Validated numerics for continuation and bifurcation of connecting orbits of maps

Transcription

1 Validated numerics for continuation and bifurcation of connecting orbits of maps Ronald Adams 1 and JD Mireles James 2 1 Embry-Riddle Aeronautical University, Department of Mathematics 2 Florida Atlantic University, Department of Mathematical Sciences July 19, 2017 Abstract Validated numerical computations are used to study one parameter branches of connecting orbits and their bifurcations The idea is to formulate the connecting orbit as a zero of a certian finite dimensional map Computer assisted methods of proof for continuation and bifurcation of zeros are applied to the map, providing global results for the underlying dynamical system We use the parameterization method to represent the local stable/unstable manifolds of the fixed point, and focus on the example of the Hénon map 1 Introduction This paper describes self validating numerical algorithms for mathematically rigorous computation of smooth branches of connecting orbits, and bifurcations of these branches We focus on discrete time dynamical systems with a single parameter More precisely, let f : R n R R n be a one parameter family of diffeomorphisms and suppose that for some fixed λ 1 R there is a hyperbolic fixed point ˆp R n, ie that f(ˆp, λ 1 ) = ˆp, and that Df(ˆp, λ 1 ) (which is an invertible matrix as f is a diffeomorphism) has no eigenvalues on the unit circle Then, by the implicit function theorem, there is a ɛ > 0 and a smooth function p: (λ 1 ɛ, λ 1 + ɛ) R n so that p(λ 1 ) = ˆp and f(p(λ)) = p(λ), RA partially supported by NSF grant DMS ADAMSR25@erauedu JMJ partially supported by NSF grant DMS and by the Alfred P Sloan Foundation grant G jmirelesjames@fauedu 1

2 for all λ (λ 1 ɛ, λ 1 + ɛ) Moreover, ɛ can be chosen small enough so that Df(p(λ), λ) is hyperbolic for all λ (λ 1 ɛ, λ 1 + ɛ) Hyperbolicity for each such λ guarantees that there are stable and unstable manifolds attached to the p(λ), and that the sum of their dimensions equals the dimension of the phase space We denote this one parameter family of manifolds by W s (λ) and W u (λ), suppressing the dependance on p(λ) and f(, λ) Recall that the manifolds depend smoothly on λ Once again fixing λ = λ 1, we say that a point x 1 R n is homoclinic for the fixed point ˆp under the map f(, λ 1 ) if lim n ± f n (x 1, λ 1 ) = ˆp Note that a homoclinic point x 1 must, by definition, lie in the intersection of the stable and unstable manifolds of p 1 : ie x 1 W s (λ 1 ) W u (λ 1 ) We say that x 1 is a transverse homoclinic point if span (T x1 W s (λ 1 ), T x1 W u (λ 1 )) = R n, ie if the stable and unstable manifolds of ˆp intersect transversally at x 1 Since the manifolds vary smoothly in λ, and since transverse intersections are robust, we can take ɛ > 0 so small that there is a transverse intersection point for each λ (λ 1 ɛ, λ 1 + ɛ) More precisely, there is a smooth function x 1 : (λ 1 ɛ, λ 1 + ɛ) R n having that x(λ 1 ) = x 1 and that x 1 (λ) W s (λ) W u (λ), for all λ (λ 1 ɛ, λ 1 + ɛ) The situation is depicted pictorially in Figure 1 Note that a smooth branch of connecting orbits continues until (a) - the transversality assumption breaks down, or (b) the hyperbolicity of the fixed point is lost Restricting to co-dimension one bifurcations of type (a), we see that a loss of transversality occurs when two connecting orbits collide and disappear as illustrated in Figure 1, ie when manifold intersections are born or when they vanish In the language of catastrophe theory we have that case (a) corresponds to a saddle-node bifurcation of the branch x(λ) which in this setting is a branch of connecting orbits Consideration of case (b) is more delicate, as the loss of hyperbolicity introduces center manifold theory into the conversation We postpone further discussion of case (b) to a future study In the remainder of the paper we focus on developing computational techniques for mathematically rigorous study of smooth branches of connecting orbits, and their bifurcations as just discussed Our methods are designed so that, in addition to proving abstract existence results, we obtain validated numerical bounds on the distance between the true and approximate connections The following developments guide our approach The method of projected boundaries connecting orbit as a zero of a higher dimensional operator: we first reformulate the connecting orbit as an 2

