6 General properties of an autonomous system of two first order ODE

Transcription

1 6 General properties of an autonomous system of two first orer ODE Here we embark on stuying the autonomous system of two first orer ifferential equations of the form ẋ 1 = f 1 (, x 2 ), ẋ 2 = f 2 (, x 2 ), (1) where f 1, f 2 C (1) (U; R), U R 2, (t), x 2 (t) are unknown functions, an t is the inepenent variable which usually enote time. System (1) can be conveniently written in the vector form ẋ = f(x), x U R 2, f : U R 2, (2) where x(t) = ( (t), x 2 (t) ), enotes transposition (all the vectors are assume to be columnvectors), f = (f 1, f 2 ) is a vector-function of two variables, which maps a subset U of R 2 to R 2, an the bol font usually enotes vectors (please be aware that in some books vectors are written in the same font as scalar variables, therefore it is the reaer s task to figure out the imensions of the object). I will assume throughout the lectures that for system (1) (or, equivalently, for system (2)) the theorem of uniqueness an existence of solutions is satisfie. To be precise, Theorem 1. Consier problem (2) together with the initial conition x(t 0 ) = x 0 U R 2 (3) an assume that f C (1) (U; R 2 ). Then there exists an ϵ > 0 such that solution to (2) (3) exists an unique for t (t 0 ϵ, t 0 + ϵ). It is important to note that the theorem is local, it guarantees the existence an uniqueness of the solutions only on a small interval of t. Accoring to the general theory we can usually exten this unique solution to some larger interval (t 0 T, t 0 + T + ), but, as simple examples show (see, e.g., Section 0 of these lecture notes), T ± o not have to be infinite. In other wors, solutions can blow up in finite time. In the rest of these lectures I will safely ignore this fact by tacitly assuming that T ± = ±, an this always will be the case for all the moels I will stuy. Therefore, I assume that problem (2) (3) has a unique solution, which I enote x(t; x 0 ), which is efine for all t (, ). There are a lot of notions pertaining to (2) that we iscusse alreay for the first orer ODE. The crucial istinctions will be clear a little later. Definition 2. The set U R 2 in which the solutions to (2) are efine is calle the phase space or the state space of system (2). Math 484/684: Mathematical moeling of biological processes by Artem Novozhilov artem.novozhilov@nsu.eu. Spring

2 By the efinition of the solutions to (2) (3) we have that they represent a curve in the phase space parameterize by the time t. These curves calle orbits or trajectories of (2). Definition 3. The set γ(x 0 ) = {x(t; x 0 ): t ± } is calle an orbit of (2) starting at the point x 0. Positive an negative semi-orbits are efine as γ + (x 0 ) = {x(t; x 0 ): t + } an respectively. γ (x 0 ) = {x(t; x 0 ): t } Some of the orbits are quite special. For example, Definition 4. A point ˆx such that γ(ˆx) = {ˆx} is calle an equilibrium point, or stationary point, or fixe point, or rest point of system (2). It shoul be obvious that Proposition 5. A point ˆx is an equilibrium if an only if f(ˆx) = 0. A very important property of the autonomous system (2) is presente in the following Proposition 6. If ϕ(t) U R 2 is a solution to (2), then ϕ(t t 0 ) is also a solution for any constant t 0. Proof. We are given that t ϕ(t) = f( ϕ(t) ), an we nee to show that t ϕ(t t 0) = f ( ϕ(t t 0 ) ) is also true. Let us make the change of the inepenent variable in the last expression: We have or, since τ t = 1, t ϕ(τ) = τ = t t 0. τ ϕ(τ) τ t = f( ϕ(τ) ), τ ϕ(τ) = f( ϕ(τ) ), which is exactly where we starte, only with τ instea of t. This means that ϕ(τ) is a solution, an hence, by returning to τ = t t 0, that ϕ(t t 0 ) is a solution. 2

3 The orbits shoul not be confuse with the integral curves: the graphs of the solutions in the space R R 2 given parametrically as ( t, (t), x 2 (t) ). Here is a simple illustration: consier the system ẋ 1 = x 2, ẋ 2 =. We can actually fin the solutions of this system in explicit form because the system is linear. Assume that we consier two ifferent initial conitions: an x 2 (here is a very common source of confusion: I use subscripts to enote both the elements of the vectors, i.e., x = (, x 2 ), an istinguish two vectors, i.e., = ( 1, x1 2 ) an x 2 = (x 2 1, x2 2 ). Try not to confuse them.) Hence we have two solutions of the system: (t) = (t; ), an x 2 (t) = x 2 (t; x 2 ). If we plot these solutions in 3D space (t,, x 2 ) we have the integral curves (see the figure, the re curves are the integral curves, an the re ots enote the initial conitions), if we plot them on the plane (, x 2 ) then we have the orbits (the blue curves in the figure), which are the curves together with the irections specifie by t (I o not put irections on the integral curves because there is a natural irection of the time increase). There is one equilibrium ˆx = (0, 0), which is shown by a blue ot, together with the corresponing integral curve, which is parallel to t-axis. Therefore, we can conclue that the orbits are the projections of the integral curves on the phase space. x 2 t t ˆx x 2 ˆx x 2 x 2 Figure 1: Integral (re) an phase curves (orbits, blue) of the system ẋ 1 = x 2, ẋ 2 =. See text for etails 3

