from delay differential equations to ordinary differential equations (through partial differential equations)
|
|
- Osborne Atkins
- 5 years ago
- Views:
Transcription
1 from delay differential equations to ordinary differential equations (through partial differential equations) dynamical systems and BCAM - Bilbao Dimitri Breda Department of Mathematics and Computer Science - University of Udine (I) December 10, 2013
2 outline basics [9,12] dynamical systems [9,10] three different views numerical analysis [1,5,11,15] from DDEs to ODEs (through PDEs) dynamical systems and BCAM 2/40
3 basics [9,12] dynamical systems [9,10] three different views numerical analysis [1,5,11,15]
4 delay: formal setting Given d 1 and τ > 0, let X := C([ τ, 0]; R d ) and F : X R d. A Retarded Functional Differential Equation (RFDE) is a relation where x t X is defined as Usual terminology: d is the dimension τ is the (maximum) delay [ τ, 0] is the delay interval x t is the state at time t X is the state space x (t) = F(x t ), x t (θ) := x(t + θ), θ [ τ, 0]. F is the Right-Hand Side (RHS, in general nonlinear, here autonomous). Common acronym: DDEs for Delay Differential Equations. from DDEs to ODEs (through PDEs) dynamical systems and BCAM 4/40
5 examples linear systems with discrete delay(s): F(ψ) = Aψ(0) + Bψ( τ) x (t) = Ax(t) + Bx(t τ) linear systems with distributed delay(s): F(ψ) = Aψ(0)+ 0 τ C(θ)ψ(θ)dθ x (t) = Ax(t)+ nonlinear delay logistic equation [13]: 0 τ C(θ)x(t+θ)dθ F(ψ) = rψ(0)[1 ψ( τ)] x (t) = rx(t)[1 x(t τ)] nonlinear Mackey-Glass equation [14]: F(ψ) = aψ( τ) 1 + [ψ( τ)] bψ(0) ax(t τ) c x (t) = 1 + [x(t τ)] bx(t) c from DDEs to ODEs (through PDEs) dynamical systems and BCAM 5/40
6 DDEs vs ODEs Contrary to ODEs for f : R d R d { x (t) = f(x(t)), t 0, x(0) = v R d, a function ϕ X is necessary for a specific solution of a DDE: { x (t) = F(x t ), t 0, x(θ) = ϕ(θ), θ [ τ, 0]. Theorem. F Lipschitz! solution and it is Lipschitz w.r.t. ϕ. A big difference: DDEs generate -dimensional dynamical systems. Other differences: no backward continuation if ϕ only continuous more smoothness of F more smoothness of x necessarily oscillations and chaos already for d = 1 (e.g., Mackey-Glass). from DDEs to ODEs (through PDEs) dynamical systems and BCAM 6/40
7 basics [9,12] dynamical systems [9,10] three different views numerical analysis [1,5,11,15]
8 local stability of equilibria Consider a linear(ized) continuous RHS L : X R d : x (t) = Lx t. (1) Looking for x(t) = e λt v, v R d \ {0}, leads to the characteristic equation Its solutions λ C are known as characteristic roots. det(λi d Le λ ) = 0. (2) Theorem. The zero solution of (1) is asymptotically stable R(λ) < 0 for all λ, unstable if R(λ) > 0 for some λ. Contrary to ODEs, (2) is a nonlinear eigenvalue problem: -many λ. Theorem. α R s.t. R(λ) < α for all λ and there are only finitely-many λ in any vertical strip of C. There exists an alternative characterization, which we follow for computation. from DDEs to ODEs (through PDEs) dynamical systems and BCAM 8/40
9 solution operator Well-posedness allows to define the solution operator T(t) : X X as T(t)ϕ = x t, t 0. It associates to the initial function ϕ in [ τ, 0] the piece of solution x in [t τ, t], shifted back to [ τ, 0]. Theorem. {T(t)} t 0 is a C 0 -semigroup of bounded linear operators: (a) T(t) is linear and bounded for all t 0 (b) T(0) = I X (c) T(t + s) = T(t)T(s) for all t, s 0 (d) {T(t)} t 0 is strongly continuous: lim t 0 T(t)ψ = ψ for all ψ X. What above is general for evolution maps of dynamical systems. What below is not always the case, but it holds for DDEs. Theorem. T(t) is compact for all t τ. from DDEs to ODEs (through PDEs) dynamical systems and BCAM 9/40
10 infinitesimal generator and -ODE C 0 -semigroups have infinitesimal generators. Here follows that associated to x (t) = Lx t. Theorem. The infinitesimal generator is the linear unbounded operator A : D(A) X X with action Aψ = ψ and domain D(A) = {ψ X : ψ X and ψ (0) = Lψ}. Theorem (abstract Cauchy problem). u(t) = T(t)ϕ = x t solves { u (t) = Au(t), t 0, u(0) = ϕ D(A). A linear DDE is an -dimensional linear ODE, governing the time-evolution of the state u(t) = x t in X. from DDEs to ODEs (through PDEs) dynamical systems and BCAM 10/40
11 spectra and stability The eigenvalues of the matrix A rule the dynamics of the linear ODE x (t) = Ax(t) (in R d ). The linear DDE x (t) = Lx t (in R d ) can be seen as u (t) = Au(t) (in X). Thanks to compactness, the dynamics depends on the spectrum σ(a) of A. Theorem. A and T(t) for t τ have only point spectrum (eigenvalues) and σ(t(t)) \ {0} = e tσ(a). The characteristic equation and dynamical systems standpoints are the same: Theorem. λ is a characteristic root λ σ(a). (but not numerically, like for ODEs: polynomial roots vs matrix eigenvalues). from DDEs to ODEs (through PDEs) dynamical systems and BCAM 11/40
12 basics [9,12] dynamical systems [9,10] three different views numerical analysis [1,5,11,15]
13 DDE { x (t) = Lx t, t 0, x(θ) = ϕ(θ), θ [ τ, 0]. from DDEs to ODEs (through PDEs) dynamical systems and BCAM 13/40
14 DDE -ODE from DDEs to ODEs (through PDEs) dynamical systems and BCAM 13/40
15 DDE -ODE from DDEs to ODEs (through PDEs) dynamical systems and BCAM 13/40
16 DDE -ODE from DDEs to ODEs (through PDEs) dynamical systems and BCAM 13/40
17 DDE -ODE from DDEs to ODEs (through PDEs) dynamical systems and BCAM 13/40
18 -ODE { u (t) = Au(t), t 0, u(0) = ϕ D(A). from DDEs to ODEs (through PDEs) dynamical systems and BCAM 13/40
19 PDE formulation So far (1) x(t) R d solves the Cauchy problem for the DDE { x (t) = Lx t, t 0, x(θ) = ϕ(θ), θ [ τ, 0], and (2) u(t) = x t X solves the abstract Cauchy problem for the -ODE { u (t) = Au(t), t 0, Consider now Since x t (θ) = x(t + θ), we have u(0) = ϕ D(A). v(t, θ) := x t (θ) = [u(t)](θ). x t (θ) x(t + θ) x(t + θ) = = = x t(θ) t t θ θ. Hence (3) v(t, θ) R d solves the initial-boundary value problem for the PDE v(t, θ) v(t, θ) =, t 0, θ [ τ, 0], t θ v(t, θ) t = Lv(t, ), t 0, θ=0 v(0, θ) = ϕ(θ), θ [ τ, 0]. from DDEs to ODEs (through PDEs) dynamical systems and BCAM 14/40
