Dynamical Systems and a Brief Introduction to Ergodic Theory

Transcription

1 Dynamical Systems an a Brief Introuction to Ergoic Theory Leo Baran Spring 2014 Abstract This paper explores ynamical systems of ifferent types an orers, culminating in an examination of the properties of the logistic map. It also introuces Ergoic theory an important results in the fiel. Contents 1 Dynamical Systems Differential Equations Flows on R Flows on R R 3 an chaos Maps The Logistic Map Ergoic Theory Measure Theory Preliminaries 1, A Few Results Dynamical Systems 4 The fiel of ynamics came to be in the 1600s by Newton s evelopments in ifferential equations an their applications to the laws of gravitation an planetary motion. He realize that fining an 1

2 exact solution for some ynamical systems (such as the three-boy problem, or escribing the exact motion of three planetary boies confine to the laws of gravitation) was essentially impossible. A coupl centuries later, Poincare evelope a novel approach to analyzing such systems. His metho was to answer qualitative questions about the system rather than evelop an exact quantitative solution. Until the mi-1900s, ynamics was largely concerne with non-linear oscillating systems an their applications to physics. The evelopment of high-spee computing le Lorenz in 1963 to iscover chaotic motion in ynamics. Since then interest in ynamics an chaos has proliferate an applications to real-worl systems have become exceeingly numerous. There are two main types of ynamical systems: ifferential equations an iterate maps. Differential equations escribe the motion of systems in continuous time, while iterate maps eal exclusively with iscrete time. 1.1 Differential Equations Consier an n-imensional space whose points (x 1 (t), x 2 (t),..., x n (t)) are functions of time. In general, an autonomous ynamical system on this n-imensional phase space is efine as the system t x 1 = f 1 (x 1, x 2,..., x n ) t x 2 = f 2 (x 1, x 2,..., x n )... t x n = f n (x 1, x 2,..., x n ) Where autonomous refers to the fact that f 1,..., f n on t epen on t. In this case, the ifferential equations that compose the system are of first orer, i.e. they only use first time erivatives of (x 1, x 2,..., x n ). 2

3 1.1.1 Flows on R 1 On the other han, in ynamical systems a first-orer system is of the form x(t) = f(x(t)) t Example 1.1. x (t) = sin(x(t)) In this case we can fin an exact solution for the system by oing some slightly sketchy separating of variables: x t = sin(x) t = x sin(x) t = x t = ln csc(x) + cot(x) + C. sin(x) If x(0) = x o, then 0 = ln csc(x 0 ) + cot(x 0 ) + C C = ln csc(x 0 ) + cot(x 0 ). Thus t = ln csc(x) + cot(x) + ln csc(x 0 ) + cot(x 0 ) = ln csc(x 0) + cot(x 0 ) csc(x) + cot(x) Most of the time we want to know what x(t) oes as t for an arbitrary initial conition x 0, however in this case (an in most systems) etermining a formula for x(t) isn t trivial. Instea we can interpret f(x(t)) as a vector fiel: t is time, x(t) is the position on R, an x t is the velocity on the line. We can graph this vector fiel while keeping in min that a positive velocity means we move in the positive irection on R an vice versa for negative. For every point at which x (t) = 0, x(t) is stationary. These points are calle fixe points. In the above graph, certain fixe points are marke with soli circles an others with hollow circles. The fixe points with soli circles at (2n+1)π, n Z are points towar which x(t) is attracte, while the hollow circles at 2nπ, n Z are points from which x(t) is repelle. These points are conveniently calle attracting an repelling fixe points, respectively. The fixe to which x(t) tens epens on 3

