Remark on Hopf Bifurcation Theorem

Transcription

1 Remark on Hopf Bifurcation Theorem Krasnosel skii A.M., Rachinskii D.I. Institute for Information Transmission Problems Russian Academy of Sciences 19 Bolshoi Karetny lane, Moscow, Russia s: November 12, 2002 In the paper we give a simple generalization of Hopf Bifurcation Theorem for the case of scalar equations of a higher order. We consider some ordinary differential equation with an equilibrium at the origin and study the degenerate case of Hopf bifurcations where for some value λ 0 of a parameter λ several roots of characteristic polynomial cross the imaginary axis in the same point. Generically the answer is as follows. Let N 1 roots cross the imaginary axis from the left to the right and N 2 roots cross the imaginary axis from the right to the left. Then, generically, N 1 N 2 branches of small cycles exist in a vicinity of an equilibrium. Partially, in the classical Hopf Bifurcation Theorem the numbers N j equals 0 and 1. MSC 2000: Primary: 34C23, Secondary: 37G10 Authors are partially supported by Russian Foundation for Basic Researches (Grants , , , ). The paper was written during the visit of A.M. Krasnosel skii at the University College, Cork, Ireland at

2 2 Contents 1 Introduction 3 2 Main result Assumptions Theorem Remarks Counterexample Equations in R m Continuous branches of cycles Equations with delays Non-sublinear nonlinearities Bifurcations at infinity Equations of control theory Final remark Proof of Theorem Preliminary Auxiliary constructions Homotopy The proof of Lemma The proof of Lemma

3 3 1. Introduction Consider the equation ( ) d L dt, λ x = f(x, x,..., x (l 1), λ) (1) of a higher order with a scalar parameter λ. The nonlinearity f(x 0, x 1,..., x l 1, λ) : R l Λ R is continuous with respect to all its variables as well as the principal linear part: the differential polynomial L(p, λ) = p l + a 1 (λ)p l a l (λ). Definition 1. A value λ 0 of the parameter is called a Hopf bifurcation point with the frequency w 0 for equation (1) if for every sufficiently small r > 0 there exists a λ = λ(r) such that equation (1) with this λ has a nonstationary periodic solution x(t; r) with a period T (r) and λ(r) λ 0, T (r) 2π/w 0, x( ; r) C l 0 as r 0. The use of an auxiliary parameter different from λ is rather standard in Hopf bifurcations (see, e.g., [1] and the original paper by Hopf therein). Statement 1. Let the polynomial L(p, λ) have a pair of simple conjugate roots µ(λ) µ(λ) that cross the imaginary axis for λ = λ 0 at the points ±w 0 i and let L(±kw 0 i, λ 0 ) 0 for all k ±1, k Z. Let the nonlinearity be sublinear: lim sup x x l 1 0 λ Λ f(x 0, x 1,..., x l 1, λ) x x l 1 = 0. (2) Then the value λ 0 is a Hopf bifurcation point with the frequency w 0 for equation (1) This statement (it is a reformulation of more general result for scalar ODE) was announced in [2] and proved in [3] by the special technics of parameter functionalization. When we say that the root µ(λ) crosses the imaginary axis for λ = λ 0, this means that the quantity (λ λ 0 ) Re µ(λ) has the same signature for all λ λ 0 sufficiently close to λ 0. No transversality conditions of the type d/dλ ( Re µ(λ) ) 0 are supposed. In this paper we consider some similar statements without the assumption on simplicity of the root µ. Instead of the simplicity we suppose that for λ = λ 0 the