20 -ODE { u (t) = Au(t), t 0, u(0) = ϕ D(A). from DDEs to ODEs (through PDEs) dynamical systems and BCAM 15/40
21 DDE { x (t) = Lx t, t 0, x(θ) = ϕ(θ), θ [ τ, 0]. from DDEs to ODEs (through PDEs) dynamical systems and BCAM 15/40
22 DDE PDE v(t, θ) t = v(t, θ) θ characteristic lines t + θ = constant. from DDEs to ODEs (through PDEs) dynamical systems and BCAM 15/40
23 DDE PDE from DDEs to ODEs (through PDEs) dynamical systems and BCAM 15/40
24 DDE PDE from DDEs to ODEs (through PDEs) dynamical systems and BCAM 15/40
25 PDE v(t, θ) v(t, θ) =, t 0, θ [ τ, 0], t θ v(t, θ) t = Lv(t, ), t 0, θ=0 v(0, θ) = ϕ(θ), θ [ τ, 0]. from DDEs to ODEs (through PDEs) dynamical systems and BCAM 15/40
26 For nonlinear F, the DDE {x (t) = F(x t ), t 0, nonlinear case x(θ) = ϕ(θ), θ [ τ, 0], is still equivalent to the PDE v(t, θ) v(t, θ) =, t 0, θ [ τ, 0], t θ v(t, θ) t = F(v(t, )), t 0, θ=0 v(0, θ) = ϕ(θ), θ [ τ, 0]. The abstract Cauchy problem reads { u (t) = A(u(t)), t 0, u(0) = ϕ D(A), with A the nonlinear differential operator Aψ = ψ and D(A) = {ψ X : ψ X and ψ (0) = F(ψ)}. From now on let d = 1 without loss of generality and remember that v(t, θ) = x t (θ) = [u(t)](θ). from DDEs to ODEs (through PDEs) dynamical systems and BCAM 16/40
27 basics [9,12] dynamical systems [9,10] three different views numerical analysis [1,5,11,15]
28 polynomial interpolation Let a function f : [ 1, 1] R be known on given nodes t i, i = 0, 1,..., N. Polynomial interpolation is a way to approximate f: find p N Π N s.t. p N (t i ) = f(t i ). Theorem.!p N Π N interpolating f for any choice of N + 1 distinct nodes. There are several ways to represent p N. We use the Lagrange form: N p N (t) = l j (t)f(t j ), where l j (t) = N k=0 k j j=0 t t k t j t k, t [ 1, 1]. Proof: p N (t i ) = f(t i ) since l j (t i ) = δ i,j, the Kronecker s delta. Interpolation is fine for smooth f and good nodes, as Chebyshev nodes are: ( ) iπ t i = cos. N from DDEs to ODEs (through PDEs) dynamical systems and BCAM 18/40
29 pseudospectral approximation PSA: whatever you do with f, do it instead with p N (dimension: finite). Example on differentiation. Assume we know y i = f(t i ) for i = 0, 1,..., N. Set y = (y 0, y 1,..., y N ) T. Let z = (z 0, z 1,..., z N ) T with z i = p N(t i ) approximating f (t i ). Differentiation and interpolation being linear, we write z = D N y, where D N R (N+1) (N+1) is the differentiation matrix d 0,0 d 0,1 d 0,N l 0(t 0 ) l 1(t 0 ) l N (t 0) d 1,0 d 1,1 d 1,N D N = = l 0(t 1 ) l 1(t 1 ) l N (t 1) d N,0 d N,1 d N,N l 0(t N ) l 1(t N ) l N (t N) from DDEs to ODEs (through PDEs) dynamical systems and BCAM 19/40
30 pseudospectral approximation PSA: whatever you do with f, do it instead with p N (dimension: finite). Example on differentiation. Assume we know y i = f(t i ) for i = 0, 1,..., N. Set y = (y 0, y 1,..., y N ) T. Let z = (z 0, z 1,..., z N ) T with z i = p N(t i ) approximating f (t i ). Differentiation and interpolation being linear, we write z = D N y, where D N R (N+1) (N+1) is the differentiation matrix d 0,0 d 0,1 d 0,N l 0(t 0 ) l 1(t 0 ) l N (t 0) d 1,0 d 1,1 d 1,N D N = = l 0(t 1 ) l 1(t 1 ) l N (t 1) d N,0 d N,1 d N,N l 0(t N ) l 1(t N ) l N (t N) d N D N from DDEs to ODEs (through PDEs) dynamical systems and BCAM 19/40
31 spectral accuracy Compare PSA on Chebyshev nodes (red) with finite differences (blue) f (t) to approximate f for f(t) = e t2. f(t + h) f(t), h = 2/N, h O(N 1 ) O(N N ) N PSA exploits all the smoothness of f: vs finite convergence order. For Chebyshev nodes, moreover, D N is known explicitly. from DDEs to ODEs (through PDEs) dynamical systems and BCAM 20/40
32 linear case: PSA of eigenvalues The idea is to reduce A to a matrix and use its eigenvalues as approximations. Recall that Aψ = ψ and D(A) = {ψ X : ψ X and ψ (0) = Lψ}. Let p N interpolate ψ X on the Chebyshev nodes in [ τ, 0] θ i = τ ( ) iπ 2 cos τ, i = 0, 1,..., N. N 2 Notice that θ 0 = 0 and θ N = τ. Then (Aψ)(θ i ) = ψ (θ i ) p N(θ i ) for i = 1,..., N while at θ 0 = 0 (i = 0) we impose to respect the condition in D(A). (Aψ)(0) = ψ (0) = Lψ Lp N from DDEs to ODEs (through PDEs) dynamical systems and BCAM 21/40
33 discretized infinitesimal generator Being L, differentiation and interpolation linear, the relation between the values ψ(θ i ) and the approximations of (Aψ)(θ i ) is given by a matrix. Indeed, if then for i = 0 while for i = 1,..., N ψ(θ) p N (θ) = (Aψ)(0) Lp N = N l j (θ)ψ(θ j ), j=0 (Aψ)(θ i ) p N(θ i ) = N Ll j ψ(θ j ) j=0 N d i,j ψ(θ j ). Therefore A is discretized by Ll 0 Ll 1 Ll N d 1,0 d 1,1 d 1,N A N = R(N+1) (N+1). d N,0 d N,1 d N,N j=0 from DDEs to ODEs (through PDEs) dynamical systems and BCAM 22/40
34 discretized infinitesimal generator Being L, differentiation and interpolation linear, the relation between the values ψ(θ i ) and the approximations of (Aψ)(θ i ) is given by a matrix. Indeed, if then for i = 0 while for i = 1,..., N ψ(θ) p N (θ) = (Aψ)(0) Lp N = N l j (θ)ψ(θ j ), j=0 (Aψ)(θ i ) p N(θ i ) = N Ll j ψ(θ j ) j=0 N d i,j ψ(θ j ). Therefore A is discretized by Ll 0 Ll 1 Ll N d 1,0 d 1,1 d 1,N A N = R(N+1) (N+1). d N,0 d N,1 d N,N d N D N from DDEs to ODEs (through PDEs) dynamical systems and BCAM 22/40 j=0
35 example Take Lψ = aψ(0) + bψ( τ) x (t) = ax(t) + bx(t τ). Since θ 0 = 0, θ N = τ and l j (θ i ) = δ i,j we get Ll 0 = a, Ll j = 0, j = 1,..., N 1, Ll N = b. Therefore a 0 0 b d 1,0 d 1,1 d 1,N 1 d 1,N A N = d N,0 d N,1 d N,N 1 d N,N Observe: only the first row of A N is influenced by the RHS of the DDE, the rest depends only on τ and on the Chebyshev nodes in [ τ, 0]. from DDEs to ODEs (through PDEs) dynamical systems and BCAM 23/40
36 convergence Theorem. Let λ be an eigenvalue of A with multiplicity m. For N sufficiently large, A N has exactly m eigenvalues λ i counted with multiplicities and ( ) N/m max λ λ C1 i C 0 i=1,...,m N with C 0, C 1 constants and C 1 proportional to λ λ 9 λ λ 5 I(λ) 10 0 λ 3 λ 1 error 10 5 λ 5 λ 7 λ λ R(λ) λ N from DDEs to ODEs (through PDEs) dynamical systems and BCAM 24/40
