Hamad Talibi Alaoui and Radouane Yafia. 1. Generalities and the Reduced System
|
|
- Jesse Greene
- 5 years ago
- Views:
Transcription
1 FACTA UNIVERSITATIS (NIŠ) Ser. Math. Inform. Vol. 22, No. 1 (27), pp SUPERCRITICAL HOPF BIFURCATION IN DELAY DIFFERENTIAL EQUATIONS AN ELEMENTARY PROOF OF EXCHANGE OF STABILITY Hamad Talibi Alaoui and Radouane Yafia Abstract. The local Hopf Bifurcation theorem states that under supercritical conditions, stability passes from the trivial branch to the bifurcated one. We give here an elementary proof of this result, based on the following steps: i) reduction to a two-dimensional system via the center manifold technic; ii) estimation of the distance between solutions of the original and the reduced equations. Thus there is no reference to the Floquet theory. Incidently, we obtain an estimate of the stability region. 1. Generalities and the Reduced System We first recall the Hopf Bifurcation theorem for retarded functional differential equations. Consider the system (1.1) u(t) = Lu t + R(α, u t ), with the following hypotheses: (H ) L : C R n is a linear continuous operator, R : R C R n is C k+1, k 1, such that R(α, ) =, for all α R and D 2 R(α, ) =, for some α R; where C = C([ r, ], R n ), n N, u t (θ) = u(t + θ) for θ [ r, ]. (H 1 ) The characteristic equation (1.2) (λ, α) =, with (λ, α) = det (λid (L + D 2 R(α, ))e λ( ) Id) of the linearized equation of equation (1.1) at the equilibrium point, has only roots with a negative real parts for α α, and exactly two roots cross Received November 3, Mathematics Subject Classification. 34Kxx, 34K18. 21
2 22 H. Talibi Alaoui and R. Yafia the imaginary axis at α = α, at points iν and iν, with multiplicity 1, ν >. More precisely, if λ(α) is the characteristic root of (1.2), bifurcated from iν, we assume that: d (H 2 ) dα Re λ(α) α=α >. Denote by (T α (t)) t the semi group associated with the linearized equation of (1.1) and by A α it s infinitesimal generator. To the continuous branch of eigenvalues λ(α), we can associate a continuous branch of eigenvectors a(α) + ib(α). Let N α = span{a(α), b(α)} and Φ α = (a(α), b(α)). There exists a 2 2 matrix B(α) of the following form ( r(α) β(α) B(α) = β(α) r(α) where λ(α) = r(α) + iβ(α) such that A α Φ α = Φ α B(α). Using the formal adjoint theory for FDEs in [5], N α = {Φ α Ψ α, ϕ, ϕ C} and C can be decomposed as C = N α S α, where Ψ α = col(e(α), f(α)) is a basis of the adjoint space N α, such that Ψ α, Φ α = I (the 2 2 identity matrix) and, is the bilinear form on C C associated with the adjoint equation of the linearized equation of (1.1) around. Using the decomposition u t = Φ α x(t)+ y t, x(t) R 2, y t S α in the variation of constants formula introduced in [5], equation (1.1) reads as (1.3) x(t) = e B(α)t x + y t = T α (t)y + ), e B(α)(t τ) Ψ α ()F (α, Φ α x(τ), y τ )dτ, d[k α (t, τ) S ]F (α, Φ α x(τ), y τ ), where F (α, ϕ, ψ) = R(α, ϕ + ψ) D 2 R(α, ϕ + ψ), K α (t, τ) S = K α (t, τ) Φ α Ψ α, K α (t, τ), K α (t, τ) = τ X α(t+θ ν)dν, X α is the n n fundamental matrix solution of the linearized equation of (1.1). Looking for solutions defined on the whole real axis and uniformly bounded on their domain, the second identity in (1.3) can be written starting from any initial point σ as y t = T α (t σ)y σ + σ d[k α(t, τ) S ]F (α, Φ α x(τ), y τ ). Letting σ, the terms T α (t σ)y(σ) approach to, while the integral has a limit, which yields an expression where the projection y has
3 Supercritical Hopf Bifurcation in Delay Differential Equations been eliminated and the system (1.3) has the form: (1.4) x(t) = e B(α)t x + y t = e B(α)(t τ) Ψ α ()F (α, Φ α x(τ), y τ )dτ, d[k α (t, τ) S ]F (α, Φ α x(τ), y τ ). In view of the center manifold theorem (see [5]), there exists a map g(α, ) defined from a neighborhood of in N α into a neighborhood of in S α such that y t = g(α, Φ α x(t)) is the equation of a local center manifold, for α close to α, where g is C k+1, g(α, ) = and D 2 g(α, ) =. Then the system (1.4) on the center manifold can be reduced to the following problem (1.5) x(t) = e B(α)t x + e B(α)(t τ) Ψ α ()F (α, Φ α x(τ), g(α, Φ α x(τ))dτ, through this transformation: (x(t), y t ) is the solution of (1.4) on the center manifold x(t) is the solution of (1.5) and y t = g(α, Φ α x(t)). In polar coordinates x = (ρ cos φ, ρ sin φ), system (1.5) reads as { ρ = r(α)ρ + h1 (α, ρ, φ), (1.6) φ = β(α) + h 2 (α, ρ, φ), where h 1 (α, ρ, φ) and h 2 (α, ρ, φ) have the form: h 1 (α, ρ, φ) = cos φf 1 (α, ρ cos φ, ρ sin φ) + sin φf 2 (α, ρ cos φ, ρ sin φ), h 2 (α, ρ, φ) = 1 ρ [ sin φf 1(α, ρ cos φ, ρ sin φ) + cos φf 2 (α, ρ cos φ, ρ sin φ)], f 1, f 2 are the components of Ψ α ()F (α, Φ α x(t), g(α, Φ α x(t))) in R 2. For ρ near to, we have φ ν t (in view of the second equation in (1.6)). Therefore t is diffeomorphic to φ, we can eliminate t between the two equations in (1.6). We obtain (1.7) dρ dφ = r(α)ρ + h 1(α, ρ, φ) β(α) + h 2 (α, ρ, φ). Let ρ(φ, c) be the solution of the last equation with initial data c.