3 W s ( 1) W s ( 2) x1( 1) W s ( 3) x1( ) x1( 1) x1( 2) =x2( 2) x1( 2) =x2( 2) x2( 1) x2( ) x2( 1) W u ( 3) W u ( 2) W u ( 1) Figure 1: Left: transverse intersection of the stable/unstable manifolds giving rise to a family of connecting orbits As the parameter changes the intersection moves closer to a tangency, at which point the intersection is broken Right: the associated fold bifurcation for the homoclinic point isolated zero of a certain nonlinear funciotn, which we call the connecting orbit operator (see Section 3 for the precise formulation of this notion) The idea is to think of the connecting orbit as the solution of a discrete time boundary value problem, with boundary conditions on the local stable/unstable manifolds This classic approach is sometimes referred to as the method of projected boundaries [13, 7, 5, 6, 12, 11] Validated numerics for the method of projected boundaries: Validated numerics refers to a whole suite of techniques in numerical analysis, which result in mathematically rigorous existence as well as tight error bounds between the true and approximate solution of the problem Many such techniques are based on a-posteriori analysis, ie we first compute an approximate numerical solution of our problem and then we analyze the results A successful validated numerical scheme involves choosing (or developing) an appropriate a-posteriori existence and uniqueness theorem (sometimes called a constructive implicit function theorem or a Newton-Kantorovich theorem), followed by careful hypothesis checking using often using interval arithmetic to manage round off errors Much more discussion, as well as more thorough review of the literature, is found in the review articles [24, 36, 34, 23], as well as the books [25, 35, 14] Validated numerical methods based on the method of projected boundaries are developed in [31, 28], and provide computer assisted proofs for connecting orbits in discrete and continuous time dynamical systems The main idea is to apply rigorous zero finding/enclosing schemes to the appropriate connecting 3

4 orbit operators In addition to producing abstract existence and enclosure results, the methods of [31, 28] provide automatically the transversality of the connection Validated numerics for rigorous branch following: A number of authors have developed validated numerical methods for studying smooth branches of zeros The studies most closely related to the present work include the references [10, 37, 4, 33, 38, 8] The interested reader will find many additional references therein We refer also the the Lecture notes [26] Validated numerics for a saddle node bifurcation: There also exist validated numerical methods for studying bifurcations of solutions of operator equations See for example the works of [33, 4, 27] Our approach is adapted directly from the last of these references The techniques discussed above comprise the main ingredients in the present work We exploit the formulation of connecting orbits as zeros of nonlinear operator equations, in concert with validated numerical methods for rigorous branch following, to prove the existence of smooth branches of connecting orbits with tight error bounds Moreover, validated numerical techniques for saddle node bifurcation analysis allow us to study branches of connecting orbits from birth do death, at least in the case that the underlying fixed point itself undergoes no stability bifurcation Remark 11 The present study is closely related to the recent work of [8], which treats heteroclinic arcs arising from the transverse intersection of a k- dimensional unstable manifold and an m-dimensional stable manifold, where k + m = n + 1 Note that these intersection arcs occur in fixed dynamical systems, ie in contrast to the parameter dependent problems considered in the present work, no parameters are involved in the work of [8] A more technical remark concerns the treatment of function compositions, which are notoriously problematic for interval arithmetics The work of [8] exploits a Lohner type scheme based on coordinate frames adapted to the expanding and contracting directions of f along the connecting orbit Such Lohner schemes are useful for mitigating the so called wrapping effect [29, 39] In the present work we exploit a multiple shooting scheme which has been used successfully in recent studies to mitigate the wrapping effect [31, 18] The idea is to introduce additional variables and solve an expanded system of equations which involves no compositions of the map f This scheme minimizes the wrapping effect while avoiding the use of special coordinates, at the cost of increasing the dimension of the problem The present work builds also on the study of of [30], where the second author developed validated numerical methods for studying one parameter families of stable/unstable manifolds The techniques developed in the reference just cited allow us to include the systems parameter in the connecting orbit operator We also mention the related work of [3, 2, 1], on existence and continuation of homoclinic tangencies in two parameter families of discrete time dynamical systems The authors prove the existence of tangencies (ie a saddle node 4