4 the initial conition. If x(0) = π π 4, then the system will ten to π as t. If x(0) = 4, then the system will ten to π an if x(0) = 0, the system will stay at x(t) = 0 t. We, being mathematicians, woul like to know about solutions to all ynamical systems of this form. Using the fact that our ynamical system is essentially a ifferential equation, we can show that solutions to nonautonomous ynamical systems exist an are unique in a certain inverval of time. Theorem 1.1. (Linear Funamental Existence an Uniqueness Theorem) Let f an g be continuous functions on (a, b) R an (a, b) an x 0 R. Then there is a unique function p(t) = x that satisfies x = f(t)x + g(t), p( ) = x 0 on (a, b). Namely, p(t) = x 0 e F (t) + e F (t) t g(τ)e F (τ) τ Where F (t) = t f(τ)τ. Proof. First we prove existence. Suppose the solution p(t) takes the form above. Take a first erivative: p (t) = x 0 e F (t) t [F (t)]+ t [e F (t) ] t g(τ)e F (τ) τ+ t [ t g(τ)e F (τ) τ]e F (t) = x 0 e F (t) f(t) f(t)e F (t) t g(τ)e F (τ) τ+g(t)e F (t) e F (t) = f(t)[x 0 e F (t) +e F (t) t g(τ)e F (τ) τ]+g(t) = f(t)p(t)+ g(t) p (t) = f(t)p(t)+g(t) p (t)+f(t)p(t) = g(t). Also p( ) = x 0 e F (t0) +e F () g(τ)e F (τ) τ = t0 x 0 e t f(τ)τ 0 = x 0. Thus p(t) is a solution to the initial value problem. Next we show that the solution is unique. Suppose q(t) is also a solution to the problem. Suppose r(t) = q(t)e F (t). Then r( ) = q( ) = x 0. Also r (t) = q (t)e F (t) q(t)f(t)e F (t) = e F (t) [q (t) q(t)f(t)] = e F (t) g(t). We know by the funamental theorem of calculus that t r (τ)τ = r(t) r( ) r(t) = r( ) + t e F (τ) g(τ)τ = x 0 + t e F (τ) g(τ)τ. We also know that r(t) = q(t)e F (t) q(t) = r(t)e F (t). Thus q(t) = e F (t) [x 0 + t e F (τ) g(τ)τ] = x 0 e F (t) + e F (t) t e F (τ) g(τ)τ. We therefore know that every solution to the problem must take this form, so the solution is unique. 4

5 Dynamical systems in two imensions become a bit more interesting Flows on R 2 We consier the general system Or t x 1(t) = f 1 (x 1 (t), x 2 (t)) t x 2(t) = f 2 (x 1 (t), x 2 (t)) t x = f(x) If our system is linear, then t x 1(t) = ax 1 (t) + bx 2 (t), where A = ( a b c ) an x = ( x 1(t) x 2 (t) ) t x 2(t) = cx 1 (t) + x 2 (t) ẋ = Ax We can learn a great eal about the behavior of this system by stuying the matrix A. We woul like to fin lines on the plane which are invariant uner A. That is, Av = λv. (Like usual in solving ifferential equations, we assume that the solution takes the form x(t) = e λt v.) This means we want to fin the eigenvalues λ j an an eigenvectors v 0 of A. We fin them in the following way: Av = λv (A λi)v = 0 = et A λi = 0. The last step is erive from the fact that v is assume not to be zero, thus (A λi) is a non-invertible matrix, which means its eterminant is zero. Next we solve et A λi = 0 et( a λ c λ b ) = 0 λ2 τλ + = 0 where τ = a + = Trace(A) an = a bc = et(a). Using the quaratic equation, we see that λ 1 = τ + τ 2 4, λ 2 = τ τ are the two eigenvalues of A corresponing to eigenvectors v 1, v 2. The general solution to the system is a linear combination of the two solutions: x(t) = c 1 e λ 1t v 1 + c 2 e λ 1t v 2. We can use the following result in linear algebra to see where this solution exists. 5

