics that set them apart from regular points in a mathematical function or equation. Firstly, at a singular point, the function or equation may become undefined or exhibit infinitely large values. This erratic behavior can be attributed to the fact that singular points often represent critical points where the derivative of a function becomes zero or undetermined.Furthermore, singular points can be classified into various categories based on their behavior. For instance, isolated singular points are those that are distinct from any nearby points. On the other hand, non-isolated singular points are surrounded by an infinite number of nearby singular points, forming intricate patterns in the mathematical landscape.Applications of Singular PointsSingular points find wide-ranging applications in diverse fields of mathematics and beyond. In physics, for instance, singular points play a crucial role in understanding the behavior of physical phenomena. One such example is the presence of singularities in black holes, where the laws of physics break down. By studying the singular points in these extreme cosmic objects, scientists can gain valuable insights into the nature of space, time, and gravity.Moreover, singular points have significant implications in engineering and optimization. In control theory, for instance, the presence of singular points can help engineers understand the stability and controllability of complex systems. By identifying and analyzing these points, engineers can develop strategies to ensure the optimal performance and efficiency of various technological systems.Significance of Singular Points in MathematicsMathematically, singular points hold immense significance as they often reveal valuable information about the behavior and solutions of equations and functions. By studying the properties of singular points, mathematicians can gain insights into the nature of a mathematical system and its underlying structure.Singular points have a direct impact on the overall solution space of equations. The presence of singular points can lead to the emergence of multiple solutions or bifurcations, where the behavior of a system changes abruptly. Understanding and characterizing these singular points through rigorous mathematical analysis are fundamental in unraveling the complexities of a wide range of mathematical phenomena.ConclusionIn conclusion, singular points are fascinating and fundamental components of mathematics with applications in various fields. These unique points display exceptional behavior, which often leads to important insights and breakthroughs in understanding equations, functions, and complex systems. By exploring the characteristics, applications, and significance of singular points, mathematicians can continue to push the boundaries of mathematical knowledge and uncover new frontiers in this ever-evolving discipline.。
运用比兴的手法写一篇英语作文700字全文共3篇示例，供读者参考篇1The Rhythm of LifeLife has a rhythm, a cadence that ebbs and flows like the tides of the ocean. Some moments crash over us with the force of towering waves, leaving us gasping for air. Others gently lap at the shores of our existence, a soothing melody in the symphony of our days.School is an entire universe unto itself, a galaxy of experiences swirling in a celestial dance. The hallways buzz with the energy of a thousand suns, students streaming through like comets trailing stardust. Each classroom is a new world to explore, the subjects unfolding like ancient maps leading to uncharted territories of knowledge.Teachers are the guides on this odyssey, beacons shining through the nebulae of uncertainty. Their lessons are the constellations by which we navigate, connecting the celestial bodies of information into coherent narratives. With each newconcept, another startling revelation unfurls like a supernova blooming in the inky cosmos.Friendships are forged in this cosmos, bonds as unbreakable as the forces binding galaxies together. Laughter and inside jokes become the