5 bifurcation point for a branch of connecting orbits), and study the behavior of such points as the two parameters are varied The authors use algebraic topological constructions based on the Conley index theory However branches of transverse homoclinic points are not studied in the reference just cited 11 Our main example We implement our method for the particular example of the classical Hénon map ( ) 1 + y λx 2 f(x, y) =, (1) bx with b = 03 The simple form of the map eases the presentation of the technical material to follow We consider the system with λ the parameter The dynamics as a function of λ are illustrated in the four frames of Figure 2, and also in Figure 5 In particular, the images illustrate the fact that homoclinic intersection points are prevalent in the Hénon system 2 The parameterization method The parameterization method is a general functional analytic framework for studying invariant manifolds For a high level introduction to the method, we refer to the series of papers [15, 16, 17, 20, 19, 21] The main idea is to formulate an operator equation describing a chart or covering map for the desired manifold This is usually done by considering a dynamical conjugacy characterizing the desired manifold The present section reviews the parameterization method only as used in the remainder of the present work: namely for one dimensional local stable/unstable manifolds of planar analytic maps The primary reference for the material in this section is [30], however the reader interested in a much more complete treatment of the parameterization method can refer to the recent book on the topic [22] In particular the book contains an exhaustive overview of the literature Consider a smooth one parameter family of diffeomorphisms f : R 2 R R 2 Let U = (λ 1 ɛ, λ 1 + ɛ) and suppose that p: U R 2 is a smooth branch of hyperbolic fixed points Let κ u, κ s : U R denote smooth branches of eigenvalues, and ξ u, ξ s : U R 2 denote assocated smooth branches of eigenvectors We focus momentarily on the unstable manifold The parameterization method seeks a function P : ( 1, 1) U R 2 satisfying the conjugacy equation f(p (θ, λ)) = P (κ u (λ)θ), (2) for all θ ( 1, 1), and subject to the first order constraints P (0, λ) = p(λ), and θ P (0, λ) = ξ u(λ) 5

6 Figure 2: The figure illustrates the dynamics of the Hénon map at three different parameter values In each of the four windows the x y axes are the phase space variables for the Hénon map The parameter λ is on the z axis The coordinate frame is based at the classical parameter values of the system, ie at λ = 0 In the top left frame we see the stable/unstable manifolds of the two fixed points of Hénon, when the parameter has λ = 02 Note for example that the red (stable) and blue (unstable) manifolds intersect transversally, giving rise to a homoclinic connecting orbit The top right frame illustrates the dynamics at λ = 0 and again there appear to be homoclinic connections (the shadow of the attractor at λ = 02 remains) The bottom left frame illustrates the dynamics at λ = 02 (with the shadows of the two previous attractors) The homoclinic connection seems to persist for all λ [ 02, 02] The bottom right frame illustrates the one parameter families of local stable/unstable manifolds for the entire parameter interval These manifolds are critical in our continuation and bifurcation analysis for connecting orbits Such a function P parameterizes a local unstable manifold for f at p Figure 3 illustrates the meaning of Equation 2 for a fixed parameter value Figure 4 illustrates the meaning of the parameter dependance 6

7 R 2 P ( ) f R 2 P ( ) P p 0 P f(p ( )) = P ( ) R R L( ) = Figure 3: graphical illustration of the conjugacy relation exploited in the formulation of the parameterization method A map P is sought which conjugates the linear dynamics in the parameter space to the nonlinear dynamics restricted to the local manifold 21 Formal series expansions To compute a numerical approximation of P solving Equation (2) we exploit power series methods If f is analytic then p, κ u,s, ξ u,s, and P depend analytically on λ So, let p(λ) = p m λ m, κ u = m=0 κ m λ m, and ξ u (λ) = m=0 denote the power series expansions We also write κ n u(λ) = κ mn λ m, m=0 ξ m λ m, to denote the power series expansion of the eigenvalue raised to the n-th power Methods for computing and validating these series expansions are discussed in [30] We desire a solution P of equation (2) having power series expansion P (θ, λ) = n=0 m=0 m=0 p mn λ m θ n (3) 7