37 extension to the nonlinear case PSA works well for the local stability analysis of equilibria through the computation of the eigenvalues of A. It is just the first step in the dynamical analysis. Next steps are stability of periodic orbits (Floquet multipliers) and detection of chaotic behaviors (Lyapunov exponents). Other directions concern with bifurcation analysis or different classes of functional equations (neutral and state-dependent, retardedadvanced, retarded partial, differential-algebraic, integro-differential, etc.). For most of the above, PSA is well-suited [2,3,4,6]. from DDEs to ODEs (through PDEs) dynamical systems and BCAM 25/40
38 extension to the nonlinear case PSA works well for the local stability analysis of equilibria through the computation of the eigenvalues of A. It is just the first step in the dynamical analysis. Next steps are stability of periodic orbits (Floquet multipliers) and detection of chaotic behaviors (Lyapunov exponents). Other directions concern with bifurcation analysis or different classes of functional equations (neutral and state-dependent, retardedadvanced, retarded partial, differential-algebraic, integro-differential, etc.). For most of the above, PSA is well-suited [2,3,4,6]. What about the nonlinear case? There is more behind the nature of PSA. Especially related to the following question: if DDEs are -ODEs, what do we loose or preserve by considering just a finite number of the latter? Observe that the interest is not in approximating a specific solution, but rather the whole dynamics. from DDEs to ODEs (through PDEs) dynamical systems and BCAM 25/40
39 PSA of nonlinear DDEs Consider { x (t) = F(x t ), t 0, x(θ) = ϕ(θ), θ [ τ, 0]. For t 0 and τ = θ N < < θ 0 = 0 the Chebyshev nodes in [ τ, 0], let ū i (t) x t (θ i ) be an approximation in θ i of the state at time t. Their interpolant v(t, θ) = N l j (θ)ū j (t) j=0 is a polynomial in θ [ τ, 0], yet a function of t 0 since so are the ū i s. Question: does v satisfy a differential equation and, if so, what is the relation with the original DDE? from DDEs to ODEs (through PDEs) dynamical systems and BCAM 26/40
40 approximated PDE It is not difficult to prove that N v(t, θ) = l j (θ)ū j (t) j=0 satisfies v(t, θ) v(t, θ) = + ε(t, θ, v), t 0, θ [ τ, 0], t θ v(t, θ) t = F( v(t, )), t 0, θ=0 N v(0, θ) = l j (θ)ϕ(θ j ), θ [ τ, 0]. j=0 It corresponds to the PDE formulation of the DDE, but for the interpolation of the initial function ϕ and the error term ε(t, θ, v), which we see in a moment. So v(t, θ) approximates v(t, θ)...and ū j (t)? from DDEs to ODEs (through PDEs) dynamical systems and BCAM 27/40
41 back to ODEs The error term is ε(t, θ, v) = l 0 (θ) ( F( v(t, )) v(t, θ) θ and it vanishes at θ i for i = 1,..., N since l j (θ i ) = δ i,j. ) θ=0 Therefore v(t, ) is the polynomial of collocation of the original PDE: v(t, θ) v(t, θ) t = θ=θi θ, i = 1,..., N, θ=θi v(t, θ) t = F( v(t, )), (i = 0), θ=0 v(0, θ i ) = ϕ(θ i ), i = 0, 1,..., N. But this is equivalent to the Cauchy problem ū 0(t) = F( v(t, )), (i = 0), t 0 ū i (t) = N d i,j ū j (t), i = 1,..., N, t 0 j=0 ū i (0) = ϕ(θ i ), i = 0, 1,..., N, for a system of N + 1 ODEs, nonlinear only in the first one. from DDEs to ODEs (through PDEs) dynamical systems and BCAM 28/40
42 again the linear case If F = L is linear, the first ODE becomes ū 0(t) = L v(t, ) = N Ll j ū j (t). j=0 Therefore ū(t) = (ū 0 (t), ū 1 (t),..., ū N (t)) T solves {ū (t) = A N ū(t), ū(0) = φ R N+1, for φ = (ϕ(θ 0 ), ϕ(θ 1 ),..., ϕ(θ N )) T. Exactly the discretization of the original abstract Cauchy problem { u (t) = Au(t), u(0) = ϕ X. from DDEs to ODEs (through PDEs) dynamical systems and BCAM 29/40
43 PDE v(t, θ) v(t, θ) =, t 0, θ [ τ, 0], t θ v(t, θ) t = F(v(t, )), t 0, θ=0 v(0, θ) = ϕ(θ), θ [ τ, 0]. from DDEs to ODEs (through PDEs) dynamical systems and BCAM 30/40
44 PDE N + 1 ODEs φ = (ϕ(θ 0 ), ϕ(θ 1 ),..., ϕ(θ N )) T from DDEs to ODEs (through PDEs) dynamical systems and BCAM 30/40
45 N + 1 ODEs ū 0(t) = F( v(t, )), (i = 0), t 0 ū i (t) = N d i,j ū j (t), i = 1,..., N, t 0 j=0 ū i (0) = ϕ(θ i ), i = 0, 1,..., N. from DDEs to ODEs (through PDEs) dynamical systems and BCAM 30/40
46 role of D N Consider the linear part ū i(t) = N d i,j ū j (t), i = 1,..., N, j=0 and recall that d 1,0 d 1,1 d 1,N d N =., D N = d N,0 d N,1 d N,N Then ũ(t) = (ū 1 (t),..., ū N (t)) T satisfies the linear inhomogeneous ODE ũ (t) = D N ũ(t) + d N ū 0 (t), whose free dynamics depends on the eigenvalues of D N. D N is independent of the RHS of the DDE, so we study the easiest case F 0: { x (t) = 0, t 0, x(θ) = ϕ(θ), θ [ τ, 0]. It has constant solution x(t) = ϕ(0). Let ϕ(0) = 0 without loss of generality. from DDEs to ODEs (through PDEs) dynamical systems and BCAM 31/40
47 properties of σ( D N ) The first ODE gives ū 0 (t) = ϕ(0) = 0 and the linear part becomes ũ (t) = D N ũ(t). (1) Also ũ(t) for t τ must vanish as N to converge to the exact solution. Theorem. D N is nonsingular. Proof: e λt w solves (1) for λ σ( D N ) with eigenvector w. The interpolant N v(t, θ) = l 0 (θ)ū 0 (t) + l j (θ)e λt w j = q(θ)e λt (q Π N ) j=1 solves the approximated PDE, hence q (θ) = λq(θ) + l 0 (θ)q (0). Finally, l 0 Π N and q Π N 1 imply λ 0. Theorem. σ( D N ) C for all positive integers N. Conjecture. R(σ( D N )) as N. from DDEs to ODEs (through PDEs) dynamical systems and BCAM 32/40
48 equilibria and stability Theorem. A constant function of value c R is an equilibrium of the DDE x (t) = F(x t ) iff the constant vector (c,..., c) T R N+1 is an equilibrium of the ODE approximation {ū 0 (t) = F( v(t, )), ũ (t) = D N ũ(t) + d N ū 0 (t). Proof: based on two properties of Lagrange interpolation: N l j (θ) = 1 θ [ τ, 0] and j=0 N d i,j = 0 i = 0, 1,..., N. j=0 The second for i = 1,..., N reads D N (1,..., 1) T + d N = 0. Theorem. Linearization and approximation commute. Proof: the only nonlinear part of the ODE is the RHS F. Theorem. The linearized ODE approximation predicts accurately the local stability properties of the equilibria of the nonlinear DDE. Proof: it gives the discretized infinitesimal generator. from DDEs to ODEs (through PDEs) dynamical systems and BCAM 33/40