lingua franca, creating a tapestry of shared experiences against the backdrop of the infinite. Disagreements may flare like supernovae, but the gravity of understanding always pulls us back into orbit.Extracurricular activities are akin to exploring new star systems, each club and sport casting its own unique light. The basketball court becomes a three-point universe, players arcing across its planes like meteors streaking through the heavens. The stage for the drama club is an entire galaxy condensed into a single plank surface, actors breathing life into cosmic roles.Adolescence itself is a singularity, a turbulent, chaotic core of confusion and metamorphosis. Emotions churn like the birth of a new star system, thoughts and feelings collapsing in on themselves until they find their own equilibrium. Awkwardness is the dark matter of this era, an invisible force exerting its influence on every interaction.Yet for every black hole period of social enormity, there are moments that shine like pulsars, guiding us forward. A kind wordfrom a friend, laughter at an inside joke, the realization of a hard-earned academic accomplishment – these are the luminous events that remind us we are not adrift, but active participants in the unfolding of something greater.High school graduation is the big bang, a spectacular event forever altering the trajectory of our cosmos. We are flung outward, hurtling into the great expanse of what comes next. College, career, trekking into the great unknown – the paths before us are as vast as the universe itself.In those final days before the cataclysm, we bask in the familiar glow of our singular galaxy. We carve our names into lockers like explorers leaving their insignias. We take silly pictures and swap yearbooks, feverishly documenting every last detail before it all changes. Final memories are created and cached like the cosmic microwave background radiation, an ever-present reminder of what once was.Then, it happens. The world around us explodes in a fantastic finale of pomp and circumstance. Diplomas are issued like spacecrafts receiving their new trajectories. Mortarboards are flung skyward, brief twinkling meteors against the vast eternal night. We cheer, cry, and embrace one another, unified in thisinstance by our shared experience before being cast asunder to the cosmic winds.And in the aftermath, we are left drifting amid the stellar streams of our future paths. The warm, steady pull of nostalgia tries its gravity on our hearts, but we keep moving forward – ever outward – propelled by the lingering echoes of that mythic resonance born from the big bang of high school graduation.The rhythm of life goes on, and weislanded celestial travelers continue orbiting through its vast, wondrous movements. Worlds await, blackholes loom, and supernovae yet undreamt of scattered like diamonds waiting to be unearthed along this infinite cosmic dance.篇2The Path of Knowledge: A Journey Through MetaphorAs students, our academic journey is paved with challenges, triumphs, and a constant thirst for understanding. We embark on a path that winds through realms of knowledge, each step revealing new vistas and perspectives. In this essay, I invite you to explore the realm of metaphor, where the abstract becomes tangible and the familiar takes on new depths.Imagine the pursuit of knowledge as a grand adventure, akin to a daring expedition into the heart of an unexplored wilderness. The classroom is our base camp, a sanctuary where we gather supplies and fortify ourselves with the wisdom of our mentors. The coursework, once daunting, becomes our trail map, guiding us through the dense foliage of concepts and theories.As we venture forth, we encounter the first obstacles – the fundamental principles that must be mastered before progressing further. These are the rugged cliffs we must scale, testing our perseverance and determination. With each foothold we gain, the vista of understanding expands, revealing new horizons to conquer.The journey