8 p( 2 ) R 2 ( 2 ) P ([ 1, 1], 2) =W u loc (p( 2)) x P ([ 1, 1], 1) =W u loc (p( 1)) p( 1 ) y R 2 ( 1 ) Figure 4: Branch of parameterized manifolds associated with a branch of hyperbolic fixed points Imposing the first order constraints gives p 0n = p n, and p 1n = ξ n, for all n N Using Equation (3) we expand Equation (2) as a power series, match like powers and solve for the n-th order coefficient This leads to homological equations describing the coefficients p mn when n 2 For example when f is the Hénon map defined in Equation (5) we have that f(p (θ, λ)) = ( 1 + m=0 n=0 p2 mnλ m θ m λ m n m=0 n=0 j=0 k=0 p1 m j,n k p1 j,k λm θ n ) b m=0 n=0 p1 mnλ m θ n + m=1 n=0 p1 m 1,nλ m θ n, and that P (κ(λ)θ, λ) = m n=0 m=0 k=0 κ m k,n ( p 1 k,n p 2 k,n ) λ m θ m Setting these equal to one another and matching like powers of λ and θ gives p 2 mn m j=0 k=0 n λp 1 m j,n kp 1 j,k = κ m k,n p k,n, and bp 1 mn + p 1 m 1,n = κ m k,n p k,n 8

9 Isolating the dependance on the (m + n)-th order term gives the homological equation for the coefficients ( ) [ ] [ ] 2λp 1 00 λ n 0 1 p 1 mn s 1 b λ n 0 p 2 = mn mn s 2, (4) mn where and where mn s 1 mn = κ m j,n p 1 jn + j=0 n m k=0 j=0 λˆδ mn jk p 1 m j,n kp 1 jk, m 1 s 2 mn = p 1 m 1,n + κ m j,n p 2 jn, ˆδ mn jk = j=0 0 if j = m and k = n 0 if j = 0 and k = 0 1 otherwise These expressions allow us to compute the Series solution P (θ, λ) to any desired finite order m + n = N The manifold surfaces illustrated in Figures 5 and 2 are computed using these formulae Suppose now that P MN (θ, λ) = M m=0 n=0 N ( p 1 mn p 2 mn ) λ m θ n, where the coefficients are the numerically computed solutions of the homological equation (4) The main result of [30] is that there exist τ, r > 0 and an analytic function H : ( r, r) ( τ, τ) R 2, so that P (θ, λ) = P MN (θ, λ) + H(θ, λ), is an exact solution of the invariance equation (2) provide explicit bounds of the form The results of [30] also sup H(θ, λ) ɛ, (θ,λ) ( r,r) ( τ,τ) where ɛ > 0 depends on N, M N, and on estimates which involve only the map f, the fixed point, and its eigenvalues/eigenvectors These bounds are quite technical and we refer the interested reader to [30] In the remainder of the present work we use the results of [30] freely and without further comment 9

10 Figure 5: The figure illustrates larger local stable/unstable manifolds for the same parameters considered in Figure 2 3 Connecting orbits: the method of projected boundaries For a smooth diffeomorphism f on R n with a hyperbolic fixed point p R n, a homoclinic orbit (x n ) n Z based at p satisfies x n+1 = f(x n ), lim x n = p n ± The authors of [5, 7, 6] use Newton s method to solve the system x n+1 = f(x n ), n = n,, n + 1 b (x n ) = b + (x n+ ) = 0 where the zero sets of b and b + are linear approximations to the local unstable and stable manifolds at the fixed point We consider a slightly modified system 10

11 P ( ) =x 1 f(x 1 )=x 2 image(q) =W s loc (p) f(x k 1 )=x k p f(x k )=Q( ) image(p )=W u loc (p) Figure 6: Graphical illustration of the multiple shooting scheme defining the connecting orbit operator and define the following connecting orbit operator F : R n(k+1) R n(k+1) f 1 (x 1 ) Q(θ) F (x, θ, φ) = f(x j ) x j+1, j = 1,, k 1 f(x k ) P (φ) Here P, Q are local parametrizations of the stable and unstable manifold, where θ W u R nu and φ W s R ns and n u + n s = n It is then apparent that if F ( x, θ, φ) = 0 then { x i } lies on a connecting orbit which is either homoclinic or heteroclinic depending on whether the local parametrization P is centered at p or another fixed point The validation of the local parametrizations P and Q can be found in [32] where as the theory is developed in [15, 16, 17] Let P be a parameterization for the stable manifold of f : R n R n at p represented by 11