6 Lemma 1.2. Suppose A is a 2 by 2 nonzero matrix with eigenvalues λ 1 an λ 2 such that λ 1 λ 2. Then the eigenvectors v 1 an v 2 that correspon to λ 1 an λ 2 are linearly inepenent. Proof. We nee to show that given a linear combination such that c 1 v 1 + c 2 v 2 = 0, c 1 = c 2 = 0. Suppose we have such a linear combination. 0 = A[c 1 v 1 + c 2 v 2 ] = c 1 λ 1 v 1 + c 2 λ 2 v 2 Also 0 = c 1 v 1 + c 2 v 2 0 = c 1 λ 1 v 1 + c 2 λ 1 v 2 Take the ifference to ge = c 1 λ 1 v 1 c 1 λ 1 v 1 + c 2 λ 2 v 2 c 2 λ 1 v 2 = c 2 v 2 (λ 2 λ 1 ). Since v 2 is an eigenvector, v 2 0. Also, since λ 1 an λ 2 are eigenvectors an not equal, (λ 1 λ 2 ) 0. Thus c 2 = 0. By a similar argument we know c 1 = 0. Thus the only linear combination of v 1 anv 2 that equals zero is one in which each linear combination coefficient is zero. This means v 1 an v 1 are linearly inepenent. Thus, as long as λ 1 λ 2, we know that v 1 an v 2 span R 2 an we can write any initial conition in the plane as a linear combination of them: x(0) = x 0 = c 1 e 0 v 1 + c 2 e 0 v 2 = c 1 v 1 + c 2 v 2 We know, then, that x(t) = c 1 e λ 1t v 1 + c 2 e λ 1t v 2 is a solution to ẋ = Ax an that it satisfies any initial conition x(0) = x 0. By the following theorem, this means that this is the only solution for the system. Theorem 1.3. General Existence an Uniqueness Theorem Consier a general nonlinear system: x 1 (t) = f 1 (x 1 (t), x 2 (t),..., x n (t)),..., x n (t) = f n (x 1 (t), x 2 (t),... x n (t)) ẋ = f(x). Assume f is continous an all of its first partial erivatives are continuous in some open,, connecte set D in R n. Then for x(0) = x 0 D, there exists a unique solution x(t) on some time interval (, ). Proof. See [5], Picar-Linelof Theorem. 6

7 The following example shows us that an interesting phenomenon can occur in two-imensional systems that is unavailable to those of a imension lower. Example 1.2. Consier the system ẋ = y, ẏ = x ẋ = ( )x. Since the eterminant of A is 1 an the trace is 0, we know that the eigenvalues are ± = ±i. The solution to the system is x(t) = c 1 e it v 1 + c 2 e it v 2 = c 1 (cos(t) + i sin(t))v 1 + c 2 (cos(t) i sin(t))v 2. Thus systems with complex eigenvalues isplay oscillitory behavior. In one imension, however, oscillations are impossible the system can either go to ± or ten to a fixe point, but it can never come back aroun to a certain value without being multi-value. There are a variety of theorems that tell us whether or not a ynamical system has a close orbit. The main two are the following: Theorem 1.4. Benixon-Dulac Theorem Let the following system be efine on a simply connecte omain D: x = f(x) with f continuously ifferentiable. If there exists a continuously ifferentiable, real-value function g(x) such that (x g(x)) has one sign throughout D, then there are no close orbits for the system lying entirely in D. Proof. Suppose there is a close orbit C lying entirely in D. Call the region insie C A. Since g(x) is continuously ifferentiable, we can apply the ivergence theorem, which says that (x g(x))a = D C (x g(x)) ns. Since we assume (xg(x)) has one sign throughout D, we know that D (xg(x))a is nonzero. However, since x is tangent to the trajectory of the system (which in this case is C) an n is perpenicular to C, x n = 0 = x g(x)) n = 0, so C (x g(x)) ns = 0, so we arrive at a contraiction an therefore no close orbit C can exist in D. Theorem 1.5. Poincare-Benixon Theorem Suppose the following: R is a compact subset of R 2, ẋ = fx where f is continuously ifferentiable on an open set containing R, R oes not contain any fixe points of the aforementione system, 7