4 4 roots ±w 0 i of the polynomial L(p, λ 0 ) have the multiplicity N > 1. Generically this means that a number of roots (more than one) for the same value of the parameter cross the imaginary axis in the same point. Along the paper we often use the notion of the rotation of vector fields ([4]) on the boundaries of some domains in Banach spaces. The notion of rotation of vector fields is equivalent to the notion of degree of a mapping; the rotation is used due to family traditions. Without additional comments we use the phase space R l of equation (1). A cycle is a close curve in R l ; it is also a periodic solution of our equations. Sometimes we say that a periodic solution defines a cycle; the meaning of these words is clear. 2. Main result 2.1. Assumptions. The main assumptions on the principal linear part are as follows. For λ = λ 0 N > 1. the polynomial L(p, λ) has the roots ±w 0 i of the multiplicity This polynomial has no other roots of the type ±kw 0 i, i.e., L(±kw 0 i, λ 0 ) 0 for all k ±1, k Z. For λ λ 0 close to λ 0 the polynomial L(p, λ) has no roots on the imaginary axis close to ±w 0 i. These assumptions means that for 0 < λ λ 0 ε in the small vicinity of w 0 i there are exactly 1 N > 1 non-imaginary roots on the complex plane C. Let for λ (λ 0 ε, λ 0 ) exactly N l roots be from the left of the imaginary axis and exactly N r roots be from the right; let for λ (λ 0, λ 0 + ε) exactly N l + roots be from the left of the imaginary axis and exactly N r + roots be from the right. Obviously, N l + + N r + = N l + N r = N. For λ = λ 0 ε there are some roots from the left and some roots from the right of the imaginary axis. Let us increase λ. For λ = λ 0 all these roots are coincide with the point w 0 i. Then for λ > λ 0 all these roots leave the imaginary axis. This process is really crossing only if N l = N r + = N or N l + = N r = N. In these cases all N roots really cross the imaginary axis and in the same direction. Otherwise, some roots µ(λ) may be in the same half-plane Re z > 0 or Re z < 0 for all values of λ λ 0. 1 All roots are calculated together with their multiplicities.

5 Theorem. Below we use the integer number M = N l N l + = N r + N r. Theorem 1. Let all assumptions of Subsection 2.1 be valid. Let M 0. Let the nonlinearity be sublinear, i.e., equality (2) hold. Then the value λ 0 is a Hopf bifurcation point with the frequency w 0 for equation (1). Let us emphasize that N > 1, the case N = 1 was considered in [2, 3]. The parameter r in the definition of Hopf bifurcations has a natural sense under the assumptions of Theorem 1. Let T = 2π/w be the period of a cycle for some λ. Then there exists a unique periodic solution of the form x(t) = r sin wt + h(wt) that defines this cycle; here the Fourier series of the 2π-periodic function h( ) does not contain the first harmonics and r > 0. In the proof below the amplitude r of the first harmonics plays the role of the additional parameter in the given definition of Hopf bifurcations. Generically under the assumptions of Theorem 1 for any r > 0 small enough there exist M distinguish cycles of the type x(t) = r sin wt + h(wt) (with different w and for different λ). 3. Remarks 3.1. Counterexample. The condition M 0 is sharp. If M = 0, then the assertion of Theorem 1 may be not valid. For example the equation x (IV) + 2(1 λ 2 )x + (1 + λ 2 ) 2 x = (x ) 3 has no nontrivial periodic solutions at all. To prove this multiply the equation by ( x and integrate over the period. The left-hand side becomes zero, therefore x (t) ) 4 dt = 0 and x = const = 0. The polynomial L(p, λ) = p 4 + 2(1 λ 2 )p 2 + (1 + λ 2 ) 2 p has 4 roots: ±λ ± i; therefore N, l N, r N+, l N+ r = 1 and M = 0. For λ = λ 0 = 0 the polynomial has the roots ±i both of the multiplicity Equations in R m. It would be interesting to obtain analogous results for general type of equations x = A(λ)x + f(x, λ), x R m. (3) If f is sublinear and simple conjugate eigenvalues µ(λ), µ(λ) cross the imaginary axis for λ = λ 0 at the points ±w 0 i and if L(±kw 0 i, λ 0 ) 0 for all k ±1, k Z, then the value λ 0 is a Hopf bifurcation point with the frequency w 0 for equation (3). Authors do not know if such results are valid without simplicity (as Theorem 1 generalizes Statement 1).