4 24 H. Talibi Alaoui and R. Yafia The periodic solution with initial value c correspond to finding c and α such that (1.8) G(α, c) =, where G is the Hopf bifurcation function (1.9) G(α, c) = 2π r(α)ρ(s, c) + h 1 (α, ρ(s, c), s) ds. β(α) + h 2 (α, ρ(s, c), s) From the Hopf bifurcation theorem (see [2], [3] and [4]), there exists ε >, and functions P (ε), ω(ε) and α(ε) (ε [, ε )) sufficiently regular such that P (ε) is ω(ε) periodic solution of equation (1.5) for α = α(ε). Furthermore, ω() := ω = 2π/ν, α() = α, α () = and the bifurcation is subcritical if α () <, and supercritical if α () >. Now, we introduce the assumption on supercritical Hopf bifurcation: (H 3 ) α () >. Remark 1.1. a) (H 3 ) means that the bifurcated branch lies in the parameter region where the trivial solution is unstable, so, we will find out, as is well known ([7]), that the system stabilizes around the nontrivial periodic solutions. b) (H 3 ) implies that α is increasing along the bifurcated branch, occasionally, we will use α instead of ε as an independent variable along the branch. Information about elements of bifurcation (α, ω, P ) are needed here. In fact, the following classical features will be enough for our purposes (see [8] and [9]): α = α + ε 2 α 2 + ε 4 α 4 + (1.1) ω = ω + ε 2 ω 2 + ε 4 ω 4 + P (ε)(t) = ε P (t). Note that the elements of Hopf bifurcation in (1.1) are obtained formally by inserting Taylor expansions of α, ω, ρ: α = α(ε) = α i ε i, ω = ω(ε) = ω i ε i, ρ = ε( ρ i ε i ) into the equation (1.8) and the integrated form of (1.6), with ρ() = ε, φ() =, ρ(ω) = ε, φ(ω) = 2π, and equating like powers of ε.
5 Supercritical Hopf Bifurcation in Delay Differential Equations Stability, Asymptotic Stability Along the Branch of Bifurcation From now on, we assume all of the hypotheses introduced before: (H ) through (H 3 ). Our first step is to give (1.4) in another form, by centering it around y t = g(α, Φ α x(t)). We set y t = z t + g(α, Φ α x(t)), that is, we consider (x, z) instead of (x, y). Inserting this into system (1.4), we obtain: x(t) = e B(α)t x + e B(α)(t τ) R 1 (α, x(τ), z τ )dτ, (2.1) z t = d[k α (t, τ) S ]R 2 (α, x(τ), z τ ), where (2.2) and R 1 (α, x(t), z t ) = Ψ α ()F (α, Φ α x(t), z t + g(α, Φ α x(t))), R 2 (α, x(t), z t ) = F (α, Φ α x(t), z t +g(α, Φ α x(t))) F (α, Φ α x(t), g(α, Φ α x(t))). Note that the bifurcating periodic solutions of (2.1) lie on z =, that is, are in fact the same as bifurcating solutions of (1.5). Extending the notations of Section 1, we denote such solutions by p = (P, ). We will normalize these solutions, considering that P () = P = (ε, ), ε >. Let (x, z ) be a data near (P, ), with x = (c, ), c >. We will denote by x the solution of (1.5) such that x () = x, and by (x #, z) the solution of (2.1) such that x # () = x, z t= = z, and let ω = ω (α, x ) denotes the first return-time for x, that is x (ω ) = (ζ, ), ζ > and ω > is the solution of the equation (2.3) φ(ω) = 2π; ω # = ω # (α, x, z ) denotes the first return-time for x #, that is the solution of the analogous equation of (2.3) resulting from polar coordinates in the first equation in (2.1). The main result of the paper is the following one. Theorem 2.1. Under hypotheses (H ) through (H 3 ), there exists ε 1 >, η >, K > such that for each ε, < ε < ε 1 ; θ, θ 1, the relations P = (ε, ), ε c θε 2 and z ηθε 3 imply i) x # (ω # ) P θ(ε 2 Kε 4 ); ii) z ω # ηθε 3.
6 26 H. Talibi Alaoui and R. Yafia Corollary 2.1. Assume (H ) through (H 3 ). Then the bifurcated orbits of equation (1.1) for α close enough to α are asymptotically stable. Proof. In view of the transformations considered above, we only have to prove the same fact for the equation (2.1). Starting with (x, z ) and θ such that ε c θε 2, z ηθε 3, denoting by (c 1, ) = x # (ω # ), z 1 = z ω # and cj, z j the corresponding terms after j rotations (c j, ) = x # (ω # j ) and φ# (ω # j ) = 2jπ, we obtain: ε c j θ(1 Kε 2 ) j ε 2, ηθε 3. z j Thus (x(t), z t ) approaches exponentially the orbit of p. Theorem 2.1 is a consequence of two intermediate results. We will first state these as lemmas, and derive the Theorem from them. Lemma 2.1. Assume the hypotheses of Theorem 2.1. Then there exists K 1 > such that x (ω ) P θ(ε 2 K 1 ε 4 ). Remark 2.1. Lemma 2.1 states the stability result for a two-dimensional system. There no doubt however that the estimation in Lemma 2.1 is the control estimate for Theorem 2.1. Precisely, our purpose here is to connect in an elementary way the result for such systems to the corresponding stability result for higher dimensional systems. Lemma 2.2. Assume the hypotheses of Theorem 2.1 and fix Then there exists K 2 >, such that: ω 1 > max(ω, ω, ω # ). i) x # (t) P (t) θk2 ε 2, t ω 1 ; ii) z t θε 3 K 2 η, t ω 1 ; iii) x # (t) x (t) θk2 ε 4 η, t ω 1. Proof of Lemma 2.1. In polar coordinates, let (ρ, φ) corresponding to the ω periodic solution P (of (1.6)) guaranteed by Hopf Bifurcation theorem and (ρ, φ ) corresponding to x with ρ() = ε and ρ () = c. From (1.9) we obtain x (ω ) P = ρ (2π) ε = c ε + G(α, c).
7 Supercritical Hopf Bifurcation in Delay Differential Equations On one hand, G(α, c) = G(α, ε) + (c ε)d c G(α, ε) (c ε)2 D 2 ccg(α, ε + ν(c ε)), for some ν [, 1]. In view of (1.8), G(α, ε) =. By differentiation with respect to ε (with α = α(ε)), we obtain D c G(α, ε) = α (ε)d α G(α, ε) = εα (ε)d α G (α, ε), where G (α, ε) = 1 G(α, ε), ε for ε, D c G(α, ), for ε =. Setting K(c, ε) = α (ε) D α G (α, ε) 1 ε 2ε 2 (c ε)d2 ccg(α, ε + ν(c ε)), we have G(α, c) = (c ε)ε 2 K(c, ε). Our next task is to look at limk(c, ε). c ε Claim 2.1. lim c ε K(c, ε) = α ()2πr (α ) ω >. Proof. In view of (1.1) and the hypotheses of Theorem 2.1 we have α (ε) = O(ε) and c ε = O(ε 2 ). Moreover, by (1.9), we have: α (ε) D α G (α, ε) α () 2πr (α ) as ε ε ω (because for c = the corresponding solution ρ = and h i = o(ρ), i = 1, 2). Using the Taylor expansion of h 1 and h 2 in terms of ρ, we have h 1 (α, ρ, s) = ρ 2 C 3 (α, s) + ρ 3 C 4 (α, s) + h 2 (α, ρ, s) = ρd 3 (α, s) + ρ 2 D 4 (α, s) + where C j and D j are polynomials of degree j with respect to cos(s), sin(s).