8 an that there exists a trajectory C that starts an stays in R for all time. Then R contains a close orbit. Proof. See Differential Equations an Dynamical Systems, Perko R 3 an chaos Again, something interesting happens when we move up a imension. The Poincare-Benixon Theorem showe us that two-imensional systems confine to compact sets not containing fixe points are guarantee to approach a close orbit as t. In three imensions, this isn t the case. Example 1.3. The Lorenz equations are efine the following: ẋ = σ(y x), ẏ = rx y xz, ż = xy bx Where σ, r, b > 0 are constants. Lorenz iscovere that this system emonstrates erratic behavior in a boune region of space over a wie range of σ, r, an b. Unlike two-imensional systems, in three imensions systems in a boune region not containing a fixe point, trajectories have the possibility of never approaching a close orbit, but wanering aroun the space for all time while never intersecting themselves, as is guarantee by the existence an uniqueness theorem (a unique solution cannot intersect itself since that woul mean two trajectories woul come from the same point). It turns out that the Lorenz system settles onto a set, calle an attractor, which we efine to be a close set A such that: 1. any trajectory of the system which starts in A stays in A for all time. 2. there exists an open set U that contains A such that if x(0) U, then as t the istance between the trajectory an A approaches zero. 3. There is no proper subset of A which satisfies the previous two conitions. 8

9 In the case of the Lorenz system, trajectories settle onto a strange attractor (see above), which is efine as an attractor which exhibits sensitive epenence on initial conitions, which is to say that two trajectories starting arbitrarily close together on the attractor rapily iverge from each other, making long term preiction of such trajectories impossible. It turns out that strange attractors are not regular two- or three-imensional sets, but fractals. We efine chaotic systems to be those with the following characteristics: 1. There exists an open set of initial conitions that leas to long-term aperioic behavior, which is to say trajectories on t settle to fixe points or close orbits as t. 2. The system is eterministic, which is to say the system has no ranom parameters. 3. The system exhibits sensitive epenence on initial conitions. 1.2 Maps One-imensional maps are of the form x n+1 = f(x n ). Similarly to ynamical systems, we efine a fixe point of a map to be a point x 0 such that f(x 0 ) = x 0. Maps can also have cycles for example, a perio-two cycle is one in which f(x 0 ) = x 1, f(x 1 ) = x The Logistic Map Example 1.4. The logistic map is efine as x n+1 = rx n (1 x n ) = f(x n ). We ll consier only positive values of x. If we restrict the parameter r such tha r 4, then the map will map {x : x [0, 1]} to itself. We want to fin the fixe points of the logistic map, which is to say points x such that x = rx (1 x ) x [(r 1) rx ] = 0. The solutions to this are x 1 = 0 an rx 2 = r 1 x 2 = r 1 r. 9

10 For 0 r 1, x 2 is negative, so x 1 is the only fixe point we care about. We want to know the behavior of trajectories with respect to the fixe point. To o this we consier a perturbation x +δ an linearly approximate the map at that point. f(x +δ n ) = x +δ n+1 = f(x )+f (x )δ n +O(δ 2 n) x +δ n+1 = x +f (x )δ n +O(δ 2 n) δ n+1 = f (x )δ n. If we can ignore the O(δ 2 ) terms, then we can etermine whether the fixe point is attracting or repelling irectly from the map δ n+1 = f (x )δ n. If f (x ) < 1, then δ n 0 as n. If f (x ) > 1, then δ n as n. If f (x ) = 1, then the O(δ 2 n) terms ictate the stability of the fixe point. In the case of the logistic map, f (0) = r 2(0) = r. For r (0, 1), the origin is an attracting fixe point. For r [1, 3], x 2 = r 1 r is non-negative an attracting: f (x r 1 2 ) = r 2 = r2 2r 2. The origin, however, is repelling, since r > 1. Thus as x, x n tens to a positive value. Things get bizarre for r greater than 3. At r = 3, a perio-two cycle comes into being: At r 3.449, a perio-four cycle begins. At r 3.544, a perio-eight cycle begins. We call these r values points of perio oubling a 0 = 3, a 1 = 3.449, etc. They eventually converge to a point a Interestingly, the perio oubling values converge in a geometric sense, meaning the ratio between successive perio oubling points shrinks by a constant: r a n a n 1 lim n a n+1 a n Keep this number in min. The above (bifurcation) iagram highlights the fixe points of the logistic map. The splitting of the graph at r = 3 correspons to the creation of the perio-two cycle. The two branches are 10