12 P (θ) = α >0 a αθ α In what s to come revisit the Hénon map with parameters a = 14, b = 03: { xn+1 = 1 ax H(a,b) 2 n + y n y n+1 = bx n (5) Recall that the computation of the coefficients a α is done by exploiting the conjugacy equation f(p (θ)) = P (Λ s θ) (6) which leads to solving the following system (homological equation): [Df(p) (λ α1 1 λα k k )Id] a α = s α, α 2, (7) the terms λ 1,, λ k are the eigenvalues of the diagonal matrix Λ s Here the function s α is in terms of the lower order coefficients a β for β < α The system (7) is a linear system of equations involving only the local data p 0, Df(p 0 ) and the stable eigenvalues λ 1,, λ k Note that in (7) a α is well defined as long as λ α1 1 λα k k λ i, that is there is no resonance in α In the case of the Hénon map the homological equation is ( n 1 ) a s n = k=1 a1 n k a1 k 0 Having computed the coefficients a α we can define the truncation error h N for the power series P determined by an N th order approximation P N given by h N = P P N With this in mind we can recast the connecting orbit operator as f 1 (x 1 ) Q Nu (θ) h u (θ) F (x, θ, φ) = f 1 (x j ) x j+1 f(x j ) x j , (8) } f(x k ) P Ns (φ) {{ } h s (φ) }{{} F N H where x R kn The idea here is to find an approximate zero x = ( x 0, θ, φ) R n(k+1) of (8) using Newton s method, that is F ( x) 0 Then using the methods in [27], in particular the radii polynomial approach we can establish existence of a true zero x = (x 0, θ, φ ) near x Now we 12

13 are ready to rigorously compute a transverse homoclinic orbit for (5) For the paramater values a = 14, b = 03 the Henón map H(14, 03) takes the form f(x, y) = f 1 (x, y), f 2 (x, y), where f 1 (x, y) = 1 14x 2 + y, f 2 (x, y) = 03x The approximating connecting orbit operator F N is computed to be Q 1 N u (θ) x 1 Q 2 N u (θ) x x x 2 x 3 03x 1 x 4 F N (x 1,, x n, θ, φ) = 1 14x 2 n 3 + x n 2 x n 1 03x n 3 x n 1 14x 2 n 1 + x n PN 1 s (φ) 03x n 1 PN 2 s (φ) Given an initial guess x with F N ( x) 0, we compute the 2(n + 2) 2(n + 2) Jacobian matrix Q Nu ( θ) θ 0 Q Nu ( θ) θ 0 28 x DF N ( x) = (9) P x n Ns ( φ) φ P 2 Ns ( φ) φ Set A = [DF N ( x)] 1, and let T (x) := x AF N (x) To apply the radii polynomial approach we compute the bounds Y, Z(r) satisfying Theorem 2 in [27] To determine Y note that T ( x) x = AF N ( x) = [DF N ( x)] 1 F N ( x) With this in mind we can compute Y = (Y 1,, Y n ) such that [T ( x) x] k Y k Furthermore we determine Z(r) such that sup DT k ( x + b)c Z k (r), k = 1,, n b,c B r(0) In order to simplify the computation of Z k we rescale the variables b and c For r > 0, let b := b r, c := c r, with this the bounds become sup DT k ( x + b) c Z k (r), k = 1,, n b, c B1(0) 13

14 To simplify some of the technical estimates we consider the splitting ( ) DT ( x + br) cr = I ADF N ( x + br) cr ( ) = (I ADF N ( x)) cr A DF N ( x + br) DF N ( x) cr ( ) = A DF N ( x + br) DF N ( x) cr where A = DF N ( x) 1 To bound the second term in the splitting we note that ( DF N ( x + br) ) ( DF N ( x) cr = D 2 F N ((1 t) x + t( x + br)) b ) cr 2, for some t (0, 1) Since the map is quadratic in (x 1,, x n ) and we have uniform bounds for the manifold parametrizations the above expression can be bounded as follows DF N ( x + br) DF N ( x) 1 n r ( sup D 2 ) F N ( )1 n 1n r 2 (10) To obtain the appropriate bounds for the manifold parametrizations in (10) we appeal to the Cauchy estimates in [32] Set Z i (r) = [( sup D 2 ) F N ( )1 n 1n, i = 1,, n ]i Therefore the radii polynomials are defined to be p i (r) = Z i (r)r 2 r + Y i For instance using Newton s method we can compute the following initial approximation x =