49 bifurcation of equilibria Concluding, the bifurcation analysis of equilibria of nonlinear DDEs can be tackled efficiently by standard tools for ODEs (e.g., AUTO [16], MATCONT [17]). A simple test. The nonlinear logistic DDE x (t) = rx(t)[1 x(t 1)] has equilibrium c = 1 for all r: asymptotically stable for r [0, π/2), unstable otherwise. At r = π/2 a Hopf bifurcation occurs and a limit cycle arises. 30 eigenvalues (N = 20) 10 0 r Hopf π/2 (MATCONT) I(λ) R(λ) N from DDEs to ODEs (through PDEs) dynamical systems and BCAM 34/40
50 more on The same should apply to periodic orbits: those of the ODE approximation are spectrally accurate approximations of the exact ones; linearization and approximation always commute; PSA of Floquet multipliers is available [3]. Ongoing work to provide theoretical support: R(σ( D N )) as N : how fast? convergence of v(t, θ) to x t (θ): spectrally accurate for smooth x? from DDEs to ODEs (through PDEs) dynamical systems and BCAM 35/40
51 more on The same should apply to periodic orbits: those of the ODE approximation are spectrally accurate approximations of the exact ones; linearization and approximation always commute; PSA of Floquet multipliers is available [3]. Ongoing work to provide theoretical support: R(σ( D N )) as N : how fast? convergence of v(t, θ) to x t (θ): spectrally accurate for smooth x? The general idea (and hope) is that standard ODE tools as applied to the ODE approximation can easily provide information on the dynamics of the original DDE. Advantages: valid for any DDE bifurcation tools for ODEs complete and efficient; not so for DDEs [18] spectral accuracy (low N) theoretical insight (dynamical systems, functional analysis, [9]). from DDEs to ODEs (through PDEs) dynamical systems and BCAM 35/40
52 the true (long-term) objective Recent sophisticated models of resource-consumer dynamics are based on delay integro-differential equations (DEs/DDEs) such as [8] h b(t) = β(x(a, S t ), S(t))F(a, S t )b(t a)da, 0 S (t) = f(s(t)) h γ(x(a, S t ), S(t))F(a, S t )b(t a)da, 0 with X and F solutions of external ODEs and discontinuities between juveniles and adults. Semigroup and stability theories are available [7], as well as the PSA of eigenvalues, but only for caricature models [2]. The ODE approximation presented for DDEs is adaptable to such models. Ongoing work: computation of eigenvalues (linear); bifurcation analysis (nonlinear). from DDEs to ODEs (through PDEs) dynamical systems and BCAM 36/40
53 research group J. BCAM P. TU Dresden (D) O. Utrecht (NL) A. de Amsterdam (NL) S. Trieste (I) R. Udine (I)...thanks for your attention! from DDEs to ODEs (through PDEs) dynamical systems and BCAM 37/40
54 bibliography 1 Berrut and Trefethen, Barycentric Lagrange interpolation, SIAM Rev., 46: , Breda, Diekmann, Maset and Vermiglio, A numerical approach for investigating the stability of equilibria for structured population models, J. Biol. Dyn., 7:4-20, Breda, Maset and Vermiglio, Approximation of eigenvalues of evolution operators for linear retarded functional differential equations, SIAM J. Numer. Anal., 50: , Breda, Maset and Vermiglio, Pseudospectral approximation of eigenvalues of derivative operators with non-local boundary condition, Appl. Numer. Math., 56: , Breda, Maset and Vermiglio, Pseudospectral differencing methods for characteristic roots of delay differential equations, SIAM J. Sci. Comput., 27: , Breda and Van Vleck, Approximating Lyapunov exponents and Sacker-Sell spectrum for retarded functional differential equations, Numer. Math., online, 2013 from DDEs to ODEs (through PDEs) dynamical systems and BCAM 38/40
55 bibliography cont d 7 Diekmann, Getto and Gyllenberg, Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars, SIAM J. Math. Anal., 39: , Diekmann, Gyllenberg, Metz, Nakaoka and de Roos Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example, J. Math. Biol., 61: , Diekmann, van Gils, Verduyn Lunel and Walther, Delay Equations - Functional, Complex and Nonlinear Analysis, Springer AMS 110, Engel and Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer GTM 194, Gottlieb, Hussaini and Orszag, Theory and applications of spectral methods, Spectral methods for partial differential equations (Hampton 1982), SIAM, 1:54, Hale and Verduyn Lunel, Introduction to functional differential equations, Springer AMS 99, Hutchinson, Circular causal systems in ecology, Ann. N.Y. Acad. Sci., 50: , 1948 from DDEs to ODEs (through PDEs) dynamical systems and BCAM 39/40
56 bibliography cont d 14 Mackey and Glass, Oscillations and chaos in physiological control systems, Science, 197: , Trefethen, Spectral methods in MATLAB, SIAM SET, AUTO, 17 MATCONT, 18 DDE-BIFTOOL, delay/ddebiftool.shtml from DDEs to ODEs (through PDEs) dynamical systems and BCAM 40/40
Reliably computing all characteristic roots of delay differential equations in a given right half plane using a spectral method
Reliably computing all characteristic roots of delay differential equations in a given right half plane using a spectral method Zhen Wu Wim Michiels Report TW 596, May 2011 Katholieke Universiteit Leuven
More informationStability Theory for Nonnegative and Compartmental Dynamical Systems with Time Delay
1 Stability Theory for Nonnegative and Compartmental Dynamical Systems with Time Delay Wassim M. Haddad and VijaySekhar Chellaboina School of Aerospace Engineering, Georgia Institute of Technology, Atlanta,
More informationNumerical bifurcation analysis of delay differential equations
Numerical bifurcation analysis of delay differential equations Dirk Roose Dept. of Computer Science K.U.Leuven Dirk.Roose@cs.kuleuven.be Acknowledgements Many thanks to Koen Engelborghs Tatyana Luzyanina
More informationLyapunov stability ORDINARY DIFFERENTIAL EQUATIONS