becomes arduous as we delve deeper into the intricate tapestry of knowledge. Complex theories and abstract ideas loom like towering mountains, their peaks shrouded in mist. Yet, undeterred, we press on, wielding the tools of critical thinking and analysis like alpinist's picks, chipping away at the rock face until we uncover the hidden truths within.Amidst this odyssey, we chance upon metaphorical oases –moments of clarity where the intricate threads of understanding weave together into a coherent tapestry. These are the eureka moments, the sudden bursts of illumination that electrify ourminds and invigorate our spirits. Like a desert traveler stumbling upon a verdant spring, we drink deeply from these wellsprings of insight, quenching our thirst for comprehension.Yet, the path is not without its pitfalls and detours. Misconceptions and fallacies lurk like treacherous quicksand, threatening to ensnare the unwary. We must tread carefully, relying on the guidance of our mentors and the wisdom of those who have traversed these trails before us. For every dead end we encounter, we backtrack and seek alternative routes, ever mindful of the ultimate destination – the summit of understanding.As we ascend higher, the air grows rarefied, and the challenges intensify. Advanced theories and complex concepts loom like imposing glaciers, their icy facades daring us to conquer them. Undaunted, we don our metaphorical crampons and ice axes, chipping away at the frozen edifices until they yield their secrets.Throughout this arduous journey, we are accompanied by fellow sojourners – our peers and classmates, each with their unique perspectives and strengths. Together, we form a cohesive team, supporting and encouraging one another through the most demanding passages. For just as no mountain can beclimbed alone, the path of knowledge is best traversed in the company of kindred spirits.And what of the summit, that elusive pinnacle we strive towards? It is the embodiment of mastery, the zenith of comprehension where all the disparate strands of knowledge converge into a harmonious whole. From this lofty vantage point, the world unfolds before us, and we bask in the glory of understanding, our toils rewarded with a panoramic view of the intellectual landscape.Yet, even as we revel in this hard-won triumph, we must not forget that the journey is never truly over. For beyond the summit, new horizons beckon, unexplored territories waiting to be charted and conquered. The thirst for knowledge is unquenchable, and the path ever-winding, inviting us to embark on new adventures, to scale greater heights, and to push the boundaries of our understanding ever further.So, let us embrace the metaphorical journey that lies ahead, armed with the tools of curiosity, perseverance, and a boundless appetite for learning. For in the realm of knowledge, the trail never ends, and the rewards are as vast as the expanse of human understanding itself.篇3The Rhythms of LifeLife dances to the beat of an ever-changing rhythm, a cosmic symphony that ebbs and flows with the tides of time. Each day unfolds like a new musical score, filled with melodies of joy and dissonance, crescendos of triumph and diminuendos of sorrow. We are the instruments in this grand orchestra, our strings vibrating with every experience, every emotion.School is a veritable concerto, a cacophony of sounds that blend into a harmonious whole. The hallways buzz like a beehive, teeming with students scurrying from one class to another, their laughter and chatter echoing off the lockers like notes bouncing off the walls of a concert hall. The classrooms are sanctuaries of learning, where knowledge flows like a river, nourishing our minds as teachers conduct us through the intricacies of each subject.Friendships are the chords that bind us together, resonating with the power of a well-tuned ensemble. Our bonds are like the strings of a violin, each strand woven together to create a rich, melodious tapestry of camaraderie. We