15 with a tolerance ɛ = 10 15, that is F N ( x) < ɛ The interval containing all admissible radii values for existence is I = [ , ] 4 Continuation of connecting orbits In many applications it is important to allow for a parameter in the map being studied, therefore we consider the problem of finding connecting orbits for f(x, λ), where f : R n R m R n Given a connecting orbit of f, which is a zero of the corresponding connecting orbit map F, that is F (x 0, λ 0 ) = 0, we can try to find a continuous parametrized function α : V R n where V R m is a neighborhood of λ 0 such that α(λ 0 ) = x 0 with f(α(λ), λ) = 0 This amounts to an application of the implicit function theorem In this section we consider examples with m = 1 and consider the following IVP dα dλ = G(α, λ) := [D xf (α, λ)] 1 F λ (α, λ), α(λ 0) = x 0, F (x 0, λ 0 ) = 0 (11) If D x F (x, λ) is invertible and G is smooth then by standard existence/uniqueness results there exists a smooth solution α : J R n where J R is a neighborhood of λ 0 From (11) D x F (x, λ) dα dλ + F (x, λ) = 0 λ d F (α, λ) = 0 dλ F (α, λ) = f(α(λ 0 ), λ 0 ) Since we assume F (α(λ 0 ), λ 0 ) = 0 we have that F (α(λ), λ) = 0 Now computing the exact form of α can be an arduous task, but using that α is a solution to the differential equation (11) we can compute an approximation to α using the standard Euler approximation and set x 1 := x 0 + hg(x 0, λ 0 ) (12) λ 1 := λ 0 + h, (13) where h > 0 is a sufficiently small step size With this choice of h, we can then expect that F (x 1, λ 1 ) 0, which would then lead to the Newton operator T (x) := x [D x F (x 1, λ 1 )] 1 F (x, λ 1 ) being a contraction mapping, that is 0 F (T (x 1 ), λ 1 ) F (x 1, λ 1 ) which shows that T (x 1 ) is a better approximation of a zero of F at λ 1 than x 1 15

16 What we have essentially done is execute a predictor-corrector method where Euler s method is used to predict a new root and Newton s method is used to improve upon the guess The goal of this section is to rigorously compute branches of connecting orbits for the map f : R n Λ R n with Λ R We also assume that f C 1 (R n Λ, R n ) Given two approximate zeros of F (x, λ) say ( x 0, λ 0 ) and ( x 1, λ 1 ), let λ s := (1 s)λ 0 + sλ 1 x s := (1 s) x 0 + s x 1 We look to obtain existence of a smooth branch of zeros x s of F (x, λ) centered about the segment { x s : s [0, 1]}, to accomplish this we use the method of radii polynomials The existence theorems used in this section can be found in [27], for completeness we state the main existence theorem below Set B r := B r ( x s ) {λ s }, and define s [0,1] (i) T : R n [0, 1] R n to be a Newton like operator T (x, s) := x AF (x, λ s ), where A M n (R) (ii) For k = 1,, n, define p k (r) := Y k +Z k (r) r where Y k and Z k (r) satisfy [T (x s, s) x s ] k Y k and sup b,c Br(0) D xt k (x s + b, s)c Z k (r) Theorem 41 If there exists r 0 > 0, such that p k (r 0 ) < 0, for all k = 1,, n, and D x f(x, λ) is invertible on B r0, then there exists a C m 1 function α : [λ 0, λ 1 ] B r0 such that f(α(λ), λ) = 0 and if (x, λ) B r0 and f(x, λ) = 0 then x = α(λ) We demonstrate the above method for the Henón map (5), and rigorously compute the continuation of homoclinic orbits of length L = 18 from λ 0 = 14, to λ 1 = 139 Using the predictor-corrector technique we compute two approximate zeros of the corresponding connecting orbit operator F N That is, 16