Lyapunov stability ORDINARY DIFFERENTIAL EQUATIONS An ordinary differential equation is a mathematical model of a continuous state continuous time system: X = < n state space f: < n! < n vector field (assigns
More informationDIFFERENTIABLE SEMIFLOWS FOR DIFFERENTIAL EQUATIONS WITH STATE-DEPENDENT DELAYS
UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLI 23 DIFFERENTIABLE SEMIFLOWS FOR DIFFERENTIAL EQUATIONS WITH STATE-DEPENDENT DELAYS by Hans-Otto Walther Careful modelling of systems governed
More informationLa complessa dinamica del modello di Gurtin e MacCamy
IASI, Roma, January 26, 29 p. 1/99 La complessa dinamica del modello di Gurtin e MacCamy Mimmo Iannelli Università di Trento IASI, Roma, January 26, 29 p. 2/99 Outline of the talk A chapter from the theory
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 informationCharacterising small solutions in delay differential equations through numerical approximations. Neville J. Ford & Sjoerd M.
ISSN 136-1725 UMIST Characterising small solutions in delay differential equations through numerical approximations Neville J. Ford & Sjoerd M. Verduyn Lunel Numerical Analysis Report No. 381 A report
More informationSTABILITY AND BIFURCATION ANALYSIS ON DELAY DIFFERENTIAL EQUATIONS. Xihui Lin. Master of Science. Applied Mathematics
University of Alberta STABILITY AND BIFURCATION ANALYSIS ON DELAY DIFFERENTIAL EQUATIONS by Xihui Lin A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements
More informationAn Arnoldi method with structured starting vectors for the delay eigenvalue problem
An Arnoldi method with structured starting vectors for the delay eigenvalue problem Elias Jarlebring, Karl Meerbergen, Wim Michiels Department of Computer Science, K.U. Leuven, Celestijnenlaan 200 A, 3001
More informationExponential stability of families of linear delay systems
Exponential stability of families of linear delay systems F. Wirth Zentrum für Technomathematik Universität Bremen 28334 Bremen, Germany fabian@math.uni-bremen.de Keywords: Abstract Stability, delay systems,
More informationContinuation and bifurcation analysis of delay differential equations
Continuation and bifurcation analysis of delay differential equations Dirk Roose 1 and Robert Szalai 2 1 Department of Computer Science, Katholieke Universiteit Leuven, Belgium 2 Department of Engineering
More informationOn feedback stabilizability of time-delay systems in Banach spaces
On feedback stabilizability of time-delay systems in Banach spaces S. Hadd and Q.-C. Zhong q.zhong@liv.ac.uk Dept. of Electrical Eng. & Electronics The University of Liverpool United Kingdom Outline Background
More informationConverse Lyapunov-Krasovskii Theorems for Systems Described by Neutral Functional Differential Equation in Hale s Form
Converse Lyapunov-Krasovskii Theorems for Systems Described by Neutral Functional Differential Equation in Hale s Form arxiv:1206.3504v1 [math.ds] 15 Jun 2012 P. Pepe I. Karafyllis Abstract In this paper
More informationLecture 4: Numerical solution of ordinary differential equations
Lecture 4: Numerical solution of ordinary differential equations Department of Mathematics, ETH Zürich General explicit one-step method: Consistency; Stability; Convergence. High-order methods: Taylor
More information7 Planar systems of linear ODE
7 Planar systems of linear ODE Here I restrict my attention to a very special class of autonomous ODE: linear ODE with constant coefficients This is arguably the only class of ODE for which explicit solution
More informationNBA Lecture 1. Simplest bifurcations in n-dimensional ODEs. Yu.A. Kuznetsov (Utrecht University, NL) March 14, 2011
NBA Lecture 1 Simplest bifurcations in n-dimensional ODEs Yu.A. Kuznetsov (Utrecht University, NL) March 14, 2011 Contents 1. Solutions and orbits: equilibria cycles connecting orbits other invariant sets
More informationObjective. Single population growth models
Objective Single population growth models We are given a table with the population census at different time intervals between a date a and a date b, and want to get an expression for the population. This
More informationChapter III. Stability of Linear Systems
1 Chapter III Stability of Linear Systems 1. Stability and state transition matrix 2. Time-varying (non-autonomous) systems 3. Time-invariant systems 1 STABILITY AND STATE TRANSITION MATRIX 2 In this chapter,
More informationLaplace Transforms Chapter 3
Laplace Transforms Important analytical method for solving linear ordinary differential equations. - Application to nonlinear ODEs? Must linearize first. Laplace transforms play a key role in important
More informationDynamics of delay differential equations with distributed delays
Dynamics of delay differential equations with distributed delays Kiss, Gábor BCAM - Basque Center for Applied Mathematics Bilbao Spain February 1, 211 Outline Delay differential equation Stability charts
More informationRational Chebyshev pseudospectral method for long-short wave equations
Journal of Physics: Conference Series PAPER OPE ACCESS Rational Chebyshev pseudospectral method for long-short wave equations To cite this article: Zeting Liu and Shujuan Lv 07 J. Phys.: Conf. Ser. 84
More information2:1 Resonance in the delayed nonlinear Mathieu equation
Nonlinear Dyn 007) 50:31 35 DOI 10.1007/s11071-006-916-5 ORIGINAL ARTICLE :1 Resonance in the delayed nonlinear Mathieu equation Tina M. Morrison Richard H. Rand Received: 9 March 006 / Accepted: 19 September
More information2 One-dimensional models in discrete time
2 One-dimensional models in discrete time So far, we have assumed that demographic events happen continuously over time and can thus be written as rates. For many biological species with overlapping generations
More informationStability and Hopf bifurcation analysis of the Mackey-Glass and Lasota equations
Stability and Hopf bifurcation analysis of the Mackey-Glass and Lasota equations Sreelakshmi Manjunath Department of Electrical Engineering Indian Institute of Technology Madras (IITM), India JTG Summer
More informationSpectral Collocation Method for Parabolic Partial Differential Equations with Neumann Boundary Conditions