share our victories andstruggles, our joys and sorrows, harmonizing our voices into a symphony of understanding and support.Exams loom like thunderclouds on the horizon, their ominous presence casting shadows of anxiety over ouronce-sunny days. Yet, we face them like warriors wielding our pens as swords, our knowledge as shields, and our determination as armor. Each test is a battle, a crescendo of tension that builds until the final note is struck, and we emerge victorious or humbled, ready to learn from the experience.Extracurricular activities are the embellishments that add color and vibrancy to our lives, like the flourishes of a virtuoso performance. Sports arenas become our stages, where we pirouette and leap like dancers, our bodies moving in perfect synchronization with the rhythm of the game. Clubs and organizations allow us to explore our passions, each pursuit a new movement in the symphony of self-discovery.Amidst the harmonies and dissonances of our academic journeys, we find solace in the embrace of nature, a sanctuary where the melodies of the earth resonate within our souls. The rustling of leaves whispers like a gentle lullaby, the babbling brook sings a soothing refrain, and the chirping of birds eruptsinto joyous arias, reminding us of the beauty that surrounds us, even in our most challenging moments.As we navigate the rhythms of life, we learn to improvise, to adapt our tempos and crescendos to the ever-changing score. Each experience shapes us, like a composer revising a masterpiece, refining and polishing our characters until we become virtuosos in the symphony of living.So let us embrace the harmonies and dissonances that life brings, for they are the notes that compose the grand opus of our existence. Let us dance with the rhythms, sway with the melodies, and crescendo with the triumphs, for in this cosmic concert, we are both the musicians and the music, the conductors and the conducted, weaving a tapestry of melodies that will resonate long after the final curtain falls.。
激光脉冲的空间分布曲线
近年来，激光技术在各个领域得到了广泛应用。
激光脉冲是激光技术中重要的一种形式，其空间分布曲线对于激光技术的研究和应用非常关键。
激光脉冲的空间分布曲线是指激光在空间中的能量分布情况。
一般来说，激光脉冲的空间分布曲线可以用高斯函数来描述。
高斯函数是一种钟形曲线，其特点是在中心点处能量最高，随着距离的增加能量逐渐减弱。
激光脉冲的空间分布曲线对于激光技术的应用非常重要。
例如，在激光加工中，激光脉冲的空间分布曲线会影响加工效果和质量。
在激光医疗中，激光脉冲的空间分布曲线会影响治疗效果和安全性。
因此，研究激光脉冲的空间分布曲线对于提高激光技术的应用效果和发展具有重要意义。
未来，随着激光技术的进一步发展，激光脉冲的空间分布曲线将会得到更加深入和精准的研究。
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超几何分布-北师大版选修2-3教案一、知识背景超几何分布（hypergeometric distribution）是离散随机变量的一种，描述从有限个物品中抽出固定数量的物品，其中有指定种类的物品数量的概率分布。
它在统计学中有广泛的应用，例如在品质控制中，抽检商品的次数以及在实验设计中选定目标人群的样本数量。
二、教学目标•理解超几何分布的概念、特点、条件和性质；•掌握超几何分布的基本计算方法和公式应用；•能够解答超几何分布的实际问题，如品质控制的样本检测等；•培养学生的逻辑思维能力和应用数学知识解决实际问题的能力。
三、教学内容1. 超几何分布的概念和特点超几何分布指从总数为N（不放回）的物品中，其中有m个种类的物品共k 个，随机抽取n个物品，其中有m0个种类的物品的个数X的分布律，用H(n, m, k)表示。
因此，超几何分布的性质为：•该分布实验不满足独立性；•分布变量的取值只能是非负整数；•总体中有k个成功物品，n个样本，成为超几何分布的参数。
2. 超几何分布的计算方法和公式超几何分布的概率函数公式为：其中，C表示组合数。
3. 超几何分布的应用品质控制中，经常需要检验样本是否达到质量标准。
对于超过某个标准值的样本，则认为该样本不符合质量要求。
超几何分布在此类问题中应用广泛。
四、教学方法•讲授法：通过讲解概念、公式和解题方法，让学生掌握超几何分布的知识；•举例法：通过实际问题，让学生在操作中掌握超几何分布的应用方法；•配套练习：在课堂上或课后布置超几何分布的练习题，检验学生掌握程度。
五、教学内容安排第一课时•教学内容：超几何分布的概念和特点；•教学重点：超几何分布的性质；•教学难点：掌握超几何分布的条件。
第二课时•教学内容：超几何分布的计算方法和公式；•教学重点：掌握超几何分布的公式和应用；•教学难点：掌握超几何分布的计算方法。
第三课时•教学内容：超几何分布的应用；•教学重点：学习超几何分布在品质控制中的应用；•教学难点：掌握超几何分布在实际问题中的运用。
中华人民共和国行业标准PSL 26-92水利水电工程技术术语标准Standard of Technical Terms on Hydroengineering1992-06-02发布1992-12-01实施中华人民共和国行业标准水利水电工程技术术语标准SL 26—92主编单位：武汉水利电力大学批准部门：中华人民共和国水利部能源部中华人民共和国水利部发布能源部关于颁发《水利水电工程技术术语标准》SL26—92的通知水科教[1992] 19号为促进水利水电科学技术的发展，统一水利水电工程技术术语，推动国内和国际技术交流，由水利水电规划设计总院委托武汉水利电力大学主编的t水利水电工程技术术语标准》，经审查现批准为水利行业标准，其名称与编号为：《水利水电工程技术术语标准》SL26—92,该标准从1992年12月1日起实施。
在实施中如有问题，请函告水利水电规划设计总院或武汉水利电力大学。
本标准由水利电力出版社出版发行。