6 Continuous branches of cycles. Under the assumptions of Theorem 1 it is possible to state the existence of local continuous branches of cycles or CBCs. A set of cycles of equation (1) forms a local CBC for λ Λ if on the boundary of any bounded open set G (0 G R l ) of a sufficiently small diameter there exists at least one point that belongs to a cycle Γ G of the equation for some λ Λ. In smooth cases a CBC looks like a 2 dimensional surface in R l. There exists at least one CBC under assumptions of Statement 1. Under the assumptions of Theorem 1 the number of such CBCs may be more than 1, generically there exist M various CBCs Equations with delays. Without essential difficulties it is possible to continue Theorem 1 for equations with delays. Consider the equation ( ) d L dt, λ x = f(x, x(t θ), λ). (4) Let all assumptions of Subsection 2.1 be valid. Let M 0. Let the nonlinearity f(x, y, λ) be sublinear in x and y. Then the value λ 0 is a Hopf bifurcation point with the frequency w 0 for equation (4). It is also possible to consider much more general equations with several delays, with delays and derivatives, with distributed delays, with delays that depend on the parameter, with delays in the left-hand side Non-sublinear nonlinearities. Instead of the sublinear nonlinearities it is possible to consider the functions f satisfying the sector estimates [ l 1 1/2, f(x 0, x 1,..., x l 1, λ) q µ j xj] 2 xj r 0, µ j 0 j=0 with q small enough. Approaches presented in [5] allow to prove the corresponding theorems and to estimate admissible values of q Bifurcations at infinity. These type of Hopf bifurcations was presented in [6] for the case of simple roots (or eigenvalues) crossing the imaginary axis. A value λ 0 of the parameter is called a Hopf bifurcation point at infinity with the frequency w 0 for equation (1) if for every sufficiently large r > 0 there exists a λ = λ(r) such that equation (1) with this λ has a nonstationary periodic solution x(t; r) with a period T (r) and λ(r) λ 0, T (r) 2π/w 0, x( ; r) C l 1 as r.

7 7 Theorem 2. Let all assumptions of Subsection 2.1 be valid. Let M 0. Let the nonlinearity be sublinear at infinity, i.e., lim x x l 1 sup λ Λ f(x 0, x 1,..., x l 1, λ) x x l 1 = 0. Then λ 0 is a Hopf bifurcation point at infinity with the frequency w 0 for equation (1). A set of cycles of equation (1) forms a CBC at infinity for λ Λ if for some ρ on the boundary of any bounded open set G {x : x ρ} in the phase space R l there exists at least one point that belongs to a cycle Γ G of the equation for some λ Λ. Generically under the assumptions of Theorem 2 there exist M such CBCs Equations of control theory. In the same way it is possible to consider the equations arising in the control theory: ( ) ( ) d d L dt, λ x = M dt, λ f(x, x,..., x (k), λ). (5) Here the real polynomials L(p, λ) = p l + a 1 (λ)p l a l (λ), M(p, λ) = b 0 (λ)p m b m (λ) are coprime and continuous in λ; their degrees l and m satisfy l > m + k. A definition of solution for equation (5) is given in most books on control theory (see, e.g., [7] and [8]). Any equation (5) is equivalent to some first order system of the form z = A(λ)z + b(λ)f(x, x,..., x (k), λ), x = (c, z), z, b, c R l. The assertion of Theorem 1 is valid for (5) with small changes in the definitions Final remark. All listed remarks can be applied in various combinations. Let us give only one example. Consider the estimate [ l 1 1/2+c, f(x 0, x 1,..., x l 1, λ) q µ j xj] 2 µj 0. (6) Let all assumptions of Subsection 2.1 be valid and M 0. Then there exists a q 0 such that for q < q 0 from estimate (6) it follows that λ 0 is a Hopf bifurcation point at infinity with the frequency w 0 for equation (1). The approach to construct the estimates for possible q 0 can be found in [5]. j=0

