探索混沌理论：微分方程与动力系统入门

Chaos

需积分: 50 107 浏览量更新于2023-05-22 2 收藏 3.2MB PDF 举报

身份认证购VIP最低享 7 折!

领优惠券(最高得80元）

《微分方程、动力系统与混沌入门》是PURE AND APPLIED MATHEMATICS系列的第60卷，由 Morris W. Hirsch、Stephen Smale 和 Robert L. Devaney 联合编著，他们分别来自加州大学伯克利分校和波士顿大学。这本书旨在深入探讨微分方程在动力系统理论中的应用，特别是对混沌现象的介绍。微分方程是数学中的基础工具，它们描述的是随着时间连续变化的系统，广泛应用于物理学、工程学、经济学等领域。在动力系统研究中，微分方程被用来刻画系统的动态行为，包括稳定的周期性模式（如简单振荡）、复杂的周期和无序运动。通过分析这些方程，学者们可以理解各种自然现象中的复杂行为，如天气模式、生物进化等。本书特别关注的是混沌理论，这是动力系统的一个分支，它揭示了某些系统如何表现出极其敏感于初始条件的性质。这种现象被称为“奇异性”，意味着即使极小的初始误差也可能导致长期预测的极大不确定性。混沌系统并非随机，而是有确定性的，但其行为看起来似乎不可预测，这为科学家和工程师提出了一系列挑战和机遇，如气候模型的精确度问题和密码学中的不可逆加密。书中可能会涉及以下关键概念： 1. **吸引子**：在混沌系统中，一个吸引子是系统长期行为的稳定核心，即使初始条件改变，所有轨道最终都会趋近这个吸引子。 2. **Lyapunov指数**：衡量混沌系统中两个初始条件的分离速度，正值表示系统是混沌的。 3. **分岔理论**：描述系统参数变化时稳定性或周期性结构的变化，是理解混沌起源的重要手段。 4. **迭代函数系统**：一种构造混沌集的方法，由有限个简单映射重复应用组成。 5. **混沌的数学证明与示例**：书中可能会提供著名的混沌模型，如洛伦兹 attractor 或双曲映射，来展示混沌行为的直观特征。 6. **混沌的物理与工程应用**：包括在天气预报、流体力学、神经网络以及工程技术中的混沌现象。 7. **控制与稳定性的挑战**：面对混沌系统的不确定性和不可预测性，设计有效的控制策略是一项具有挑战性的任务。《微分方程、动力系统与混沌入门》不仅是一本理论著作，还可能包含丰富的例子、练习和应用案例，以帮助读者深化理解和实际操作能力。通过阅读这本书，读者将获得对动力系统中复杂行为背后数学原理的深刻理解，并认识到混沌这一概念在现代科学和工程中的重要地位。

资源详情

资源推荐

First-Order Equations

The purpose of this chapter is to develop some elementary yet important

examples of ﬁrst-order differential equations. These examples illustrate some

of the basic ideas in the theory of ordinary differential equations in the simplest

possible setting.

We anticipate that the ﬁrst few examples in this chapter will be familiar to

readers who have taken an introductory course in differential equations. Later

examples, such as the logistic model with harvesting, are included to give the

reader a taste of certain topics (bifurcations, periodic solutions, and Poincaré

maps) that we will return to often throughout this book. In later chapters, our

treatment of these topics will be much more systematic.

1.1 The Simplest Example

The differential equation familiar to all calculus students

= ax

is the simplest differential equation. It is also one of the most important.

First, what does it mean? Here x = x(t) is an unknown real-valued function

of a real variable t and dx/dt is its derivative (we will also use x



or x



(t)

for the derivative). Also, a is a parameter; for each value of a we have a

2 Chapter 1 First-Order Equations

different differential equation. The equation tells us that for every value of t

the relationship



(t) = ax(t)

is true.

The solutions of this equation are obtained from calculus: If k is any real

number, then the function x(t ) = ke

is a solution since



(t) = ake

= ax(t ).

Moreover, there are no other solutions. To see this, let u(t ) be any solution and

compute the derivative of u(t )e

−at



u(t )e

−at



= u



(t)e

−at

+ u(t)(−ae

−at

)

= au(t)e

−at

− au(t)e

−at

= 0.

Therefore u(t )e

−at

is a constant k,sou(t ) = ke

. This proves our assertion.