17 F N ( x 0, 14) 0, F N ( x 1, 139) 0 : x 0 = , x 1 = Following the radii polynomial approach we first determine the bound Y k by noting that [T (x s, s) x s ] k = [AF ( x s, λ s )] k, here we use the numerically computed inverse A D x F N ( x 0, λ 0 ) 1 To simplify this expression we define g : [0, 1] R 18 by g(s) = F ( x s, λ s ), expanding about zero yields g(s) = g(0) + g (0)s g (0)s g (s )s 3 = F ( x 0, λ 0 ) + [D x F ( x 0, λ 0 )( x 1 x 0 ) + D λ F ( x 0, λ 0 )(λ 1 λ 0 )] s + 1 [ D 2 2 x F ( x 0, λ 0 )( x 1 x 0, x 1 x 0 ) + 2D x,λ F ( x 0, λ 0 )( x 1 x 0 )(λ 1 λ 0 ) +DλF 2 ( x 0, λ 0 )(λ 1 λ 0 ) 2] s [ D 3 6 x F ( x s, λ s )( x 1 x 0 ) 3 +2D xx,λ F ( x δ, λ s )( x 1 x 0, x 1 x 0 )(λ 1 λ 0 ) + 2D x,λλ F ( x s, λ s )( x 1 x 0 )(λ 1 λ 0 ) 2 + D 3 λf ( x s, λ s )(λ 1 λ 0 ) 3] s 3, where s (0, s) The reason for the third order expansion is due to the fact that for i = 3,, L 2 the i th coordinate function F (i) is at most cubic in s To deal with the parametrizations P and Q we use the Cauchy estimate found 17

18 in [32], in particular Lemma 29 Returning to the Y k bounds, set y (0) := F ( x 0, λ 0 ) y (1) := D x F ( x 0, λ 0 )( x 1 x 0 ) + D λ F ( x 0, λ 0 )(λ 1 λ 0 ) y (2) := 1 [ D 2 2 x F ( x 0, λ 0 )( x 1 x 0, x 1 x 0 ) + 2D x,λ F ( x 0, λ 0 )( x 1 x 0 )(λ 1 λ 0 ) +DλF 2 ( x 0, λ 0 )(λ 1 λ 0 ) 2] y (3) := 1 [ D 3 6 x F ( x s, λ s )( x 1 x 0 ) 3 + 2D xx,λ F ( x s, λ δ )( x 1 x 0, x 1 x 0 )(λ 1 λ 0 ) +2D x,λλ F ( x s, λ δ )( x 1 x 0 )(λ 1 λ 0 ) 2 + D 3 λf ( x s, λ s )(λ 1 λ 0 ) 3] Therefore, T (x s, s) x s = AF ( x s, λ s ) Ay (0) Ay + (1) + Ay (2) Ay + (3) Hence we take Y to be Y = (Y 1, Y 2, Y 3, Y 4 ) satisfying Ay (0) Ay + (1) + Ay (2) Ay + (3) Y, the above bounds are computed using interval arithmetic Next let us consider the Z k bounds in Theorem 41 D x T ( x s + b, s)c = [I AD x F ( x s + b, λ s )] c = [I AD x F ( x 0 + ( x 1 x 0 )s + b, λ 0 + (λ 1 λ 0 )s)] c = [I AD x F ( x 0, λ 0 ) + AD x F ( x 0, λ 0 ) AD x F ( x 0 + ( x 1 x 0 )s + b, λ 0 + (λ 1 λ 0 )s)] c (14) Again we make use of the fact that for i = 3,, L 2, F i is at most cubic in s, therefore we consider the Taylor expansion of D x F (x, λ) centered at ( x 0, λ 0 ) D x F ( x 0 + ( x 1 x 0 )s + b, λ 0 + (λ 1 λ 0 )s) = D x F ( x 0, λ 0 ) + D 2 xxf ( x 0, λ 0 )(( x 1 x 0 )s + b) + D 2 xλf ( x 0, λ 0 )(λ 1 λ 0 )s D3 xxxf ( x s, λ s )(( x 1 x 0 )s + b, ( x 1 x 0 )s + b) + D 3 xxλf ( x s, λ s )( x 1 x 0 )s + b)(λ 1 λ 0 )s D3 xλλf ( x s, λ s )(λ 1 λ 0 ) 2 s 2 Plugging this into (14) we get D x T ( x s + b, s)c = [I AD x F ( x 0, λ 0 )] c + A [ D 2 xxf ( x 0, λ 0 )(( x 1 x 0 )s + b) ] c + A [ D 2 xλf ( x 0, λ 0 )(λ 1 λ 0 )s ] c A [ D 3 xxxf ( x s, λ s )(( x 1 x 0 )s + b, ( x 1 x 0 )s + b) ] c + A [ D 3 xxλf ( x s, λ s )(( x 1 x 0 )s + b)(λ 1 λ 0 )s ] c A [ D 3 xλλf ( x s, λ s )(λ 1 λ 0 ) 2 s 2] c 18