8 8 4. Proof of Theorem Preliminary. At the beginning let us rescale time in equation (1). Consider the equation L (w d ) dt, λ x = f(x, wx,..., w l 1 x (l 1), λ). (7) Every 2π-periodic solution x(t) of (7) defines (2π/w)-periodic solution x(wt) of (1). We search 2π-periodic solutions of equation (7) in the form x(t) = r sin t + h(t). (8) Here h(t) is unknown 2π-periodic function having no harmonics sin t and cos t in its Fourier series. To prove Theorem 1 we prove that for any positive sufficiently small r there exist the values w = w r > 0, λ = λ r, and the function h = h r (t) such that x(t) = r sin t + h(t) satisfies (7). Any non-constant periodic solution x(t) of any autonomous equation is included in the continuum of the shifts x(t+θ), all functions from this continuum corresponds to a single cycle. We find solution in the form (8); it is possible to find it as r sin(t + φ) + h(t). The choice of the phase φ = 0 and r > 0 fixes a single function from the continuum. At the beginning in our equation we had the parameter λ and the unknown function x(t) of the unknown period 2π/w. The unknown function x(t) had the unknown component h(t) and the phase φ and the amplitude r of the first harmonics r sin(t+φ). Any solution is embedded in a continuum with respect to φ that defines the cycle. From this moment and up to the end of the proof we choose another set of the variables and unknowns. We fix the period as 2π. The amplitude r of the first harmonics becomes a new parameter; the phase φ = 0 is fixed; we have new unknown w and the parameter λ also becomes an unknown. As we see below, such choice of variables leads us to equations that can be studied by topological methods. The special role of the first harmonics is rather natural: they define cycles for f Auxiliary constructions. Let C be the space of continuous functions with the usual norm: x C = max{ x(t) : t [0, 2π]}; let C l 1 be the space l 1 times differentiable functions x(t) : [0, 2π] R with the norm x C l 1 = ( x C,..., x (l 1) C ) ; let L 2 be the usual space of functions x(t) : [0, 2π] R with the usual norm; let Wl 2 be the space of 2π-periodic functions with absolutely

9 9 continuous derivatives of the order l 1 and square integrable derivative of the order l. Consider the projector Px(t) = 1 π 2 0 cos(t s) x(s) ds onto the plane E spanned on the functions sin t and cos t and the projector Q = I P onto the subspace E. This subspace consists from all the functions x such that Px = 0. In L 2 these projectors are orthogonal. The projectors P and Q act in all considered spaces. Consider the differential operator L = L(w d/dt, λ). For λ = λ 0 and w = w 0 the kernel of this operator with 2π-periodic boundary conditions is the plane E. Denote as H = H w (λ) the linear operator that maps any function y L 2 to unique solution x = Hy E Wl 2 of the equation Lx = Qy(t). The existence of x(t) follows from Qy E, the uniqueness follows from x E. Operators H : L 2 Wl 2 are completely continuous with respect to all their variables, HQ = H, in L 2 their norms permit the uniform estimate H w (λ) L 2 L 2 sup L(wni, λ) 1, (w, λ) D. n=0,2,3,... Here and below D is a fixed sufficiently small vicinity of the point (w 0, λ 0 ) on the plane (w, λ). Under assumptions of Theorem 1 it is possible to choose D such that L(wki, λ) 0 for k = 0, 2, 3,... if (w, λ) D. The operators H w (k) (λ) = dk dt H w(λ) k for k = 0, 1,..., l 1 also act in L 2 and completely continuous. All the operators H w (k) (λ) map continuously L 2 into C, H (k) w (λ) L 2 C ν, (w, λ) D. (9) 4.3. Homotopy. To prove Theorem 1 it is sufficient to show that for any r > 0 there exist w > 0, λ Λ, and y(t) E such that the function x(t) = r sin t+hy(t) is 2π-periodic solution of equation (7). Put Y (t) = f ( r sin t + Hy, w(r sin t + Hy),..., w l 1 (r sin t + Hy) (l 1), λ ). Without loss of generality soppose that the function f is bounded for all values of its variables. Therefore the operator (w, λ, y) Y (t) is completely continuous with