8 28 H. Talibi Alaoui and R. Yafia Substituting these expansions for h 1 and h 2 in the integral expression of (1.7) we obtain (2.4) ρ (φ) = c + 1 β(α) φ where for each j, D j = D j/β(α). In view of (2.4), we have D 2 ccg(α, c) = 1 β(α) [ r(α)ρ + ( C 3 D 3r(α) ) (ρ ) 2 + ( C 4 C 3 D 3 + r(α) ( ] (D 3) 2 D 4)) (ρ ) 3 + ds, 2π [ r(α)d 2 ccρ + ( C 3 D 3r(α) ) D 2 cc(ρ ) 2 ( + C 4 C 3 D 3 + r(α) ( (D 3) 2 D 4) ) ] Dcc(ρ 2 ) 3 + ds. From (2.4) once again, we can deduce that limd c ρ = 1; c ε limdcc(ρ 2 ) 2 = 2, c ε limdcc(ρ 2 ) j = for j 3 c ε (since r(α ) = and limρ = ). c ε Therefore, we may conclude that limdccg(α, 2 c) = c ε = 1 2π β(α ) 2 β(α ) 2π C 3 (α, s)dcc(ρ 2 ) 2 ε= ds c= C 3 (α, s)ds = (since C 3 is a polynomial in (cos(s), sin(s)) of degree 3). Coming back to K, this yields the result. By continuity property of K, we may deduce that ( ) K(c, ε) K 1 >, for (c, ε) in a neighborhood of (, ) and some K 1 >. We may conclude the proof of Lemma 2.1: taking ε 1 small enough, we can assert that for each pair (c, ε) satisfying the requirements stated in Theorem 2.1, condition ( ) holds.
9 Supercritical Hopf Bifurcation in Delay Differential Equations Therefore, we have x (ω ) P c ε (1 ε 2 K 1 ). This complete the proof of Lemma 2.1. Proof of Lemma 2.2. According to (1.5) and the first equation of (2.1), we have (2.5) x # (t) P (t) = e B(α)t (x P ) [ ] + e B(α)(t τ) R 1 (α, x # (τ), z τ ) R 1 (α, P (τ), ) dτ, (2.6) R 1 (α, x # (τ), z τ ) R 1 (α, P (τ), ) R 1 (α, x # (τ), z τ ) R 1 (α, x # (τ), ) + R 1 (α, x # (τ), ) R 1 (α, P (τ), ), R 1 (α, x # (τ), z τ ) R 1 (α, x # (τ), ) D 3 R 1 (α, x # (2.7) (τ), µz τ ) z τ, and (2.8) R 1 (α, x # (τ), ) R 1 (α, P (τ), ) D 2 R 1 (α, P (τ) + ν(x # (τ) P (τ)), ) x # (τ) P (τ) for some µ, ν [, 1]. From the hypotheses of Theorem 2.1, if ε then (x, z ) (, ) and lim ε x# (t) =, limz t =. Therefore, x # (t) = o(1), z t = o(1) as ε and ε (2.9) { D3 R 1 (α, x # (t), µz t ) = o(1), D 2 R 1 (α, P (t) + ν(x # (t) P (t)), ) = o(1). According to (2.5) through (2.9), with t ω 1, we obtain } x # (t) P (t) M 1 { x P +o(1) max τ t x# (τ) P (τ) +o(1) max z τ τ t with M 1 = max t ω 1 e B(α)t. We conclude that (2.1) x # (t) P (t) M { } x P + o(1) max z τ, τ t
10 3 H. Talibi Alaoui and R. Yafia where M > is a constant. The second equation in (2.1) gives z t T α (t)z + R 2 (α, x # (τ), z τ ) = d[k α (t, τ) S ]R 2 (α, x # (τ), z τ ). R 2 (α, x # (τ), z τ ) R 2 (α, x # (τ), ) D 3 R 2 (α, x # (τ), µz τ ) z τ. } Then z t N 1 { z + o(1) max z τ, where N 1 = max T α (t), and τ t t ω 1 we deduce (2.11) z t N z. With the estimate given in Theorem 2.1, (2.1) and (2.11) lead to i) and ii) of Lemma 2.2. Now, from (1.5) and the first equation in (2.1), we obtain: x # (t) x (2.12) (t) e B(α)(t τ) (R 1 (α, x # (τ), z τ ) R 1 (α, x (τ), )) dτ. R 1 (α, x # (τ), z τ ) R 1 (α, x (τ), ) D 3 R 1 (α, x # (τ), µz τ ) z τ + D 2 R 1 (α, x (τ) + ν(x # (τ) x (τ)), ) x # (τ) x (τ) for some µ, ν [, 1]. Finally, from (2.9), there exists K > such that R 1 (α, x # (τ), z τ ) R 1 (α, x (τ), ) [ K ε max x # (τ) x (τ) τ t ] z τ τ t + max with a constant K >, and from (2.12), we obtain x # (t) x (t) εk max z τ (with K > ), τ t
11 Supercritical Hopf Bifurcation in Delay Differential Equations which, together with ii), gives iii). Proof of Theorem 2.1. i) In inequality (2.13) x # (ω # ) P x # (ω # ) x (ω # ) + x (ω # ) x (ω ) + x (ω ) P the first and the third addend on the right hand side can be estimated using Lemma 2.1 and Lemma 2.2, respectively. In order to estimate the second addend, we need to evaluate ω # ω. Claim 2.2. ω ω # ηθk ε 3. Proof. Near the bifurcation point, equation (1.5) varies more slowly in magnitude than in direction. This is a classical observation, best perceived d in polar coordinates representation: dt φ β(α d ); dtρ = o(ρ). So, for some ε >, C, C, C > we have: C x x (t) C x, d dt φ (2.14) C, for t ω 1, ε ε. Define ψ = angle(x # (ω # ), x (ω # )) = angle(x (ω ), x (ω # )) = φ (ω # ). Using Lemma 2.2 iii) and (2.14), we have tan ψ = tan φ (ω # ) x (ω # ) x # (ω # ) x (ω ) θk 2ε 3 η C. On the other hand tan ψ φ (ω # ) C ω ω #. This yields the desired estimate: ω ω # ηθk ε 3. The second addend of (2.13) can now be estimated using x (ω # ) x (ω ) sup dx (t) dt ω ω #. <t<ω 1 From equation (2.2) we have R 1 (α, x, ) = o(x), and from the proof of Claim 2.2 we have, dx (t) dt C C ε, for some constant C. Thus, in view of the claim 2.2, we obtain x (ω # ) x (ω ) ηθε 4 K 4.