1992年6月2日目次I 水利水电工程勘测SURVEY AND INVESTIGATIONFOR HYDROENGINEERING工程测量1 工程测量基础2 测量仪器3 工程测量4 摄影测量5 遥感技术6 地图编绘与制印工程地质7 地质基础，8 水文地质9 工程地质水文测验10 一般术语11 水文调查12 水文测站和站网13 水文观测14 近代水文测验技术15 水文资料整编岩土力学16 岩土的物理性质17 岩土的变形性质18 岩土的强度特性19 岩土的渗透性质20 岩土中应力及岩土体变形计算21 岩土体稳定分析及承载力22 岩土现场测试Ⅱ水利水电工程规划PLANNING OF HYDROENGINEERING水文计算及水文预报1 河流及流域特征2 水文分析计算3 水文预报水资源开发利用4 水资源开发利用5 地下水资源开发利用航道整治规划6 航道整治规划防洪规划7 防洪规划水能利用规划8 水能利用规划灌溉排水规划9 土壤一作物一大气系统10 灌溉用水量的分析和计算11 灌溉水源12 灌水技术13 灌溉系统14 治涝排渍15 圩垸区及感潮河段治理16 排水系统17 灌溉排水试验及管理水土保持规划18 水土保持规划河流泥沙及河道整治规划19 河流泥沙运动力学20 河道形态与河床演变2l 河道整治22 水库泥沙23 河流模拟环境影响与库区移民24 环境影响与库区移民经济评价25 经济评价Ⅲ水工建筑物HYDROSTRUCTURES水力学1 水静力学2 水运动学及水动力学3 层流与紊流4 水流阻力和能头损失5 管流6 明槽流(明渠流)7 堰流及孔口出流8 建筑物下游消能9 波浪10 渗流水工建筑物11 水工建筑物的类别及荷载12 坝13 水闸14 溢洪道15 水工隧洞16 涵洞与涵管17 取水建筑物18 河道整治建筑物19 渠系建筑物20 通航、过木、过鱼建筑物21 地基处理水电站22 水电站23 引水系统及尾水系统建筑物水泵站24 抽水装置25 泵站Ⅳ水力机械与电气设备HYDRAULIC MACHINERY ANDELECTRIC EQUIPMENT水力机械1 水轮发电机组2 水泵电动机机组3 水力机组调节系统4 水力机组辅助系统5 水力机组测试6 水力机组的安装和试运行水工金属结构及安装7 钢结构8 闸门，阀门9 钢管、拦污栅及清理设备10 启闭机及起重机11 钢桥12 升船机及船厢13 埋件、连接件14 金属结构安装电力工程一次部分15 电力系统16 电力系统运行17 电力系统计算18 主要电气设备19 主接线及配电装置20 过电压21 厂用电、近区供电与施工用电电力工程二次部分22 励磁系统23 自动化及远动化24 继电保护25 控制与信号26 直流系统、二次设备及器具27 通信V 水利水电工程施工CONSTRUCTION OF HYDRAULICENGINEEBING施工组织l 施工组织施工导流2 施工导流土石方工程3 土石方工程混凝土工程4 混凝土工程施工工程设施5 施工工程设施概算、预算、决算6 概算、预算、决算施工管理7 施工管理中文索引I 水利水电工程勘测Ⅱ水利水电工程规划Ⅲ水工建筑物Ⅳ水力机械与电气设备V 水利水电工程施工编制说明附加说明I 水利水电工程勘测SURVEY AND IHVESTIGATION FORHYDROENGINEEBING工程测量Engineering Survey1 工程测量基础Fundomentals of engineering survey1.1 坐标与高程Coordinate and elevation1.1.1 大地水准面geoid与平均海(水)面(无波浪、潮汐、水流和大气压变化引起的扰动)重合并延伸到大陆和岛屿内部所形成的一个封闭的水准面。
Chaotic singular mapsM.G.Cosenza 1and O.Alvarez-LLamoza 2,11Centro de F´ısica Fundamental,Universidad de Los Andes,M´e rida,Venezuela 2Departamento de F´ısica,FACYT,Universidad de Carabobo,Valencia,VenezuelaWe consider a family of singular maps as an example of a simple model of dynamical systemsexhibiting the property of robust chaos on a well defined range of parameters.Critical boundaries separating the region of robust chaos from the region where stable fixed points exist are calculated on the parameter space of the system.It is shown that the transitions to robust chaos in these systems occur either through the routes of type-I or type-III intermittency and the critical boundaries for each type of transition have been determined on the phase diagram of the system.The simplicity of these singular maps and the robustness of their chaotic dynamics make them useful ingredients in the construction of models and in applications that require reliable operation under chaos.Keywords:Singular maps,Robust chaos,Intermittency.1.INTRODUCTION Many practical uses of the phenomenon of chaos have been proposed in recent years,as for instance,in commu-nications [1,2],in enhancing mixing in chemical processes [3],in avoiding electromagnetic interferences [4],in cryp-tography [5],in stabilizing plasma fusion [6],etc.In such applications it is necessary to obtain reliable operation of chaotic systems.It is known that most chaotic attractors of smooth sys-tems are embedded with a dense set of periodic windows for any range of parameter values.Therefore in practical systems functioning in chaotic mode,a slight fluctuation of a parameter may drive the system out of chaos.On the other hand,it has been shown that some dynamical systems can exhibit robust chaos [7,8,9].A chaotic at-tractor is said to be robust if,for its parameter values,there exist a neighborhood in the parameter space with absence of periodic windows and the chaotic attractor is unique [7].Robustness is an important property in ap-plications that require reliable operation under chaos in the sense that the chaotic behavior cannot be destroyed by arbitrarily small perturbations of the system parame-ters.For example,robust chaos has efficiently been used in communications schemes [10].In this article we study a family of singular maps as an example of a simple model of dynamical systems that shows robust chaos on a finite interval of their parameter values.In Section 2we introduce this family of maps and investigate their dynamical properties,both analytically and numerically.It is found that the transitions to robust chaos in these systems occur either through the routes of type-I or type-III intermittency [11].The region where robust chaos takes place is characterized on the space of parameters of the maps.Conclusions are presented in Section 3.2.SINGULAR MAPSAs a simple model of a dynamical system displaying robust chaos,we consider the following family of singularmapsx n +1=f (x n )=b −|x n |z ,(1)where n ∈Z ,|z |<1,and b is a real parameter.The exponent z describes the order of the singularity at the origin that separates two piecewise smooth branches of the map Eq.