10 10 respect to all its variables. Consider in the space R R E a completely continuous deformation Θ(w, λ, y, ξ) = ( Θ w, Θ λ, Θ y ) (ξ is the parameter of the deformation) with the components Θ w (w, λ, y, ξ) = r Re L(wi, λ) ξ π 2π Y (t) sin t dt, 0 2π Θ λ (w, λ, y, ξ) = r Im L(wi, λ) ξ Y (t) cos t dt, π (10) Θ y (w, λ, y, ξ) = y(t) ξqy (t). 0 If w > 0, λ Λ, and y(t) E is the singular point of this deformation for ξ = 1, then the function x(t) = r sin t + Hy(t) is a 2π-periodic solution of equation (7). Lemma 1. All singular points (w, λ, h) of the deformation Θ(w, λ, h, ξ) define 2π-periodic solutions x(t) = r sin t + Hy(t) of the equations L (w d ) dt, λ x = ξf(x, wx,..., w l 1 x (l 1), λ). For ξ = 1 the singular points of the deformation define 2π-periodic solutions of (7). This lemma follows directly from the definition of the operators H. Lemma 2. There exists an r 0 > 0 such that for any r < r 0 all singular points (w, λ, y), (w, λ) D of the deformation Θ(w, λ, h, ξ) satisfy the estimates Hy L 2, Hy C l 1 r. k = 0, 1,..., l 1. (11) This lemma follows from the equality Θ y = 0 and from the sublinearity of f. Consider small ε w, ε λ > 0 such that the rectangle Q = w w 0 ε w, λ λ 0 ε λ belongs to the set D. Let Q be the boundary of Q. Put α = inf L(wi, λ) > 0. (w,λ) Q From the sublinearity of f it follows that for any q > 0 there exist r(q) such that for any r < r(q) singular points (w, λ, y) of the deformation satisfy the estimate Y (t) L 2 qr. (12) Fix some q < α π/2. Let r < min{r 0, r(q)}. We search the singular points in the set Ω = { (w, λ) Q, y L 2 2r }. (13)

11 11 Lemma 3. Deformation (10) is nondegenerate on the boundary of set (13). On the boundary Ω of Ω the deformation Θ is non-degenerate, i.e., it is a homotopy. Therefore the rotation γ = γ(θ, Ω) of the vector field Θ is well-defined for any ξ [0, 1] and γ does not depend on ξ. For ξ = 0 this deformation has very simple form Θ w = r Re L(wi, λ), Θ λ = r Im L(wi, λ), Θ y = y. The components Θ w and Θ λ depend on w and λ only; the component Θ y depends on y. According to standard Rotation Product Formulae ([4]) the value γ coincide with the index of the planar vector field Φ(w, λ) = ( Re L(wi, λ), Im L(wi, λ) ) at (w 0, λ 0 ). Lemma 4. M 0. The index of the vector field Φ(w, λ) at the point (w 0, λ 0 ) equals The existence of singular points of the homotopy for all ξ follows from the general principles of degree theory. For ξ = 1 these singular points define the cycles of (7). The last step to prove Theorem 1 is that the singular point (w r, λ r, y r ) satisfies w r w 0, λ r λ 0, Hy C l 0. This follows from (12) and Lemma 2 due to the definition of the homotopy Θ The proof of Lemma 3. The boundary Q consists from two parts: { } { } Ω Q = (w, λ) Q, y L 2 2r, and Ω y = (w, λ) Q, y L 2 = 2r,. On Ω Q at least one of the components Θ w and Θ λ is non-degenerate: the relations r 2 α 2 r 2 L(wi, λ) 2 = ξ2 π 2 2 π Y (t) 2 L 2 2 π r2 q 2 [( 2π 2 ( 2π Y (t) sin t dt) ) 2 ] Y (t) cos t dt contradict with q < α π/2. On the part Ω Q the component Θ y is non-degenerate due to Lemma 2.

