Two dimensional dynamics

MINISTRY OF EDUCATION AND TRAINING HANOI PEDAGOGICAL UNIVERSITY DEPARTMENT OF MATHEMATICS NGUYEN THI HUYEN TWO-DIMENSIONAL DYNAMICS BACHELOR THESIS Hanoi – 2019 MINISTRY OF EDUCATION AND TRAINING HANOI PEDAGOGICAL UNIVERSITY DEPARTMENT OF MATHEMATICS NGUYEN THI HUYEN TWO-DIMENSIONAL DYNAMICS BACHELOR THESIS Major: Analysis SUPERVISOR: Dr TRAN VAN BANG Hanoi – 2019 Thesis Assurance I assure for this is my research thesis which is completed under the guidance of Dr.Tran Van Bang I hereby declare that this thesis is my own work and to the best of my knowledge, it contains no material previously published or written by another person, nor material which to a substantial extent has been accepted for the award of any other degree or diploma at any educational institution, except where due acknowledgement is made in the thesis I also assure that all the help for this thesis has been acknowledge and that the results presented in the thesis has been identified clearly Ha Noi, May, 2019 Student Nguyen Thi Huyen Bachelor thesis NGUYEN THI HUYEN Thesis Acknowledgement This thesis is conducted at the Department of Mathematics, Ha Noi Pedagogical University The lecturers have imparted valuable knowledge and facilitated for me to complete the course and the thesis Firstly, I would like to express my deep respect and sincere gratitude to my supervisor Dr.Tran Van Bang for the continuous support of my study as well as related research, for his patience, motivation and immense knowledge Without his precious guidance in all the time of research, it would not be possible to complete this thesis Besides my advisor, I would like to take this opportunity to thank to all teachers of the Department of Mathematics, Hanoi Pedagogical University 2, the teachers in the Analysis group as well as the teachers involved Due to time, capacity and conditions are limited, so the thesis can not avoid errors So I am looking forward to receiving valuable comments from teachers and friends Ha Noi, May, 2019 Student Nguyen Thi Huyen Contents Notation Preface Preliminaries 1.1 Differential equation 1.2 Flows 1.3 Limit sets and trajectories 1.4 Stability 1.5 Linearization and Hyperbolicity 11 Two-Dimensional Dynamics 15 2.1 Linear systems in R2 15 2.2 The effect of nonlinear terms 23 2.3 Trivial linearization 35 2.4 The Poincare index 37 2.5 Dulac’s criterion 39 2.6 The Poincare - Bendisxon Theorem 40 Conclusion 45 References 46 Bachelor thesis NGUYEN THI HUYEN Notation x˙ Derivates with respect to time t γ(x) The trajectory through x γ + (x) The positive semi-trajectory through x γ − (x) The negative semi-trajectory through x Λ(x) The w−limit set of x A(x) The α-limit set of x  a b  T rD Trace of matrix D =  J Jacobian matrix Γ Simple closed curve (or periodic orbit) T Period IΓ Poincare index L A local transversal E Bounded domain Cr Function is continuously differentiable r times  c d Bachelor thesis NGUYEN THI HUYEN Preface As with other scientific major, differential equations appear on the basis of the development of science, engineering, and the demands of reality Differential equations are an important major of mathematics and it is considered as a bridge between theory and application In almost situations, the differential equations describe the time dependence of a point in a geometrical space, then it is usually called a dynamic system Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a lake At any given time, a dynamical system has a state given by a tuple of real numbers (a vector) that can be represented by a point in an appropriate state space (a geometrical manifold) The evolution rule of the dynamical system is a function that describes what future states follow from the current state Often the function is deterministic, that is, for a given time interval only one future state follows from the current state However, some