4 You can see in the figure that the orbits o not intersect. This is not an obvious observation in general, but it is true ue to the fact that the system is autonomous. Proposition 7. Two orbits of (2) either o not have common points or coincie. Proof. Assume that x 0 U R 2 is a point that belongs to two orbits. This implies that there are two solutions ϕ(t) an ψ(t) an t 1 an t 2 such that Consier ϕ(t 1 ) = ψ(t 2 ). χ(t) = ϕ(t + (t 1 t 2 )). This function, ue to Proposition 6, is also a solution. Moreover, the orbit corresponing to χ(t) coincies with the the orbit corresponing to ϕ(t) because of its efinition (we just use a ifferent parametrization on the orbit, shifte by the value t 1 t 2 ). On the other han, χ(t 2 ) = ϕ(t 1 ) = ψ(t 2 ), which yiels, ue to theorem of existence an uniqueness of the solutions that χ an ψ coincie. Therefore the orbits efine by ϕ(t) an ψ(t) coincie. To repeat myself, the last proposition is not true for non-autonomous systems. This is why, while stuying autonomous systems, we can restrict our attention to the phase space an the orbits in this space. Now consier the group property of the solutions to (2). Proposition 8. If x(t; x 0 ) is a solution to (2), then for any t 1, t 2 R. Proof. Consier two solutions to (2): By the efinition of solution: We have that x(t 2 + t 1 ; x 0 ) = x ( t 2 ; x(t 1 ; x 0 ) ) = x ( t 1 ; x(t 2 ; x 0 ) ) ϕ 1 (t) = x ( t; x(t 1 ; x 0 ) ), ϕ 2 (t) = x(t + t 1 ; x 0 ). x(0; x 0 ) = x 0. ϕ 1 (0) = ϕ 2 (0), which, by theorem of uniqueness an existence implies that they coincie. If we plug t 2 instea of t in these function, we obtain first of the require equalities. The secon one is prove in a similar way. 4

5 x2 Figure 2: Vector fiel of the system ẋ 1 = x 2, ẋ 2 = (cf. the previous figure) The last proposition in particular shows that x ( t; x(t; x 0 ) ) = 0. In the language of algebra this means that the solution to (2), which is often calle the flow of the system, forms a group acting on the phase (or state) space U. We will come back to this interpretation in ue course. Another useful viewpoint at the system (2) is to recognize that the right han sie actually efines a vector fiel at any point x U, i.e., for each point x U there is a vector ( f 1 (x), f 2 (x) ), which is tangent to the orbit at this particular point an points in the irection of the time increase. I illustrate this concept with the same simple system ẋ 1 = x 2, ẋ 2 =. Note that the only point at which the vector fiel is not efine is the equilibrium ˆx = (0, 0). Using the notion of the vector fiel we obtain almost immeiate Proposition 9. Any orbit of (2) ifferent from an equilibrium point is a smooth curve. An here by smooth curve I mean a curve that has a tangent vector at any point. Moreover, Proposition 10. Any orbit of (2) belongs to one of three types: a smooth curve without selfintersections, a close smooth curve (cycle), or a point. The solution corresponing to the cycle is a perioic function of t. 5

6 Proof. If the orbit is not a point then, accoring to the previous proposition, it is a smooth curve. The smooth curve is either close or not. If it is close an because of the fact that at any point of this curve there is non-zero constant with respect to t vector fiel, then there is a constant T > 0 such that the solution corresponing to the close orbit satisfies x(t + T ; x 0 ) = x ( t; x(t ; x 0 ) ) = x(t; x 0 ), i.e., a perioic function of the perio T. It is impossible to raw all the orbits in the phase space. However, we can get a general impression of the orbits behavior by looking at several key orbits, such as equilibria, cycles, an orbits connecting equilibria. Definition 11. Partitioning the phase space into orbits is calle the phase portrait. We alreay raw a number of phase portraits in the case of one first orer ODE. Then it was just a partitioning of the x-axis into orbits. Here we have the plane as our phase space, therefore we have much more possibilities for the mutual positioning of the orbits (an therefore this problem is more complex than the one-imensional case). You shoul look at the phase portrait of the Lotka Volterra system to make sure you unerstan the meanings of the many concepts introuce in this lecture. An final remark: I was talking about systems of two first orer ifferential equations. However, nowhere in the proof I use the fact that the phase space is two imensional. All the statements in this lecture hol for any imension with an obvious caveat that alreay in three imensions the phase portraits are quite ifficult to present (although we will see some example), an in imension four an higher this nice geometric interpretation is of almost no use. 6