11 the two values between which the map oscillates. As we can see, past the critical value a, utter panemonium ensues. Interesting features of this bifurcation iagram inclue an interval of periothree behavior ami the chaos an self-similar perio-oubling (each branch splitting is similar to the overall structure). The reason for the latter is obvious once we consier what successive iterates of the function look like. Above we have a graph of f(x, R 0 ) for R 0 < 3. Next to it is a graph of f 2 (x, R 1 ) = f(f(x, R 1 )) for 3 < R 1 < = a 1. The points where these graphs cross the line y = x correspon to fixe points of the map. Stable fixe points of f 2 are points of a perio-two cycle. One notices that the region insie the box in the graph on the right is an inverte copy of the graph on the left. Each time we iterate f twice more, another two-cycle is create at a local maximum/mimimum, as emonstrate below. (Left) graph of f 4 (x) Next we consier the same system, but for the map x n+1 = g(x n ) = r sin( n ) for 0 r 1 an 0 x 1. The graph of this function in the given parameters looks very similar to that of the logistic map for small r. In fact both are sai to be unimoal, or have a single quaratic maximum in the region we re consiering. If we look at the bifurcation iagrams of these two maps, we fin 11

12 that they are qualitatively exactly the same. It turns out that if we calculate δ = lim n a n a n 1 a n+1 a n (where a n is the point of the nth perio oubling) for the sine map, we arrive at the same value for the logistic map of δ In fact, this constant (ubbe the Feigenbaum constant) is the same for any unimoal map 3. 2 Ergoic Theory Ergoic theory is the stuy of measurable ynamics it uses measure theory to stuy the behavior of ynamical systems on abstract spaces. The original motivation for the fiel came from statistical mechanics, specifically the problem of moeling the evolution of n particles an representing each at a certain time as a point in R 6, with three imensions for the position of each particle an three more for momentum vectors in each irection. 2.1 Measure Theory Preliminaries 1,2 Definition 2.1. An algebra Σ on a nonempty set X is a collection of subsets of X which is close uner complements an a finite number of unions, which is to say if A Σ then X \ A Σ an if A 1, A 2, A 3...A n Σ then A 1 A 2 A 3... A n Σ, respectively. Definition 2.2. A σ-algebra Σ is a collection of subsets of X which is close uner complements an countably many unions. Definition 2.3. A measure space is a triplet (X, Σ, µ) in which X is a nonempty set, Σ is a σ-algebra on X an µ : Σ [0, ] is a function such that 1. µ(a) 0 A Σ 2. µ( ) = 0 where is the empty set 3. µ is σ-aitive, which is to say for all countable collections {A n } of isjoint sets in Σ, µ( n A n ) = n µ(a n ) Definition 2.4. If (X, Σ, µ) is a measure space an µ(x) = 1, then µ is calle a probability measure an (X, Σ, µ) is calle a probability space. 12