systems are stochastic, in that random events also affect the evolution of the state variables Chapter Preliminaries In this chapter, we will recall the differential equation including of flows, trajectory and the stability 1.1 Differential equation Consider the differential equation in the form x˙ = f (x, t), x ∈ Rn , f : Rn × R → Rn , where the dot denotes differentiation with respect to time t A particularly simple example of differential equation is the linear differential equation x˙ = Ax, (1.1) where A is an n × n matrix with constant coefficients If the initial condition at t = is x0 then the equation has solutions x = etA x0 where tA e ∞ = k=0 (tA) k! k = I + tA + (tA) 2! + + (tA) k! k + Bachelor thesis NGUYEN THI HUYEN Theorem 1.1.1 (Local existence and uniqueness) Suppose x˙ = f (x, t) and f : Rn × R −→ Rn is continuously differentiable Then there exists maximal t1 > 0, t2 > such that a solution x(t) with x(t0 ) = x0 exists and is unique for all t ∈ (t0 − t1 , t0 + t1 ) Theorem 1.1.2 (Continuity of solutions) Suppose that f is C r (r times continuously differentiable) and r ≥ 1, in some neighbourhood of (x0 , t0 ) Then there exists > and δ > such that if |x −x0 | < , there is a unique solution x(t) defined on [t0 −δ, t0 +δ] with x(t0 ) = x Solutions depend continuously on x and on t 1.2 Flows In this section, we see that solutions to differential equations can be represented as curves in some appropriate space Consider the Autonomous’s equation x˙ = f (x), x ∈ Rn (1.2) Definition 1.2.1 The curve (x1 (t), , xn (t)) in Rn is an integral curve of equation (1.2) iff (x˙ (t), , x˙ n (t)) = f (x1 (t), , xn (t)) for all t ∈ I On the other words, (x1 (t), , xn (t)) is solution of (1.2) on I Thus the tangent to the integral curve at (x1 (t0 ), , xn (t0 )) is f (x1 (t0 ), , xn (t0 )) Example 1.2.2 Consider the differential equation x˙ = −x; x:I→R Bachelor thesis NGUYEN THI HUYEN We have x˙ + x = then the integral curve is x = ce−t Definition 1.2.3 Consider x˙ = f (x) The solution of this differential equation defines a flow, ϕ(x, t) such that ϕ(x, t) is solution of the equation (1.2) with the initial condition x(0) = x Hence d ϕ(x, t) = f (ϕ(x, t)) dt for all t and ϕ(x, 0) = x Then the solution x(t) with x(0) = x0 is ϕ(x0 , t) Lemma 1.2.4 (Properties of the flow) (i) ϕ(x, 0) = x; (ii) ϕ(x, t + s) = ϕ(ϕ(x, t), s) = ϕ(ϕ(x, s), t) = ϕ(x, s + t) Example 1.2.5 Consider the equation x˙ = Ax with x(0) = x0 The solution of equation is x = x0 etA Then the flow ϕ(x, t) = xetA We will go to check properties of the flow, we have: (i) ϕ(x, 0) = xe0 = x (ii) We have ϕ(x, t) = etA x then ϕ(x, t + s) =xe(t+s)A , ϕ(ϕ(x, t), s) =ϕ(x, t)esA = xetA esA = xe(t+s)A , ϕ(ϕ(x, s), t) =ϕ(x, s)etA = xesA etA = xe(t+s)A , ϕ(x, s + t) =xe(s+t)A Bachelor thesis NGUYEN THI HUYEN and Q Particularly, P (r, θ) ∼ o(r) and Q (r, θ) ∼ o(1) So choosing r sufficiently small so that |P (r, θ)| < δr for any δ > such that + δ < −λ If r sufficiently small we find that r˙ (λ + for sufficiently small r with λ + r(t) + δ)r + δ = −k < Thus r˙ −kr and so r0 e−kt This implies that the origin is asymptotically stable Consider the θ˙ equation θ˙ = −sin2 θ + Q (r, θ) where Q (r, θ) ∼ o(1) For sufficiently small r and all δ > there exists η > and r1 > such that if r < r1 and θ lies outside S1 = {(r, θ)| |θ| < δ} and S3 = {(r, θ)| |θ − π| < δ} then θ˙ < −η Hence the motion is inwards and clockwise outside S1 and S3 Inside these sectors, there are cases: (a) Case 1: Their solutions tend to the origin remaining in the sectors for all the time (b) Case 2: The nonlinear terms conspire to push the trajectories through the sectors, eventually coming out the other side Hence, we obtain the cases as shown in Fig 2.14 If the nonlinear terms satisfy the "big O" conditions only the first case is possible, the improper node still is an improper node 32 Bachelor thesis NGUYEN