We have therefore found all possible solutions of this differential equation. We

call the collection of all solutions of a differential equation the general solution

of the equation.

The constant k appearing in this solution is completely determined if the

value u

of a solution at a single point t

is speciﬁed. Suppose that a function

x(t ) satisfying the differential equation is also required to satisfy x(t

) = u

Then we must have ke

= u

, so that k = u

−at

. Thus we have determined

k, and this equation therefore has a unique solution satisfying the speciﬁed

initial condition x(t

) = u

. For simplicity, we often take t

= 0; then k = u

There is no loss of generality in taking t

= 0, for if u(t) is a solution with

u(0) = u

, then the function v(t ) = u(t − t

) is a solution with v(t

) = u

It is common to restate this in the form of an initial value problem:



= ax, x(0) = u

A solution x(t) of an initial value problem must not only solve the differential

equation, but it must also take on the prescribed initial value u

at t = 0.

Note that there is a special solution of this differential equation when k = 0.

This is the constant solution x(t) ≡ 0. A constant solution such as this is called

an equilibrium solution or equilibrium point for the equation. Equilibria are

often among the most important solutions of differential equations.

The constant a in the equation x



= ax can be considered a parameter.

If a changes, the equation changes and so do the solutions. Can we describe

4 Chapter 1 First-Order Equations

by a solid dot), while any other solution moves up or down the x-axis, as

indicated by the arrows in Figure 1.1.

The equation x



= ax is stable in a certain sense if a = 0. More precisely,

if a is replaced by another constant b whose sign is the same as a, then the

qualitative behavior of the solutions does not change. But if a = 0, the slightest

change in a leads to a radical change in the behavior of solutions. We therefore

say that we have a bifurcation at a = 0 in the one-parameter family of equations



= ax.

1.2 The Logistic Population Model

The differential equation x



= ax above can be considered a simplistic model

of population growth when a>0. The quantity x(t ) measures the population

of some species at time t. The assumption that leads to the differential equa-

tion is that the rate of growth of the population (namely, dx/dt ) is directly

proportional to the size of the population. Of course, this naive assumption

omits many circumstances that govern actual population growth, including,

for example, the fact that actual populations cannot increase with bound.

To take this restriction into account, we can make the following further

assumptions about the population model:

1. If the population is small, the growth rate is nearly directly proportional

to the size of the population;

2. but if the population grows too large, the growth rate becomes negative.

One differential equation that satisﬁes these assumptions is the logistic

population growth model. This differential equation is



= ax



1 −



Here a and N are positive parameters: a gives the rate of population growth

when x is small, while N represents a sort of “ideal” population or “carrying

capacity.” Note that if x is small, the differential equation is essentially x



= ax

[since the term 1 − (x/N ) ≈ 1], but if x>N, then x



< 0. Thus this simple

equation satisﬁes the above assumptions. We should add here that there are

many other differential equations that correspond to these assumptions; our

choice is perhaps the simplest.

Without loss of generality we will assume that N = 1. That is, we will

choose units so that the carrying capacity is exactly 1 unit of population, and

x(t ) therefore represents the fraction of the ideal population present at time t .

剩余431页未读，继续阅读

libin_swjtu

粉丝: 0
资源: 1

会员权益专享

探索混沌理论：微分方程与动力系统入门

Ordinary Differential Equations and Dynamical Systems

常微分方程和动力学系统Ordinary Differential Equations And Dynamical Systems

Introduction to Differential Equations with Dynamical Systems

an introduction to stochastic differential equations version微盘

differential equations and linear algebra pdf

journal of differential equations与journal of functional analysis哪个期刊好？

neural ordinary differential equations

elementary differential equations and 答案

partial differential equations solutions

xuerong mao stochastic differential equations and their applications

matlab fvm,The Finite Volume Method in Computational Fluid Dynamics: An Advanced Introduction with O...

bouc-wen hysteresis for nonlinear discrete systems

Filippov solution

ordinary differential equations

handbook of linear partial differential equations

computational particial differential equations using matlab

numerical solutions of stochastic differential equations

Support of character vectors and strings will be removed in a future release. Use sym objects to define differential equations instead. matlab中的错误

a deep learning library for solving differential equations

用神经网络代码实现 Monte Carlo fPINNs: Deep learning method for forward and inverse problems involving high dimensional fractional partial differential equations

会员权益专享

最新资源