12 32 H. Talibi Alaoui and R. Yafia To reach the conclusion of i) in Theorem 2.1, choose η small enough for K 2 η < K 1 /4 (Lemma 2.1 and Lemma 2.2) and K 4 η < K 1 /4; define K = K 1 /2, and replace addend of right hand side of (2.13) by their corresponding estimates. From Lemma 2.2 we conclude ii). R E F E R E N C E S 1. A. Casal and M. Freedman: Linsted-Poincaré approach to bifurcation problems for differential-delay equations. IEEE Trans. Automatic Control 25 (198). 2. S. N. Chow and J. K. Hale: Methods of Bifurcation Theory. Springer- Verlag New-York, Heidelberg, Berlin, S. N. Chow and J. Mallet-Paret: Integral averaging and bifurcation. J. Differential Equations 26 (1977), T. Faria and L. T. Magalhas: Normal form for retarded functional differential equations with parameters and applications to hopf singularity. J. Differential Equations 122 (1995), J. K. Hale and S. M. Verduyn Lunel: Introduction to Functional Differential Equations. Springer-Verlag, New York, B. D. Hassard, N. D. Kazarinoff and Y-H. Wan: Theory and Applications of Hopf Bifurcation. Cambridge University Press, J. E. Marsden and M. McCracken: The Hopf Bifurcation and its Applications. Applied Math. Series, Vol. 19, Springer-Verlag, New York, H. Talibi and Casal: A perturbation approach to bifurcation branches of periodic solutions for some functional differential equations. Facta Univ. Ser. Math. Inform. 4 (1989), H. Talibi: Contribution à l Etude du Problème de Bifurcation de Hopf dans le Cadre des Equations Differentielles à Retard. Thèse d état, Universit Mohammed V-Rabat, Université Chouaib Doukkali Faculté des Sciences Département de Mathématiques et Informatique B.P: 2, El Jadida, Morocco talibi@math.net Université Ibn Zohr Faculté Polydisciplinaire B.P: 638, Ouarzazate, Morocco yafia1@yahoo.fr
5.2.2 Planar Andronov-Hopf bifurcation
138 CHAPTER 5. LOCAL BIFURCATION THEORY 5.. Planar Andronov-Hopf bifurcation What happens if a planar system has an equilibrium x = x 0 at some parameter value α = α 0 with eigenvalues λ 1, = ±iω 0, ω
More informationHopf Bifurcation Analysis of a Dynamical Heart Model with Time Delay
Applied Mathematical Sciences, Vol 11, 2017, no 22, 1089-1095 HIKARI Ltd, wwwm-hikaricom https://doiorg/1012988/ams20177271 Hopf Bifurcation Analysis of a Dynamical Heart Model with Time Delay Luca Guerrini
More informationTibor Krisztin. Gabriella Vas
Electronic Journal of Qualitative Theory of Differential Equations 11, No. 36, 1-8; http://www.math.u-szeged.hu/ejqtde/ ON THE FUNDAMENTAL SOLUTION OF LINEAR DELAY DIFFERENTIAL EQUATIONS WITH MULTIPLE
More informationMath Ordinary Differential Equations
Math 411 - Ordinary Differential Equations Review Notes - 1 1 - Basic Theory A first order ordinary differential equation has the form x = f(t, x) (11) Here x = dx/dt Given an initial data x(t 0 ) = x
More informationAndronov Hopf and Bautin bifurcation in a tritrophic food chain model with Holling functional response types IV and II
Electronic Journal of Qualitative Theory of Differential Equations 018 No 78 1 7; https://doiorg/10143/ejqtde018178 wwwmathu-szegedhu/ejqtde/ Andronov Hopf and Bautin bifurcation in a tritrophic food chain
More information{ =y* (t)=&y(t)+ f(x(t&1), *)
Journal of Differential Equations 69, 6463 (00) doi:0.006jdeq.000.390, available online at http:www.idealibrary.com on The Asymptotic Shapes of Periodic Solutions of a Singular Delay Differential System
More information6 Linear Equation. 6.1 Equation with constant coefficients
6 Linear Equation 6.1 Equation with constant coefficients Consider the equation ẋ = Ax, x R n. This equating has n independent solutions. If the eigenvalues are distinct then the solutions are c k e λ
More informationODE Final exam - Solutions
ODE Final exam - Solutions May 3, 018 1 Computational questions (30 For all the following ODE s with given initial condition, find the expression of the solution as a function of the time variable t You
More informationMATH 205C: STATIONARY PHASE LEMMA
MATH 205C: STATIONARY PHASE LEMMA For ω, consider an integral of the form I(ω) = e iωf(x) u(x) dx, where u Cc (R n ) complex valued, with support in a compact set K, and f C (R n ) real valued. Thus, I(ω)
More informationStability of Limit Cycle in a Delayed IS-LM Business Cycle model
Applied Mathematical Sciences, Vol. 2, 28, no. 5, 2459-247 Stability of Limit Cycle in a Delayed IS-LM Business Cycle model A. Abta, A. Kaddar H. Talibi Alaoui Université Chouaib Doukkali Faculté des Sciences
More information(8.51) ẋ = A(λ)x + F(x, λ), where λ lr, the matrix A(λ) and function F(x, λ) are C k -functions with k 1,
2.8.7. Poincaré-Andronov-Hopf Bifurcation. In the previous section, we have given a rather detailed method for determining the periodic orbits of a two dimensional system which is the perturbation of a
More informationPhase Synchronization
Phase Synchronization Lecture by: Zhibin Guo Notes by: Xiang Fan May 10, 2016 1 Introduction For any mode or fluctuation, we always have where S(x, t) is phase. If a mode amplitude satisfies ϕ k = ϕ k
More informationA note on linear differential equations with periodic coefficients.