Applied Mathematical Sciences, Vol. 1, 2007, no. 5, 211-218 Spectral Collocation Method for Parabolic Partial Differential Equations with Neumann Boundary Conditions M. Javidi a and A. Golbabai b a Department
More informationSIMPLE CHAOTIC FLOWS WITH ONE STABLE EQUILIBRIUM
International Journal of Bifurcation and Chaos, Vol. 23, No. 11 (2013) 1350188 (7 pages) c World Scientific Publishing Company DOI: 10.1142/S0218127413501885 SIMPLE CHAOTIC FLOWS WITH ONE STABLE EQUILIBRIUM
More informationNumerical Bifurcation Analysis of Physiologically Structured Populations: Consumer-Resource, Cannibalistic and Trophic Models
Bulletin of Mathematical Biology manuscript No. (will be inserted by the editor) Numerical Bifurcation Analysis of Physiologically Structured Populations: Consumer-Resource, Cannibalistic and Trophic Models
More informationCollocation approximation of the monodromy operator of periodic, linear DDEs
p. Collocation approximation of the monodromy operator of periodic, linear DDEs Ed Bueler 1, Victoria Averina 2, and Eric Butcher 3 13 July, 2004 SIAM Annual Meeting 2004, Portland 1=Dept. of Math. Sci.,
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 informationExistence of permanent oscillations for a ring of coupled van der Pol oscillators with time delays
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 14, Number 1 (2018), pp. 139 152 Research India Publications http://www.ripublication.com/gjpam.htm Existence of permanent oscillations
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 informationSection 5.4 (Systems of Linear Differential Equation); 9.5 Eigenvalues and Eigenvectors, cont d
Section 5.4 (Systems of Linear Differential Equation); 9.5 Eigenvalues and Eigenvectors, cont d July 6, 2009 Today s Session Today s Session A Summary of This Session: Today s Session A Summary of This
More informationLecture 5. Numerical continuation of connecting orbits of iterated maps and ODEs. Yu.A. Kuznetsov (Utrecht University, NL)
Lecture 5 Numerical continuation of connecting orbits of iterated maps and ODEs Yu.A. Kuznetsov (Utrecht University, NL) May 26, 2009 1 Contents 1. Point-to-point connections. 2. Continuation of homoclinic
More informationSolution via Laplace transform and matrix exponential
EE263 Autumn 2015 S. Boyd and S. Lall Solution via Laplace transform and matrix exponential Laplace transform solving ẋ = Ax via Laplace transform state transition matrix matrix exponential qualitative
More informationSummary of topics relevant for the final. p. 1
Summary of topics relevant for the final p. 1 Outline Scalar difference equations General theory of ODEs Linear ODEs Linear maps Analysis near fixed points (linearization) Bifurcations How to analyze a
More informationNumerical Continuation of Bifurcations - An Introduction, Part I
Numerical Continuation of Bifurcations - An Introduction, Part I given at the London Dynamical Systems Group Graduate School 2005 Thomas Wagenknecht, Jan Sieber Bristol Centre for Applied Nonlinear Mathematics
More informationBIFURCATION PHENOMENA Lecture 4: Bifurcations in n-dimensional ODEs
BIFURCATION PHENOMENA Lecture 4: Bifurcations in n-dimensional ODEs Yuri A. Kuznetsov August, 2010 Contents 1. Solutions and orbits: equilibria cycles connecting orbits compact invariant manifolds strange
More informationON A NORMALIZATION TECHNIQUE FOR CODIMENSION TWO BIFURCATIONS OF EQUILIBRIA OF DELAY DIFFERENTIAL EQUATIONS
ON A NORMALIZATION TECHNIQUE FOR CODIMENSION TWO BIFURCATIONS OF EQUILIBRIA OF DELAY DIFFERENTIAL EQUATIONS.5.4.3.2.1 x 2.1.2.3.4.5.5.4.3.2.1.1.2.3.4.5 x 1 SEBASTIAAN G. JANSSENS Master Thesis Under supervision
More informationExistence of Secondary Bifurcations or Isolas for PDEs
Existence of Secondary Bifurcations or Isolas for PDEs Marcio Gameiro Jean-Philippe Lessard Abstract In this paper, we introduce a method to conclude about the existence of secondary bifurcations or isolas
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 informationStability of flow past a confined cylinder
Stability of flow past a confined cylinder Andrew Cliffe and Simon Tavener University of Nottingham and Colorado State University Stability of flow past a confined cylinder p. 1/60 Flow past a cylinder
More information21 Linear State-Space Representations
ME 132, Spring 25, UC Berkeley, A Packard 187 21 Linear State-Space Representations First, let s describe the most general type of dynamic system that we will consider/encounter in this class Systems may
More informationLMI Methods in Optimal and Robust Control
LMI Methods in Optimal and Robust Control Matthew M. Peet Arizona State University Lecture 20: LMI/SOS Tools for the Study of Hybrid Systems Stability Concepts There are several classes of problems for
More informationNumerical Methods for ODEs. Lectures for PSU Summer Programs Xiantao Li
Numerical Methods for ODEs Lectures for PSU Summer Programs Xiantao Li Outline Introduction Some Challenges Numerical methods for ODEs Stiff ODEs Accuracy Constrained dynamics Stability Coarse-graining
More informationOscillatory Mixed Di erential Systems
Oscillatory Mixed Di erential Systems by José M. Ferreira Instituto Superior Técnico Department of Mathematics Av. Rovisco Pais 49- Lisboa, Portugal e-mail: jferr@math.ist.utl.pt Sra Pinelas Universidade
More informationCalculus and Differential Equations II
MATH 250 B Second order autonomous linear systems We are mostly interested with 2 2 first order autonomous systems of the form { x = a x + b y y = c x + d y where x and y are functions of t and a, b, c,
More informationOSCILLATION AND ASYMPTOTIC STABILITY OF A DELAY DIFFERENTIAL EQUATION WITH RICHARD S NONLINEARITY
2004 Conference on Diff. Eqns. and Appl. in Math. Biology, Nanaimo, BC, Canada. Electronic Journal of Differential Equations, Conference 12, 2005, pp. 21 27. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu
More informationUnderstand the existence and uniqueness theorems and what they tell you about solutions to initial value problems.