THI HUYEN Figure 2.14: Nonlinear improper node (v) Foci For  C= ρ ω −ω ρ   The nonlinear equations are r˙ = ρr + P (r, θ), and θ˙ = ω + Q(r, θ) Without loss of generality, we assume that ρ < anf ω > Choosing r sufficiently small, we deduce that |P (r, θ)| < r for some < < −ρ and |Q(r, θ)| < δ < ω Then there exists positive numbers k1 and k2 such that r˙ −k1 r and θ˙ 33 k2 Bachelor thesis NGUYEN THI HUYEN Thus, r(t) r0 e−k1 t which implies that the origin is asymptotically stable and θ(t) → ∞ as t → ∞ In conclusion, the origin is a focus (vi) Centres   ρ ω  with ρ = then the equation becomes For C =  −ω ρ r˙ = P (r, θ) and θ˙ = ω + Q(r, θ) By the argument used above for the focus, the θ equation is unclearly for sufficiently small r : θ(t) → ∞ as t → ∞ if ω > We consider trajectories which start on the y-axis with coordinate (x, y) = (0, y0 ) in small neighbourhood of origin There are cases as shown in Fig 2.15 Since θ(t) increases to infinity, there must exist some finite time after the trajectories strickes y-axis in y > at (0, y1 ) (a) If y1 < y0 (in Fig 2.15a) then all trajectories start on (0, y) with y1 < y < y0 must return to y-axis below y1 (b) If y1 > y0 (in Fig 2.15b), the opposite is true (c) If y1 = y0 , the trajectory is periodic Hence, the origin is either a stable focus , an unstable focus, a centre or an infinite sequence of isolated periodic orbits which accumulate on the origin Example 2.2.2 Consider r˙ = −r3 , θ˙ = 34 Bachelor thesis NGUYEN THI HUYEN Figure 2.15: Nonlinear centres Integrating the r equation, we get r(t) = (2t + c)− , where c is a positive constant So r tends to zero as t → ∞ and similarly, to the θ equation we get θ(t) = θ0 + t Hence, the origin is a stable nonlinear node 2.3 Trivial linearization Finding the phase portrait in such cases, we can apply some approaches as follows: Way 1: Look at dy dx on trajectories If this can be solved the equation then we can be sketched Way 2: Determine the trajectories in phase space where x˙ = and y˙ = From this, sketch the curve in phase space Way 3: Changing into polar coordinates where r˙ = and θ˙ = From this, sketch the curve Way 4: Look at the first integral or some invariant curve 35 Bachelor thesis NGUYEN THI HUYEN Example 2.3.1 Consider the equation x˙ = x, y˙ = y dy y2 then = dx x Integrating we get − = log x + c y for some constant c or −1 e y = ec x Since if x = then x˙ = and if y = then y˙ = so both the x− and y−axes are invariant Therefore, we receive the phase portrait as shown in Fig 2.16 It is called half saddle and half node Figure 2.16: Example 2.3.1 36 Bachelor thesis 2.4 NGUYEN THI HUYEN The Poincare index Definition 2.4.1 Consider the system   x˙ = f1 (x, y)  y˙ = f2 (x, y) At each point vector filed (f1 (x, y), f2 (x, y)) defines an angle f2 (x, y) f1 (x, y) ψ = tan−1 (2.13) Let Γ be any simple closed curve in the plane, then moving around Γ we see that ψ changes continuously When we comeback to the original starting point the value of ψ has change by a multiple of 2π This multiple which may be positive or negative is called the Poincare index of Γ, IΓ We represent by IΓ = 2π dψ = 2π Γ Since d −1 dx tan = 1+x2 d tan−1 f2 (x, y) f1 (x, y) Γ , we find d tan−1 IΓ = 2π Γ f2 (x,y) f1 (x,y) = f1 df2 −f2 df1 f1 +f2 f1 df2 − f2 df1 , f1 + f2 which is an integer On the other words, we can write dfi = ∂fi ∂fi dx + dy ∂x ∂y 37 so (2.14) Bachelor thesis NGUYEN THI HUYEN Lemma 2.4.2 (Some properties of the Poincare index) Let Γ be an simple closed curve in the plane a If Γ is a closed curve received from Γ by a continuous deformation not crossing any stationary points, then IΓ = IΓ b The index of a periodic orbits is +1 c The index of node, center or focus is +1 d The index of a saddle is −1 e The index is additive, so if Γ3 = Γ1 + Γ1 in the sense defined in Fig 2.17a then IΓ3 = IΓ1 + IΓ2 Figure 2.17: Addition of curves f, The index of a