(1).These maps are unbounded,that is,x n ∈(−∞,∞).The Schwarzian derivative of the family of maps Eq.(1)is always positive,i.e.,Sf =ff −32ff 2=1−z 22x >0.(2)for |z |<1.Thus maps defined by Eq.(1)do not belong to the standard universality classes of unimodal maps and do not satisfy Singers theorem [12].As a consequence,these singular maps do not exhibit a sequence of period-doubling bifurcations.Instead,the condition Sf >0leads to the occurrence of an inverse period-doubling bi-furcation,where a stable fixed point on one branch of the singular map losses its stability at some critical value of the parameter b to yield robust chaos.It should be noted that robust chaos has also been discovered in smooth,continuous one-dimensional maps [13].Figure 1shows the bifurcation diagrams of the iterates of map Eq.(1)as a function of the parameter b for two different values of the singularity exponent z .Figure 1reveals robust chaos,i.e.,the absence of windows of stable periodic orbits and coexisting attractors,on a well defined interval of the parameter b for each value of z .The transition to chaos at the boundaries of the robust chaotic interval occurs by intermittency.Intermittent chaos is characterized by the display of long sequences of periodiclike behavior,called the laminar phases,inter-rupted by comparatively short chaotic bursts.The phe-nomenon has been extensively studied since the original work of Pomeau and Manneville [11]classifying type-I,-II,and -III instabilities when the Floquet multipliers of the local Poincar´e map associated to the system crosses the unit circle.Type-I intermittency occurs by a tangent bifurcation when the Floquets multiplier for the Poincar´e map crosses the circle of unitary norm in the complexa r X i v :0805.0452v 2 [n l i n .C D ] 20 M a y 2008FIG.1:Bifurcation diagrams of the iterates of the map Eq.(1)as a function of the parameter b for two values of the orderof the singularity z,showing robust chaos.Type-I or type-IIIintermittencies appear at the boundaries of the robust chaosintervals.(a)z=−0.5;(b)z=0.5.plane through+1;type-II intermittency is due to a Hopfsbifurcation which appears as two complex eigenvalues ofthe Floquets matrix cross the unitary circle offthe realaxis;and type-III intermittency is associated to an in-verse period doubling bifurcation whose Floquets multi-plier is−1.Two stablefixed points satisfying f(x∗)=x∗and|f (x∗)|<1exist for each value of z:x∗−<0and x∗+>0,both are seen in Figure1.For z∈(−1,0),thefixed pointx∗−becomes unstable at the parameter valueb−(z)=|z|z1−z−|z|11−z,(3)through an inverse period doubling bifurcation that givesrise to chaos via type-III intermittency,while thefixedpoint x∗+originates from a tangent bifurcation at thevalueb+(z)=|z|z1−z+|z|11−z,(4)and the transition to chaos at this value of b takes placethrough type-I intermittency.On the other hand,forFIG.2:Some unstable periodic orbits of the singular mapwith z=−0.25,indicated by dotted lines,as a function ofb.The stablefixed points x∗−and x∗+are plotted with solidlines.At the parameter value b−=0.9896,thefixed pointx∗−becomes unstable through an inverse period-doubling bi-furcation,giving raise to the unstablefixed point x−.At thevalue b+=1.6494,a tangent bifurcation takes place and thepair of points x∗+(stable)and x+(unstable)are born.Theperiod-2unstable orbit x1and x2,satisfying f(x1)=f(x2),are shown.z∈(0,1)the behavior of thefixed points is inter-changed:x∗−experiences a tangent bifurcation at theparameter value b−(z)and a type-I intermittent tran-sition to chaos occurs;while thefixed point x∗+under-goes an inverse period-doubling bifurcation at the valueb+(z),setting the scenario for a type-III intermittenttransition to chaos.There exist several unstable period-m orbits{x1,x2,...,x m}satisfying f(m)(x j)=x j and|ddxf(m)(x j)|=mj=1|f (x j)|<1in the chaotic intervalb∈[b+(z),b−(z)].Figure2shows some unstable periodicorbits of the singular map with z=−0.25as a functionof the parameter b.Figure3shows the critical boundaries b−(z)and b+(z)for the transition to chaos.These boundaries separatethe region on the parameter plane(b,z)where robustchaos takes place from the region where stablefixedpoints of the maps Eq.