A note on linear differential equations with periodic coefficients. Maite Grau (1) and Daniel Peralta-Salas (2) (1) Departament de Matemàtica. Universitat de Lleida. Avda. Jaume II, 69. 251 Lleida, Spain.
More informationResearch Article Almost Periodic Viscosity Solutions of Nonlinear Parabolic Equations
Hindawi Publishing Corporation Boundary Value Problems Volume 29, Article ID 873526, 15 pages doi:1.1155/29/873526 Research Article Almost Periodic Viscosity Solutions of Nonlinear Parabolic Equations
More informationGlobal continuation of forced oscillations of retarded motion equations on manifolds
Global continuation of forced oscillations of retarded motion equations on manifolds Pierluigi Benevieri, Alessandro Calamai, Massimo Furi and Maria Patrizia Pera To the outstanding mathematician Andrzej
More informationON THE STABILITY OF THE QUASI-LINEAR IMPLICIT EQUATIONS IN HILBERT SPACES
Khayyam J. Math. 5 (219), no. 1, 15-112 DOI: 1.2234/kjm.219.81222 ON THE STABILITY OF THE QUASI-LINEAR IMPLICIT EQUATIONS IN HILBERT SPACES MEHDI BENABDALLAH 1 AND MOHAMED HARIRI 2 Communicated by J. Brzdęk
More informationNOTE ON THE NODAL LINE OF THE P-LAPLACIAN. 1. Introduction In this paper we consider the nonlinear elliptic boundary-value problem
2005-Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 14, 2006, pp. 155 162. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
More informationAn introduction to Birkhoff normal form
An introduction to Birkhoff normal form Dario Bambusi Dipartimento di Matematica, Universitá di Milano via Saldini 50, 0133 Milano (Italy) 19.11.14 1 Introduction The aim of this note is to present an
More informationPositive solutions for a class of fractional boundary value problems
Nonlinear Analysis: Modelling and Control, Vol. 21, No. 1, 1 17 ISSN 1392-5113 http://dx.doi.org/1.15388/na.216.1.1 Positive solutions for a class of fractional boundary value problems Jiafa Xu a, Zhongli
More informationA comparison of delayed SIR and SEIR epidemic models
Nonlinear Analysis: Modelling and Control, 2011, Vol. 16, No. 2, 181 190 181 A comparison of delayed SIR and SEIR epidemic models Abdelilah Kaddar a, Abdelhadi Abta b, Hamad Talibi Alaoui b a Université
More informationEXISTENCE THEOREMS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXXVIII, 2(29), pp. 287 32 287 EXISTENCE THEOREMS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES A. SGHIR Abstract. This paper concernes with the study of existence
More informationRemark on Hopf Bifurcation Theorem
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, 101447 Moscow, Russia E-mails:
More informationStability and Hopf Bifurcation for a Discrete Disease Spreading Model in Complex Networks
International Journal of Difference Equations ISSN 0973-5321, Volume 4, Number 1, pp. 155 163 (2009) http://campus.mst.edu/ijde Stability and Hopf Bifurcation for a Discrete Disease Spreading Model in
More informationStability and bifurcation of a simple neural network with multiple time delays.
Fields Institute Communications Volume, 999 Stability and bifurcation of a simple neural network with multiple time delays. Sue Ann Campbell Department of Applied Mathematics University of Waterloo Waterloo
More informationAn inverse scattering problem for short-range systems in a time-periodic electric field. François Nicoleau
An inverse scattering problem for short-range systems in a time-periodic electric field. François Nicoleau Laboratoire Jean Leray UMR CNRS-UN 6629 Département de Mathématiques 2, rue de la Houssinière
More informationFLUCTUATIONS IN A MIXED IS-LM BUSINESS CYCLE MODEL
Electronic Journal of Differential Equations, Vol. 2008(2008), No. 134, pp. 1 9. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) FLUCTUATIONS
More informationASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NAVIER STOKES EQUATIONS. Tian Ma. Shouhong Wang
DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Volume 11, Number 1, July 004 pp. 189 04 ASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NAVIER STOKES EQUATIONS Tian Ma Department of
More informationBessel Functions Michael Taylor. Lecture Notes for Math 524
Bessel Functions Michael Taylor Lecture Notes for Math 54 Contents 1. Introduction. Conversion to first order systems 3. The Bessel functions J ν 4. The Bessel functions Y ν 5. Relations between J ν and
More informationON THE EXISTENCE OF HOPF BIFURCATION IN AN OPEN ECONOMY MODEL
Proceedings of Equadiff-11 005, pp. 9 36 ISBN 978-80-7-64-5 ON THE EXISTENCE OF HOPF BIFURCATION IN AN OPEN ECONOMY MODEL KATARÍNA MAKOVÍNYIOVÁ AND RUDOLF ZIMKA Abstract. In the paper a four dimensional
More informationDelayed loss of stability in singularly perturbed finite-dimensional gradient flows
Delayed loss of stability in singularly perturbed finite-dimensional gradient flows Giovanni Scilla Department of Mathematics and Applications R. Caccioppoli University of Naples Federico II Via Cintia,
More informationDirection and Stability of Hopf Bifurcation in a Delayed Model with Heterogeneous Fundamentalists
International Journal of Mathematical Analysis Vol 9, 2015, no 38, 1869-1875 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/1012988/ijma201554135 Direction and Stability of Hopf Bifurcation in a Delayed Model
More informationBIFURCATION AND OF THE GENERALIZED COMPLEX GINZBURG LANDAU EQUATION
BIFURCATION AND OF THE GENERALIZED COMPLEX GINZBURG LANDAU EQUATION JUNGHO PARK Abstract. We study in this paper the bifurcation and stability of the solutions of the complex Ginzburg Landau equation(cgle).
More information8.1 Bifurcations of Equilibria
1 81 Bifurcations of Equilibria Bifurcation theory studies qualitative changes in solutions as a parameter varies In general one could study the bifurcation theory of ODEs PDEs integro-differential equations
More informationDISCONTINUOUS DYNAMICAL SYSTEMS AND FRACTIONAL-ORDERS DIFFERENCE EQUATIONS
Journal of Fractional Calculus Applications, Vol. 4(1). Jan. 2013, pp. 130-138. ISSN: 2090-5858. http://www.fcaj.webs.com/ DISCONTINUOUS DYNAMICAL SYSTEMS AND FRACTIONAL-ORDERS DIFFERENCE EQUATIONS A.