closed curve is the sum of the indices of the station- ary points inside the curve g, The index of a closed curve containing no stationary points is h, The index of a curve is unchanged if (f1 , f2 ) is replaced by (−f1 , −f2 ) Corollary 2.4.3 If Γ is a periodic orbit then Γ encloses at least one stationary point If the stationary points are hyperbolic then Γ encloses 2n + stationary points (for some n remainder are sinks or sources 38 0) n of which are saddles, the Bachelor thesis 2.5 NGUYEN THI HUYEN Dulac’s criterion Lemma 2.5.1 In mathematics, the Bendixson–Dulac theorem on dynamical systems states that if there exists a C function ϕ(x, y) such that the expression ∂(ϕf ) ∂(ϕg) + ∂x ∂y has the same sign almost everywhere in a simply connected region of the plane, then the plane autonomous system dx = f (x, y), dt dy = g(x, y) dt has no nonconstant periodic solutions lying entirely within the region Almost everywhere means everywhere except possibly in a set of measure 0, such as a point or line Proof Without loss of generality, let there exist a function ϕ(x, y) such that ∂(ϕf ) ∂x + ∂(ϕg) ∂y > in simply connected region R Let C be a closed trajectory of the plane autonomous system in R2 Let D be the interior of C then by Green’s theorem, we have ∂(ϕf ) ∂(ϕg) + dxdy = ∂x ∂y D (−ϕgdx + ϕf dy) = C ϕ(−ydx ˙ + xdy) ˙ C But on C, dx = xdt ˙ and dy = ydt ˙ dt so the integral evaluates to This is a contradiction, so there can be no such closed trajectory C 39 Bachelor thesis NGUYEN THI HUYEN Example 2.5.2 Show that this system has no closed orbits anywhere x˙ = y y˙ = −x − y + x2 + y Solution: Let us try ϕ(x, y) = then ∂ ∂ (y) + (−x − y + x2 + y ) = −1 + 2y ∂x ∂y Although this shows that there is no closed orbit contained in either half-plane y < or y > 12 , it does not rule out the existence of a closed orbit in the whole plane since there may be such an orbit which crosses the line y = 12 Now let us try ϕ(x, y) = eαx ∂ αx ∂ αx e (−x − y + x2 + y ) (e y) + ∂x ∂y = αyeαx − eαx + 2yeαx = eαx [(α + 2)y − 1] Choosing α = −2 reduces the expression to −e−2x which is negative everywhere Hence there are no closed orbits 2.6 The Poincare - Bendisxon Theorem Definition 2.6.1 A local transversal L is a line segment which all trajectories cross from the same side as shown Fig 2.18 40 Bachelor thesis NGUYEN THI HUYEN Figure 2.18: L is a local transversal Lemma 2.6.2 If x0 is not a stationary point then it is always possible to construct a local transversal in a neighbourhood of x0 Proof Assume that x0 is not a stationary point, we can choose coordinates y in the direction of the flow at x0 and z is orthogonal to y Then the flow has the form y˙ = a + O(|(y, z) − x0 |), z˙ = O(|(y, z) − x0 |), (2.15) where a is a non-zero constant Now choosing a line through x0 in the y direction Because a = so all trajectories cross this line closed to x0 so in the same direction By the property of local transversal in R2 then the successive intersections of trajectory move monotonically along L Lemma 2.6.3 If a trajectory γ(x) intersects a local transversal L several times, the successive crossing points move monotonically along L Proof We consider two successive crossings In Fig 2.19, γ(x) leaves region and can not return So the next crossing is futher away Corollary 2.6.4 If x ∈ Λ(x) is not a stationary point and x ∈ γ(x0 ), then γ(x) is a closed curve 41 Bachelor thesis NGUYEN THI HUYEN Figure 2.19: Proof We have x ∈ γ(x0 ) and Λ(x) = Λ(x0 ) From x ∈ Λ(x0 ) implies x ∈ Λ(x) We choose L is a local transversal through x then there exists an increasing sequence (tj ), tj > tj−1 > such that ϕ(x, tj ) → x as j → ∞ and ϕ(x, tj ) ∈ L Moreover ϕ(x, 0) = x Assume that ϕ(x, t1 ) = x the successive intersections of ϕ(x, t) with L are bound away from x by (2.6.3) This is contradiction Hence ϕ(x, t1 ) = x and γ(x) is periodic orbits Now, we can state and prove