Review Outline To review for the final, look over the following outline and look at problems from the book and on the old exam s and exam reviews to find problems about each of the following topics.. Basics
More informationSemigroup Generation
Semigroup Generation Yudi Soeharyadi Analysis & Geometry Research Division Faculty of Mathematics and Natural Sciences Institut Teknologi Bandung WIDE-Workshoop in Integral and Differensial Equations 2017
More informationReaction-Diffusion Models and Bifurcation Theory Lecture 11: Bifurcation in delay differential equations
Reaction-Diffusion Models and Bifurcation Theory Lecture 11: Bifurcation in delay differential equations Junping Shi College of William and Mary, USA Collaborators/Support Shanshan Chen (PhD, 2012; Harbin
More informationSolution of Linear State-space Systems
Solution of Linear State-space Systems Homogeneous (u=0) LTV systems first Theorem (Peano-Baker series) The unique solution to x(t) = (t, )x 0 where The matrix function is given by is called the state
More informationMATH 251 Final Examination December 16, 2015 FORM A. Name: Student Number: Section:
MATH 5 Final Examination December 6, 5 FORM A Name: Student Number: Section: This exam has 7 questions for a total of 5 points. In order to obtain full credit for partial credit problems, all work must
More informationBifurcation in a Discrete Two Patch Logistic Metapopulation Model
Bifurcation in a Discrete Two Patch Logistic Metapopulation Model LI XU Tianjin University of Commerce Department of Statistics Jinba road, Benchen,Tianjin CHINA beifang xl@63.com ZHONGXIANG CHANG Tianjin
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 informationFinding periodic orbits in state-dependent delay differential equations as roots of algebraic equations
Finding periodic orbits in state-dependent delay differential equations as roots of algebraic equations Jan Sieber arxiv:1010.2391v8 [math.ds] 26 Feb 2015 July 6, 2017 In this paper we prove that periodic
More informationAvailable online at ScienceDirect. Procedia IUTAM 19 (2016 ) IUTAM Symposium Analytical Methods in Nonlinear Dynamics
Available online at www.sciencedirect.com ScienceDirect Procedia IUTAM 19 (2016 ) 11 18 IUTAM Symposium Analytical Methods in Nonlinear Dynamics A model of evolutionary dynamics with quasiperiodic forcing
More informationA Numerical Solution of the Lax s 7 th -order KdV Equation by Pseudospectral Method and Darvishi s Preconditioning
Int. J. Contemp. Math. Sciences, Vol. 2, 2007, no. 22, 1097-1106 A Numerical Solution of the Lax s 7 th -order KdV Equation by Pseudospectral Method and Darvishi s Preconditioning M. T. Darvishi a,, S.
More informationLinear ODEs. Existence of solutions to linear IVPs. Resolvent matrix. Autonomous linear systems
Linear ODEs p. 1 Linear ODEs Existence of solutions to linear IVPs Resolvent matrix Autonomous linear systems Linear ODEs Definition (Linear ODE) A linear ODE is a differential equation taking the form
More informationOn Sontag s Formula for the Input-to-State Practical Stabilization of Retarded Control-Affine Systems
On Sontag s Formula for the Input-to-State Practical Stabilization of Retarded Control-Affine Systems arxiv:1206.4240v1 [math.oc] 19 Jun 2012 P. Pepe Abstract In this paper input-to-state practically stabilizing
More informationAUTOMATIC CONTROL. Andrea M. Zanchettin, PhD Spring Semester, Introduction to Automatic Control & Linear systems (time domain)
1 AUTOMATIC CONTROL Andrea M. Zanchettin, PhD Spring Semester, 2018 Introduction to Automatic Control & Linear systems (time domain) 2 What is automatic control? From Wikipedia Control theory is an interdisciplinary
More informationHopf bifurcations, and Some variations of diffusive logistic equation JUNPING SHIddd
Hopf bifurcations, and Some variations of diffusive logistic equation JUNPING SHIddd College of William and Mary Williamsburg, Virginia 23187 Mathematical Applications in Ecology and Evolution Workshop
More informationAbstract. This thesis concerns the development of a method for the detection of small
Abstract This thesis concerns the development of a method for the detection of small solutions to delay differential equations. The detection of small solutions is important because their presence has
More informationNumerical Continuation and Normal Form Analysis of Limit Cycle Bifurcations without Computing Poincaré Maps
Numerical Continuation and Normal Form Analysis of Limit Cycle Bifurcations without Computing Poincaré Maps Yuri A. Kuznetsov joint work with W. Govaerts, A. Dhooge(Gent), and E. Doedel (Montreal) LCBIF
More informationETNA Kent State University
Electronic Transactions on Numerical Analysis. Volume 35, pp. 2-26, 29. Copyright 29,. ISSN 68-963. ETNA GAUSSIAN DIRECT QUADRATURE METHODS FOR DOUBLE DELAY VOLTERRA INTEGRAL EQUATIONS ANGELAMARIA CARDONE,
More informationStability of an abstract wave equation with delay and a Kelvin Voigt damping
Stability of an abstract wave equation with delay and a Kelvin Voigt damping University of Monastir/UPSAY/LMV-UVSQ Joint work with Serge Nicaise and Cristina Pignotti Outline 1 Problem The idea Stability
More informationSyed Abbas. June 19th, 2009
Almost Department of Mathematics and Statistics Indian Institute of Technology Kanpur, Kanpur, 208016, India June 19th, 2009 Harald Bohr Function Almost Bohr s early research was mainly concerned with
More informationIdentification Methods for Structural Systems
Prof. Dr. Eleni Chatzi System Stability Fundamentals Overview System Stability Assume given a dynamic system with input u(t) and output x(t). The stability property of a dynamic system can be defined from
More informationWHERE TO PUT DELAYS IN POPULATION MODELS, IN PARTICULAR IN THE NEUTRAL CASE
CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 11, Number 2, Summer 2003 WHERE TO PUT DELAYS IN POPULATION MODELS, IN PARTICULAR IN THE NEUTRAL CASE K. P. HADELER AND G. BOCHAROV ABSTRACT. Hutchinson s
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 informationA REMARK ON THE GLOBAL DYNAMICS OF COMPETITIVE SYSTEMS ON ORDERED BANACH SPACES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 A REMARK ON THE GLOBAL DYNAMICS OF COMPETITIVE SYSTEMS ON ORDERED BANACH SPACES KING-YEUNG LAM
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 informationComplex Dynamic Systems: Qualitative vs Quantitative analysis
Complex Dynamic Systems: Qualitative vs Quantitative analysis Complex Dynamic Systems Chiara Mocenni Department of Information Engineering and Mathematics University of Siena (mocenni@diism.unisi.it) Dynamic
More informationDYNAMICS IN 3-SPECIES PREDATOR-PREY MODELS WITH TIME DELAYS. Wei Feng