More informationNonlinear stability of semidiscrete shocks for two-sided schemes
Nonlinear stability of semidiscrete shocks for two-sided schemes Margaret Beck Boston University Joint work with Hermen Jan Hupkes, Björn Sandstede, and Kevin Zumbrun Setting: semi-discrete conservation
More informationRelative Controllability of Fractional Dynamical Systems with Multiple Delays in Control
Chapter 4 Relative Controllability of Fractional Dynamical Systems with Multiple Delays in Control 4.1 Introduction A mathematical model for the dynamical systems with delayed controls denoting time varying
More informationSYMPLECTIC GEOMETRY: LECTURE 5
SYMPLECTIC GEOMETRY: LECTURE 5 LIAT KESSLER Let (M, ω) be a connected compact symplectic manifold, T a torus, T M M a Hamiltonian action of T on M, and Φ: M t the assoaciated moment map. Theorem 0.1 (The
More informationStability of Feedback Solutions for Infinite Horizon Noncooperative Differential Games
Stability of Feedback Solutions for Infinite Horizon Noncooperative Differential Games Alberto Bressan ) and Khai T. Nguyen ) *) Department of Mathematics, Penn State University **) Department of Mathematics,
More informationUNCERTAINTY FUNCTIONAL DIFFERENTIAL EQUATIONS FOR FINANCE
Surveys in Mathematics and its Applications ISSN 1842-6298 (electronic), 1843-7265 (print) Volume 5 (2010), 275 284 UNCERTAINTY FUNCTIONAL DIFFERENTIAL EQUATIONS FOR FINANCE Iuliana Carmen Bărbăcioru Abstract.
More informationSpike-adding canard explosion of bursting oscillations
Spike-adding canard explosion of bursting oscillations Paul Carter Mathematical Institute Leiden University Abstract This paper examines a spike-adding bifurcation phenomenon whereby small amplitude canard
More informationAn homotopy method for exact tracking of nonlinear nonminimum phase systems: the example of the spherical inverted pendulum
9 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June -, 9 FrA.5 An homotopy method for exact tracking of nonlinear nonminimum phase systems: the example of the spherical inverted
More informationSt. Petersburg Math. J. 13 (2001),.1 Vol. 13 (2002), No. 4
St Petersburg Math J 13 (21), 1 Vol 13 (22), No 4 BROCKETT S PROBLEM IN THE THEORY OF STABILITY OF LINEAR DIFFERENTIAL EQUATIONS G A Leonov Abstract Algorithms for nonstationary linear stabilization are
More informationConvergence Rate of Nonlinear Switched Systems
Convergence Rate of Nonlinear Switched Systems Philippe JOUAN and Saïd NACIRI arxiv:1511.01737v1 [math.oc] 5 Nov 2015 January 23, 2018 Abstract This paper is concerned with the convergence rate of the
More informationAn extremal problem in Banach algebras
STUDIA MATHEMATICA 45 (3) (200) An extremal problem in Banach algebras by Anders Olofsson (Stockholm) Abstract. We study asymptotics of a class of extremal problems r n (A, ε) related to norm controlled
More informationMohamed Akkouchi WELL-POSEDNESS OF THE FIXED POINT. 1. Introduction
F A S C I C U L I M A T H E M A T I C I Nr 45 2010 Mohamed Akkouchi WELL-POSEDNESS OF THE FIXED POINT PROBLEM FOR φ-max-contractions Abstract. We study the well-posedness of the fixed point problem for
More information(2m)-TH MEAN BEHAVIOR OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS UNDER PARAMETRIC PERTURBATIONS
(2m)-TH MEAN BEHAVIOR OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS UNDER PARAMETRIC PERTURBATIONS Svetlana Janković and Miljana Jovanović Faculty of Science, Department of Mathematics, University
More informationHOPF BIFURCATION FOR TUMOR-IMMUNE COMPETITION SYSTEMS WITH DELAY
Electronic Journal of Differential Equations Vol. 4 (4) No. 7 pp. 3. ISSN: 7-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu HOPF BIFURCATION FOR TUMOR-IMMUNE
More informationA Model of Evolutionary Dynamics with Quasiperiodic Forcing
paper presented at Society for Experimental Mechanics (SEM) IMAC XXXIII Conference on Structural Dynamics February 2-5, 205, Orlando FL A Model of Evolutionary Dynamics with Quasiperiodic Forcing Elizabeth
More informationBreakdown of Pattern Formation in Activator-Inhibitor Systems and Unfolding of a Singular Equilibrium
Breakdown of Pattern Formation in Activator-Inhibitor Systems and Unfolding of a Singular Equilibrium Izumi Takagi (Mathematical Institute, Tohoku University) joint work with Kanako Suzuki (Institute for
More informationWe denote the space of distributions on Ω by D ( Ω) 2.
Sep. 1 0, 008 Distributions Distributions are generalized functions. Some familiarity with the theory of distributions helps understanding of various function spaces which play important roles in the study
More informationNORM DEPENDENCE OF THE COEFFICIENT MAP ON THE WINDOW SIZE 1
NORM DEPENDENCE OF THE COEFFICIENT MAP ON THE WINDOW SIZE Thomas I. Seidman and M. Seetharama Gowda Department of Mathematics and Statistics University of Maryland Baltimore County Baltimore, MD 8, U.S.A
More informationProf. Krstic Nonlinear Systems MAE281A Homework set 1 Linearization & phase portrait
Prof. Krstic Nonlinear Systems MAE28A Homework set Linearization & phase portrait. For each of the following systems, find all equilibrium points and determine the type of each isolated equilibrium. Use
More informationLinear ODE s with periodic coefficients
Linear ODE s with periodic coefficients 1 Examples y = sin(t)y, solutions Ce cos t. Periodic, go to 0 as t +. y = 2 sin 2 (t)y, solutions Ce t sin(2t)/2. Not periodic, go to to 0 as t +. y = (1 + sin(t))y,
More informationA Brunn Minkowski theory for coconvex sets of finite volume
A Brunn Minkowski theory for coconvex sets of finite volume Rolf Schneider Abstract Let C be a closed convex cone in R n, pointed and with interior points. We consider sets of the form A = C \ K, where
More information1 Math 241A-B Homework Problem List for F2015 and W2016
1 Math 241A-B Homework Problem List for F2015 W2016 1.1 Homework 1. Due Wednesday, October 7, 2015 Notation 1.1 Let U be any set, g be a positive function on U, Y be a normed space. For any f : U Y let
More informationNONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS
Fixed Point Theory, Volume 9, No. 1, 28, 3-16 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html NONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS GIOVANNI ANELLO Department of Mathematics University
More informationBASIC MATRIX PERTURBATION THEORY
BASIC MATRIX PERTURBATION THEORY BENJAMIN TEXIER Abstract. In this expository note, we give the proofs of several results in finitedimensional matrix perturbation theory: continuity of the spectrum, regularity
More informationConverse Lyapunov theorem and Input-to-State Stability
Converse Lyapunov theorem and Input-to-State Stability April 6, 2014 1 Converse Lyapunov theorem In the previous lecture, we have discussed few examples of nonlinear control systems and stability concepts
More informationApplied Mathematics Letters
Applied Mathematics Letters 24 (211) 219 223 Contents lists available at ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml Laplace transform and fractional differential
More informationLYAPUNOV-RAZUMIKHIN METHOD FOR DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT. Marat Akhmet. Duygu Aruğaslan
DISCRETE AND CONTINUOUS doi:10.3934/dcds.2009.25.457 DYNAMICAL SYSTEMS Volume 25, Number 2, October 2009 pp. 457 466 LYAPUNOV-RAZUMIKHIN METHOD FOR DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT
More informationHOPF BIFURCATION FOR A 3D FILIPPOV SYSTEM
Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis 16 (2009) 759-775 Copyright c 2009 Watam Press http://www.watam.org HOPF BIFURCATION FOR A 3D FILIPPOV SYSTEM M. U.