the Poincare-Bendixson Theorem 42 Bachelor thesis NGUYEN THI HUYEN Theorem 2.6.5 (Poincare-Bendixson) Suppose that γ(x0 ) enters and does not leave some closed, bounded domain E and that there are no stationary points on E Then there exists at least one periodic orbit in E, and this orbit is in the ω - limit set of x0 Proof Since γ(x0 ) enters and does not leave some closed, bounded domain E so Λ(x0 ) is non-empty and it is contained in E Taking a point x ∈ Λ(x0 ) that x is not a stationary, we have cases: (i) If x is in γ(x0 ), applying (2.6.4), we get γ(x) is periodic (ii) If x ∈ / γ(x0 ) Because x is in Λ(x0 ) and γ + (x0 ) ⊂ Λ(x0 ) so γ + (x0 ) ⊂ E Hence x has a limit point in E, says x∗ in E If x∗ ∈ γ + (x0 ) so γ + (x0 ) is closed curve We choose L is a local transversal through x∗ Since x∗ ∈ Λ(x) then γ(x) and L intersects at points p1 , p2 , which accumulate monotonically on x∗ But pi ∈ Λ(x0 ) implies that γ(x0 ) pass arbitrarily closed to pi continuously to pi + and then pi again Thus, the intersections of L and γ(x0 ) not move monotonically along L which is contradiction Therefore x ∈ γ(x0 ) or we can say that there is at least sone periodic orbit in E Note 2.6.1 We can apply the Poincare - Bendixson Theorem by the following way: (i) Firstly, we find a region E on R2 which has no stationary points (ii) We find at least one orbit enters but not leave Then E must contain at least one periodic orbit 43 Bachelor thesis NGUYEN THI HUYEN Example 2.6.6 Consider system   x˙ = x(x2 + y − 2x − 3) − y (I)  y˙ = y(x2 + y − 2x − 3) + x This system has only stationary point, that is the origin We find the region E on R2 We have divergence of the right hand side is 33 4x + 4y − 6x − = (x − ) + y − 16 2 Thus, inside circle with center I( 34 , 0) and R = √ 33 We can define the sign of this expression so there is not a closed curve, which is in circle The other orbit can lie outside or intersect this circle We put x = r cos θ , y = r sin θ Substituting into system (I) we get:   r˙ = r(r2 − 2r cos θ − 3) (II)  θ˙ = If r < we have r˙ < 0, if r > we also have r˙ > We define the annular region E = {(r, θ)|1 < r < 3} Since the only stationary point is the origin, there are no stationary point in E Applying the Poincare-Bendixson theorem, E must contain at least a periodic orbit 44 Bachelor thesis NGUYEN THI HUYEN Conclusion In this thesis, we have presented systematically the following results: Recall the definitions and some properties about autonomous differential equations in the plane We have introduced with a discussion of the stationary points of two-dimensional dynamics Moreover, we have presented the stability of linear system in R2 and the effect of nonlinear terms We have shown the detail about some properties of the Poinecare index, Poincare-Bendixon’s theorem , Dulac’s criterion and take some examples related to them Finally, thesis goes to develop methods for proving the existence of periodic orbits in such systems by applying the Poincare-Bendixson theorem 45 Bibliography [A] References in Vietnamese [1] Nguyễn Thế Hoàn- Phạm Phu, Cơ sở phương trình vi phân lí thuyết ổn định, NXB Giáo dục Việt Nam, 2010 [2] Cung Thế Anh, Cơ sở lí thuyết phương trình vi phân, NXB Đai học sư phạm Hà Nội, 2015 [B] References in English [3] Paul Glendinning, Stability and Instability and Chaos, USA, 1994 46 ... EDUCATION AND TRAINING HANOI PEDAGOGICAL UNIVERSITY DEPARTMENT OF MATHEMATICS NGUYEN THI HUYEN TWO- DIMENSIONAL DYNAMICS BACHELOR THESIS Major: Analysis SUPERVISOR: Dr TRAN VAN BANG Hanoi – 2019 Thesis... Stability 1.5 Linearization and Hyperbolicity 11 Two- Dimensional Dynamics 15 2.1 Linear systems in R2 15 2.2 The effect of nonlinear... and y are hyperbolic saddles (with x = y possibly) then W s (x) ∩ W u (y) = ∅ 14 Chapter Two- Dimensional Dynamics 2.1 Linear systems in R2 In this section, we consider the differential equation