(1)exist.The transition tochaos via type-I intermittency takes place at the pa-rameter boundaries b I(z)=b+(z)for z∈(−1,0),andb I(z)=b−(z)for z∈(0,1).On the other hand,the tran-sition to chaos via type-III intermittency occurs at thecritical parameter values b III(z)=b−(z)for z∈(−1,0),and b III(z)=b+(z)for z∈(0,1).The boundaries b I(z)and b III(z)on the space of parameters(b,z)are indi-cated in Figure3.The width of the interval for robustchaos on the parameter b for a given|z|<1is∆b(z)=b+(z)−b−(z)=2|z|11−z.(5)Figure4shows the Lyapunov exponentλas a functionof the parameter b for the family of maps Eq.(1),forFIG.3:Critical boundaries b −(z )and b +(z )of the robust chaos region for the singular maps on the space of parameters (b,z ).The thick,dark line indicates the boundary b III (z )for the transition to chaos via type-III intermittency.The thin,light line corresponds to the boundary b I (z )for the onset of type-I intermittency.two values of z ,calculated asλ=1T Tn =1log |f (x n )|,(6)with T =5×104iterates after discarding 5000tran-sients for each parameter value.The boundaries b −(z )and b +(z )correspond to the values λ=0.The Lyapunov exponent is positive on the robust chaos interval ∆b (z ).The transition to chaosthrough type-I intermittency is smooth,as seen in Figure 4.In contrast,the transition to chaos via type III intermittency is manifested by a dis-continuity of the derivative of the Lyapunov exponent at the parameter values corresponding to the critical bound-ary b III (z ).This discontinuity is due to the sudden loss of stability of the fixed point associated to the inverse period doubling bifurcation that occurs at the boundary b III (z ).The Lyapunov exponent can be regarded as an order parameter that characterizes the transition to chaos via type-I or type-III intermittency.This transition can be very abrupt in the case of type-III intermittency,as seen in Figure 4.3.CONCLUSIONSWe have introduced a family of singular maps as an example of a simple model of dynamical systems exhibit-ing robust chaos on a well defined range of parameters.The behavior of these maps has been characterized as a phase diagram in the space of their parameters,show-ing a region where robust chaos takes place and regionswhere stable fixed points occur.We have shown that the transitions to robust chaos in these systems occur either through the routes of type-I or type-III intermit-tency and have calculated the critical boundaries for each type of transition on the phase diagram of the systems.The simplicity of these singular maps and the robustness of their chaotic dynamics make them useful ingredients in the construction of models and in applications that require the property of chaos.FIG.4:Lyapunov exponent λas a function of the parameter b for two values of z ,calculated over 5×104iterations after neglecting 5×103iterates representing transient behavior for each value of b .(a)z =−0.5;(b)z =0.5.AcknowledgmentsThis work was supported by Consejo de Desarrollo Cient´ıfico,Human´ıstico y Tecnol´o gico of the Universi-dad de Los Andes,M´e rida,under grant No.C-1396-06-05-B and by FONACIT,Venezuela,under grant No.F-2002000426.[1]S.Hayes,C.Grebogi,E.Otto,Phys.Rev.Lett.70:3031-3034(1993).[2]A.Argyris,D.Syvridis,rger,V.Annovazzi-Lodi,P.Colet,I.Fischer,J.Garc´ıa-Ojalvo,C.R.Mirasso;L.Pesquera,K.A.Shore,Nature438:343-346(2005). 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