More informationEquilibrium analysis for a mass-conserving model in presence of cavitation
Equilibrium analysis for a mass-conserving model in presence of cavitation Ionel S. Ciuperca CNRS-UMR 58 Université Lyon, MAPLY,, 696 Villeurbanne Cedex, France. e-mail: ciuperca@maply.univ-lyon.fr Mohammed
More informationConvexity of the Reachable Set of Nonlinear Systems under L 2 Bounded Controls
1 1 Convexity of the Reachable Set of Nonlinear Systems under L 2 Bounded Controls B.T.Polyak Institute for Control Science, Moscow, Russia e-mail boris@ipu.rssi.ru Abstract Recently [1, 2] the new convexity
More informationOn Some Mean Value Results for the Zeta-Function and a Divisor Problem
Filomat 3:8 (26), 235 2327 DOI.2298/FIL6835I Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat On Some Mean Value Results for the
More informationMath 113 (Calculus 2) Exam 4
Math 3 (Calculus ) Exam 4 November 0 November, 009 Sections 0, 3 7 Name Student ID Section Instructor In some cases a series may be seen to converge or diverge for more than one reason. For such problems
More informationOn a Two-Point Boundary-Value Problem with Spurious Solutions
On a Two-Point Boundary-Value Problem with Spurious Solutions By C. H. Ou and R. Wong The Carrier Pearson equation εü + u = with boundary conditions u ) = u) = 0 is studied from a rigorous point of view.
More informationLECTURE 15: COMPLETENESS AND CONVEXITY
LECTURE 15: COMPLETENESS AND CONVEXITY 1. The Hopf-Rinow Theorem Recall that a Riemannian manifold (M, g) is called geodesically complete if the maximal defining interval of any geodesic is R. On the other
More informationStability of Stochastic Differential Equations
Lyapunov stability theory for ODEs s Stability of Stochastic Differential Equations Part 1: Introduction Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH December 2010
More informationEXISTENCE OF THREE WEAK SOLUTIONS FOR A QUASILINEAR DIRICHLET PROBLEM. Saeid Shokooh and Ghasem A. Afrouzi. 1. Introduction
MATEMATIČKI VESNIK MATEMATIQKI VESNIK 69 4 (217 271 28 December 217 research paper originalni nauqni rad EXISTENCE OF THREE WEAK SOLUTIONS FOR A QUASILINEAR DIRICHLET PROBLEM Saeid Shokooh and Ghasem A.
More informationTaylor and Laurent Series
Chapter 4 Taylor and Laurent Series 4.. Taylor Series 4... Taylor Series for Holomorphic Functions. In Real Analysis, the Taylor series of a given function f : R R is given by: f (x + f (x (x x + f (x
More informationEntrance Exam, Differential Equations April, (Solve exactly 6 out of the 8 problems) y + 2y + y cos(x 2 y) = 0, y(0) = 2, y (0) = 4.
Entrance Exam, Differential Equations April, 7 (Solve exactly 6 out of the 8 problems). Consider the following initial value problem: { y + y + y cos(x y) =, y() = y. Find all the values y such that the
More informationApplied Math Qualifying Exam 11 October Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems.
Printed Name: Signature: Applied Math Qualifying Exam 11 October 2014 Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems. 2 Part 1 (1) Let Ω be an open subset of R
More informationAveraging Method for Functional Differential Equations
Averaging Method for Functional Differential Equations Mustapha Lakrib and Tewfik Sari Abstract. In this paper, the method of averaging which is well known for ordinary differential equations is extended,
More informationControl for stability and Positivity of 2-D linear discrete-time systems
Manuscript received Nov. 2, 27; revised Dec. 2, 27 Control for stability and Positivity of 2-D linear discrete-time systems MOHAMMED ALFIDI and ABDELAZIZ HMAMED LESSI, Département de Physique Faculté des
More informationζ (s) = s 1 s {u} [u] ζ (s) = s 0 u 1+sdu, {u} Note how the integral runs from 0 and not 1.