12 12 λ λ λ λ Q Q Q Q (w 0,λ 0 ) (w 0,λ 0 ) (w 0,λ 0 ) w w w w σ >0, σ + >0 σ <0, σ + <0 σ <0, σ + >0 σ >0, σ + <0 γ j = 0 γ j = 0 γ j = 1 γ j = 1 Fig. 1. The values γ j The proof of Lemma 4. Below we use the simple formula for the planar rotation (it is also called the winding number) γ = γ(φ, Γ) of non-degenerate complex fields along the simple close curve. The plane is considered as the set of complex numbers with the operation of multiplying. If Φ(z) = Φ 1 (z)φ 2 (z), then the rotations γ(φ 1, Γ) and γ(φ 2, Γ) are also well-defined and γ(φ, Γ) = γ(φ 1, Γ) + γ(φ 2, Γ). (14) This formula follows from the properties of the multiplication of the complex numbers: Arg(z 1 z 2 ) = Arg(z 1 ) + Arg(z 1 ) and from the definition of the winding number. Formula (14) is valid if Γ is the boundary of any domain (not necessary for simple curves). Denote by µ j (λ) the roots of L(p, λ) that are in the vicinity of the point w 0 i C. The functions µ j are continuous in λ; µ j (λ 0 ) = w 0 i. The choice of µ j (λ) is not unique; let us fix one such choice. The polynomial L(p, λ) has the representation L(p, λ) = L 0 (p, λ) L 0 (iw, λ) 0 for (w, λ) D. Now (14) implies The values Ind Φ(w, λ) = (w0,λ 0 ) N ( p 2 2 Re µ j (λ)p + µ j (λ) µ j (λ) ), (15) j=1 N Ind ( µ j (λ) µ j (λ) w 2 2 Re µ j (λ)wi ) (w0,λ 0 ). j=1 γ j = Ind ( Ψ j (w, λ) ) (w0,λ 0 ), Ψ j (w, λ) = µ j (λ) µ j (λ) w 2 2 Re µ j (λ)wi

13 13 depend on the signatures σ and σ + of Re µ j (λ) for λ < λ 0 and for λ > λ 0 correspondingly. On Fig. 1 the vector fields Ψ j are shown for different values of the signatures σ and σ +. The fields are drawn on the boundary of some rectangle Q Q. We fix the width of Q the same as the width of Q and put the height so small that the quantities µ j (λ) 2 w 2 are positive on the left-hand side of rectangle and negative on the right-hand side. If σ σ + > 0, then γ j = 0; if σ < 0, σ + > 0, then γ j = 1; if σ > 0, σ + < 0, then γ j = 1. Instead of the drawing of the pictures it is possible to present direct computations, they are rather cumbersome. Therefore, N γ j = M; j=1 this completes the proof of the lemma.

14 14 References 1. J. Marsden, M. McCracken, Hopf Bifurcation and its Applications. Springer, New York, Krasnosel skiĭ M.A., Kozjakin V.S. The method of parameter functionalization in the problem of bifurcation points, Soviet Math. Dokl., 22, 2, 1980, Kozyakin V.S., Krasnosel skii M.A. The method of parameter functionalization in the Hopf bifurcation problem. Nonlinear Anal. 11, 2, 1987, Krasnosel skiĭ M.A., Zabreĭko P.P. Geometrical Methods of Nonlinear Analysis. Berlin Heidelberg New York Tokyo, Springer, Krasnosel skii A.M., Rachinskii D.I. On existence of cycles in autonomous systems. Doklady Mathematics, submitted 6. Krasnosel skii A.M., Krasnosel skii M.A. Large-amplitude cycles in autonomous systems with hysteresis. Soviet Math. Dokl., 32, 1, 1985, Vidyasagar M. Nonlinear system analysis. Prentice Hall, Englewood Cliffs, Khalil H.K. Nonlinear systems. Macmillan Publishing Company, 1992.