13 Definition 2.5. (X, Σ) an (X, Σ ) are measure spaces. A function f : X X is measurable if for every A Σ, f 1 (A ) Σ. Definition 2.6. A measure preserving transformation is a quartet (X, Σ, µ, T ) such that (X, Σ, µ) is a measure space an T : X X is a measurable function such that A Σ, µ(t 1 (A)) = µ(a). T is sai to preserve the measure µ. Definition 2.7. A probability preserving transformation is a measure preserving transformation on a probability space. 2.2 A Few Results 3 Theorem 2.1. Monotonicity of Measure Let (X, Σ, µ) be a measure space. Then µ is monotone, which is to say if A, B Σ an A B, then µ(a) µ(b). Proof. Assume A, B Σ an A B. By efinition µ is an aitive function for isjoint sets, so if A B = 0, then µ(a B) = µ(a) + µ(b). We know B = B (B A) = B ((B \A) A) = ((B \A) B) ((B \A) A) = (B \A) (A B) = (B \ A) A, the last equality holing since A B. Since (B \ A) A = 0, B is the union of two isjoint sets. Thus µ(b) = µ(b \ A) + µ(a) µ(a) since by efinition µ(c) 0 C Σ. Definition 2.8. A measure-preserving transformation T : X X on a measure space (X, Σ, µ) is recurrent if for every measurable set A Σ with positive measure there is a null set N A (null in this context means the outer measure of N is zero) such that x A \ N there is an integer n > 0 such that T n (x) = [T T... T ](x) A. Informally, T is recurrent means for any positive measure set A, almost every point in A returns to A at some point in the future. Theorem 2.2. Poincare Recurrence Theorem Let (X, Σ, µ) be a measure space an T : X X a measure-preserving transformation. Then T is a recurrent transformation. Proof. We want to show that for a set A Σ, µ({x A : Nsuch that f n (x) A n > N}) = 0. Define A n = j=n f j (A). Thus if j i, A i A j.by efinition, µ(f 1 (A)) = µ(a) since f is a measure preserving transformation. We can write A i = f j i (A j ), so µ(a i ) = µ(a j ) since applying 13

14 f 1 numerous times oesn t change the measure. We know that A A 0 = A f 1 (A)..., thus A \ A n A 0 \ A n. Since we know µ has monotone measure, µ(a \ A n ) µ(a 0 \ A n ). Notice that A 0 = (A 0 \ A n ) A 0 A n ) = (A 0 \ A n ) A n (see proof for monotonicity of measure), therefore µ(a 0 ) = µ(a 0 \ A n ) + µ(a n ) since (A 0 \ A n ) an A n are isjoint. µ(a 0 ) = µ(a 0 \ A n ) + µ(a n ) µ(a 0 \ A n ) = µ(a 0 ) µ(a n ) = 0. Hence µ(a \ A n ) = 0 for any n > 0. Since (A \ A n ) is the set of x A such that for all j > n, f j (x) A, we are one. Theorem 2.3. Birkhoff Ergoic Theorem Let (X, Σ, µ) be a probability space an T a measure-preserving transformation on the space. If f : X R is an integrable function an T is ergoic, which is to say for every A Σ with positive measure, µ( n=1 T n (A)) = 1. Then n 1 1 lim f(t i (x)) = n n i=0 fµ almost everywhere. Proof. See [3], page 177. The previous theorem is the highlight of Ergoic theory. What the theorem says is that given an ergoic transformation an invariant measure, then the time average is equal to the space average. If we pick any point in the space an calculate the average of f along the point s orbit an take the limit as time goes to, it will be equal to the average value fo the function at all points in the space. Thus any orbit of the system will take up the entire space. References [1] M. Brin an G. Stuck, Introuction to Dynamical Systems, Cambrige, [2] G. Follan, Real Analysis, Wiley, [3] C. E. Silva, Invitation to Ergoic Theory, American Mathematical Society,

15 [4] Strogatz, S. H. Nonlinear Dynamics an Chaos, Perseus Books, [5] Teschl, G. Orinary Differential Equations an Dynamical Systems. American Mathematical Society,