DISCRETE AND CONTINUOUS Website: www.aimsciences.org DYNAMICAL SYSTEMS SUPPLEMENT 7 pp. 36 37 DYNAMICS IN 3-SPECIES PREDATOR-PREY MODELS WITH TIME DELAYS Wei Feng Mathematics and Statistics Department
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 informationStrong stability of neutral equations with dependent delays
Strong stability of neutral equations with dependent delays W Michiels, T Vyhlidal, P Zitek, H Nijmeijer D Henrion 1 Introduction We discuss stability properties of the linear neutral equation p 2 ẋt +
More informationNonlinear Systems and Control Lecture # 12 Converse Lyapunov Functions & Time Varying Systems. p. 1/1
Nonlinear Systems and Control Lecture # 12 Converse Lyapunov Functions & Time Varying Systems p. 1/1 p. 2/1 Converse Lyapunov Theorem Exponential Stability Let x = 0 be an exponentially stable equilibrium
More informationDepartment of Mathematics IIT Guwahati
Stability of Linear Systems in R 2 Department of Mathematics IIT Guwahati A system of first order differential equations is called autonomous if the system can be written in the form dx 1 dt = g 1(x 1,
More informationHalf of Final Exam Name: Practice Problems October 28, 2014
Math 54. Treibergs Half of Final Exam Name: Practice Problems October 28, 24 Half of the final will be over material since the last midterm exam, such as the practice problems given here. The other half
More informationKey words. delay differential equations, characteristic matrix, stability of periodic orbits
CHARACTERISTIC MATRICES FOR LINEAR PERIODIC DELAY DIFFERENTIAL EQUATIONS JAN SIEBER AND ROBERT SZALAI Abstract. Szalai et al. (SIAM J. on Sci. Comp. 28(4), 2006) gave a general construction for characteristic
More informationOscillation Onset In. Neural Delayed Feedback
Oscillation Onset In Neural Delayed Feedback Andre Longtin Complex Systems Group and Center for Nonlinear Studies Theoretical Division B213, Los Alamos National Laboratory Los Alamos, NM 87545 Abstract
More informationcomputer session iv: Two-parameter bifurcation analysis of equilibria and limit cycles with matcont
computer session iv: Two-parameter bifurcation analysis of equilibria and limit cycles with matcont Yu.A. Kuznetsov Department of Mathematics Utrecht University Budapestlaan 6 3508 TA, Utrecht May 16,
More informationDynamics of Modified Leslie-Gower Predator-Prey Model with Predator Harvesting
International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:13 No:05 55 Dynamics of Modified Leslie-Gower Predator-Prey Model with Predator Harvesting K. Saleh Department of Mathematics, King Fahd
More informationIntroduction to Applied Nonlinear Dynamical Systems and Chaos
Stephen Wiggins Introduction to Applied Nonlinear Dynamical Systems and Chaos Second Edition With 250 Figures 4jj Springer I Series Preface v L I Preface to the Second Edition vii Introduction 1 1 Equilibrium
More informationOn quasiperiodic boundary condition problem
JOURNAL OF MATHEMATICAL PHYSICS 46, 03503 (005) On quasiperiodic boundary condition problem Y. Charles Li a) Department of Mathematics, University of Missouri, Columbia, Missouri 65 (Received 8 April 004;
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 informationGlobal Stability Analysis on a Predator-Prey Model with Omnivores
Applied Mathematical Sciences, Vol. 9, 215, no. 36, 1771-1782 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ams.215.512 Global Stability Analysis on a Predator-Prey Model with Omnivores Puji Andayani
More informationOne dimensional Maps
Chapter 4 One dimensional Maps The ordinary differential equation studied in chapters 1-3 provide a close link to actual physical systems it is easy to believe these equations provide at least an approximate
More informationON A NORMALIZATION TECHNIQUE FOR CODIMENSION TWO BIFURCATIONS OF EQUILIBRIA OF DELAY DIFFERENTIAL EQUATIONS
ON A NORMALIZATION TECHNIQUE FOR CODIMENSION TWO BIFURCATIONS OF EQUILIBRIA OF DELAY DIFFERENTIAL EQUATIONS SEBASTIAAN G. JANSSENS Master Thesis Under supervision of DR. YURI A. KUZNETSOV Co-supervised
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 informationNumerical techniques: Deterministic Dynamical Systems
Numerical techniques: Deterministic Dynamical Systems Henk Dijkstra Institute for Marine and Atmospheric research Utrecht, Department of Physics and Astronomy, Utrecht, The Netherlands Transition behavior
More informationNUMERICAL SOLUTION OF DELAY DIFFERENTIAL EQUATIONS VIA HAAR WAVELETS
TWMS J Pure Appl Math V5, N2, 24, pp22-228 NUMERICAL SOLUTION OF DELAY DIFFERENTIAL EQUATIONS VIA HAAR WAVELETS S ASADI, AH BORZABADI Abstract In this paper, Haar wavelet benefits are applied to the delay
More informationOn the periodic logistic equation
On the periodic logistic equation Ziyad AlSharawi a,1 and James Angelos a, a Central Michigan University, Mount Pleasant, MI 48858 Abstract We show that the p-periodic logistic equation x n+1 = µ n mod
More informationChem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Fall Semester 2006 Christopher J. Cramer. Lecture 5, January 27, 2006
Chem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Fall Semester 2006 Christopher J. Cramer Lecture 5, January 27, 2006 Solved Homework (Homework for grading is also due today) We are told
More informationELEC 3035, Lecture 3: Autonomous systems Ivan Markovsky
ELEC 3035, Lecture 3: Autonomous systems Ivan Markovsky Equilibrium points and linearization Eigenvalue decomposition and modal form State transition matrix and matrix exponential Stability ELEC 3035 (Part
More informationNumerical solution of linear time delay systems using Chebyshev-tau spectral method
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 93-9466 Applications and Applied Mathematics: An International Journal (AAM) Vol., Issue (June 07), pp. 445 469 Numerical solution of linear time
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 informationMTH 464: Computational Linear Algebra
MTH 464: Computational Linear Algebra Lecture Outlines Exam 4 Material Prof. M. Beauregard Department of Mathematics & Statistics Stephen F. Austin State University April 15, 2018 Linear Algebra (MTH 464)
More informationHOPF BIFURCATION CALCULATIONS IN DELAYED SYSTEMS
PERIODICA POLYTECHNICA SER. MECH. ENG. VOL. 48, NO. 2, PP. 89 200 2004 HOPF BIFURCATION CALCULATIONS IN DELAYED SYSTEMS Gábor OROSZ Bristol Centre for Applied Nonlinear Mathematics Department of Engineering
More informationDynamical Systems & Lyapunov Stability
Dynamical Systems & Lyapunov Stability Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University Outline Ordinary Differential Equations Existence & uniqueness Continuous dependence
More information