Problem Sheet 3. From Theorem 3. we have ζ (s) = + s s {u} u+sdu, (45) valid for Res > 0. i) Deduce that for Res >. [u] ζ (s) = s u +sdu ote the integral contains [u] in place of {u}. ii) Deduce that for
More information1 2 predators competing for 1 prey
1 2 predators competing for 1 prey I consider here the equations for two predator species competing for 1 prey species The equations of the system are H (t) = rh(1 H K ) a1hp1 1+a a 2HP 2 1T 1H 1 + a 2
More informationMath 312 Lecture Notes Linear Two-dimensional Systems of Differential Equations
Math 2 Lecture Notes Linear Two-dimensional Systems of Differential Equations Warren Weckesser Department of Mathematics Colgate University February 2005 In these notes, we consider the linear system of
More informationStudies on Rayleigh Equation
Johan Matheus Tuwankotta Studies on Rayleigh Equation Intisari Dalam makalah ini dipelajari persamaan oscilator tak linear yaitu persamaan Rayleigh : x x = µ ( ( x ) ) x. Masalah kestabilan lokal di sekitar
More informationStabilization of Distributed Parameter Systems by State Feedback with Positivity Constraints
Stabilization of Distributed Parameter Systems by State Feedback with Positivity Constraints Joseph Winkin Namur Center of Complex Systems (naxys) and Dept. of Mathematics, University of Namur, Belgium
More informationHopf Bifurcation Analysis of Pathogen-Immune Interaction Dynamics With Delay Kernel
Math. Model. Nat. Phenom. Vol. 2, No. 1, 27, pp. 44-61 Hopf Bifurcation Analysis of Pathogen-Immune Interaction Dynamics With Delay Kernel M. Neamţu a1, L. Buliga b, F.R. Horhat c, D. Opriş b a Department
More informationCLASSIFICATIONS OF THE FLOWS OF LINEAR ODE
CLASSIFICATIONS OF THE FLOWS OF LINEAR ODE PETER ROBICHEAUX Abstract. The goal of this paper is to examine characterizations of linear differential equations. We define the flow of an equation and examine
More informationON THE PROBLEM OF LINEARIZATION FOR STATE-DEPENDENT DELAY DIFFERENTIAL EQUATIONS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 5, May 1996 ON THE PROBLEM OF LINEARIZATION FOR STATE-DEPENDENT DELAY DIFFERENTIAL EQUATIONS KENNETH L. COOKE AND WENZHANG HUANG (Communicated
More informationMassera-type theorem for the existence of C (n) -almost-periodic solutions for partial functional differential equations with infinite delay
Nonlinear Analysis 69 (2008) 1413 1424 www.elsevier.com/locate/na Massera-type theorem for the existence of C (n) -almost-periodic solutions for partial functional differential equations with infinite
More informationThermosolutal Convection at Infinite Prandtl Number with or without rotation: Bifurcation and Stability in Physical Space
1/29 Thermosolutal Convection at Infinite Prandtl Number with or without rotation: Bifurcation and Stability in Physical Space Jungho Park Department of Mathematics New York Institute of Technology SIAM
More informationPermanence Implies the Existence of Interior Periodic Solutions for FDEs
International Journal of Qualitative Theory of Differential Equations and Applications Vol. 2, No. 1 (2008), pp. 125 137 Permanence Implies the Existence of Interior Periodic Solutions for FDEs Xiao-Qiang
More informationAllen Cahn Equation in Two Spatial Dimension
Allen Cahn Equation in Two Spatial Dimension Yoichiro Mori April 25, 216 Consider the Allen Cahn equation in two spatial dimension: ɛ u = ɛ2 u + fu) 1) where ɛ > is a small parameter and fu) is of cubic
More informationONLINE APPENDIX TO: NONPARAMETRIC IDENTIFICATION OF THE MIXED HAZARD MODEL USING MARTINGALE-BASED MOMENTS
ONLINE APPENDIX TO: NONPARAMETRIC IDENTIFICATION OF THE MIXED HAZARD MODEL USING MARTINGALE-BASED MOMENTS JOHANNES RUF AND JAMES LEWIS WOLTER Appendix B. The Proofs of Theorem. and Proposition.3 The proof
More informationOn the Stability of the Best Reply Map for Noncooperative Differential Games
On the Stability of the Best Reply Map for Noncooperative Differential Games Alberto Bressan and Zipeng Wang Department of Mathematics, Penn State University, University Park, PA, 68, USA DPMMS, University
More information5 Measure theory II. (or. lim. Prove the proposition. 5. For fixed F A and φ M define the restriction of φ on F by writing.
5 Measure theory II 1. Charges (signed measures). Let (Ω, A) be a σ -algebra. A map φ: A R is called a charge, (or signed measure or σ -additive set function) if φ = φ(a j ) (5.1) A j for any disjoint
More informationLecture Notes for Math 524
Lecture Notes for Math 524 Dr Michael Y Li October 19, 2009 These notes are based on the lecture notes of Professor James S Muldowney, the books of Hale, Copple, Coddington and Levinson, and Perko They
More informationNonlinear Systems and Control Lecture # 19 Perturbed Systems & Input-to-State Stability
p. 1/1 Nonlinear Systems and Control Lecture # 19 Perturbed Systems & Input-to-State Stability p. 2/1 Perturbed Systems: Nonvanishing Perturbation Nominal System: Perturbed System: ẋ = f(x), f(0) = 0 ẋ
More informationThe exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag
J. Math. Anal. Appl. 270 (2002) 143 149 www.academicpress.com The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag Hongjiong Tian Department of Mathematics,
More informationRecall that if X is a compact metric space, C(X), the space of continuous (real-valued) functions on X, is a Banach space with the norm
Chapter 13 Radon Measures Recall that if X is a compact metric space, C(X), the space of continuous (real-valued) functions on X, is a Banach space with the norm (13.1) f = sup x X f(x). We want to identify
More informationComparison of the Weyl integration formula and the equivariant localization formula for a maximal torus of Sp(1)
Comparison of the Weyl integration formula and the equivariant localization formula for a maximal torus of Sp() Jeffrey D. Carlson January 3, 4 Introduction In this note, we aim to prove the Weyl integration
More informationAdditive resonances of a controlled van der Pol-Duffing oscillator
Additive resonances of a controlled van der Pol-Duffing oscillator This paper has been published in Journal of Sound and Vibration vol. 5 issue - 8 pp.-. J.C. Ji N. Zhang Faculty of Engineering University
More informationWe denote the derivative at x by DF (x) = L. With respect to the standard bases of R n and R m, DF (x) is simply the matrix of partial derivatives,
The derivative Let O be an open subset of R n, and F : O R m a continuous function We say F is differentiable at a point x O, with derivative L, if L : R n R m is a linear transformation such that, for
More informationExamples include: (a) the Lorenz system for climate and weather modeling (b) the Hodgkin-Huxley system for neuron modeling
1 Introduction Many natural processes can be viewed as dynamical systems, where the system is represented by a set of state variables and its evolution governed by a set of differential equations. Examples
More informationExponentially Accurate Semiclassical Tunneling Wave Functions in One Dimension
Exponentially Accurate Semiclassical Tunneling Wave Functions in One Dimension Vasile Gradinaru Seminar for Applied Mathematics ETH Zürich CH 8092 Zürich, Switzerland, George A. Hagedorn Department of
More informationA Class of Fractional Stochastic Differential Equations
Vietnam Journal of Mathematics 36:38) 71 79 Vietnam Journal of MATHEMATICS VAST 8 A Class of Fractional Stochastic Differential Equations Nguyen Tien Dung Department of Mathematics, Vietnam National University,
More informationB5.6 Nonlinear Systems
B5.6 Nonlinear Systems 4. Bifurcations Alain Goriely 2018 Mathematical Institute, University of Oxford Table of contents 1. Local bifurcations for vector fields 1.1 The problem 1.2 The extended centre
More information