19 Using the above expression we can identify the linear and quadratic coefficients of the radii polynomials, for r > 0, let b := b r, c := c r Z (1) := I AD x F ( x 0, λ 0 ) A D 2 xx F ( x 0, λ 0 )( x 1 x 0 ) 1 18 and + A D 2 xλf ( x 0, λ 0 )(λ 1 λ 0 ) A D 3 xxλ F ( x s, λ s )( x 1 x 0 )(λ 1 λ 0 ) A D 3 xλλ F ( x s, λ s )(λ 1 λ 0 ) D 2 A 3 xxx F ( x s, λ s )(( x 1 x 0 )s + b, ( x 1 x 0 )s + b) 1 18, Z (2) := A D 2 xx F ( x 0, λ 0 ) A D 3 xxλ F ( x s, λ s )(λ 1 λ 0 ) Note that for the third order terms D xxx F, D xxλ F, and D xλλ F the only non constant entries come from the local parametrization functions We need to handle the corresponding truncation error functions h i N = P i PN i, and gi N = Q i Q i N for i = 1, 2 We again appeal to Lemma 29 in [32] D 3 xxλ h i N( x s, λ s ) 8π 3 ν 2 τσ 2 νσ τ h i N ν (15) D 3 xλλ h i N( x s, λ s ) 8π 3 ντ 2 σ ν στ 2 D 3 xxxh i N ( x s, λ s ) 8π3 ν 3 σ 3 ν h i N h i N ν (16) ν, (17) Here ν and τ are the radii of convergence for P N (θ, λ), σ ν satisfies e σν ν = x 0(L 1)+ν 2 and τ satisfies e στ τ = x0(l 1)+τ 2 We can compute analogous bounds for the truncation error gn i Therefore using the above bounds for Z (1) and Z (2) we have for all s [0, 1], sup D x T ( x s + b, s)c Z(r) := Z (1) r + Z (2) r 2 b,c B r(0) The radii polynomials are then p k (r) = Z (2) k r2 + (Z (1) k 1)r + Y k, for k = 1,, L The interval containing all admissible radii values for existence is I = [ , ] Choosing r 0 = 003, we get that for each s [0, 1] there is a homoclinic orbit x s B r0 ( x s ), that is F (x s, λ s ) = 0 19

20 5 Saddle-Node Bifurcation In this section we consider the problem of rigourously computing saddle-node bifurcation points for the connecting orbit map F Consider a C 1 function F : R n R R n which gives rise to a parameter dependent map F (x, λ) = F λ (x), λ R A parameter value λ R is a bifurcation point if given any ɛ > 0 there exists λ R such that λ λ < ɛ and F λ (x) is not topologically equivalent to F λ(x) One can search for bifurcation points by observing changes in the eigenvalue structure of the linearization of F Connecting orbits are solutions to F (x, λ) = 0 If F ( x, λ) = 0 and D x F ( x, λ) is invertible then by the implicit function theorem there is a smooth family of connecting orbits x(λ) in a neighborhood of λ Where as if D x F ( x, λ) is singular then there must be a zero eigenvalue, here we consider the simplest possible case where the multiplicity of the zero eigenvalue is one Definition 51 Consider f : R n R R n A saddle-node for f is a point ( x, λ) R n R such that (i) f( x, λ) = 0, and (ii) zero is an eigenvalue of D x F ( x, λ) with algebraic multiplicity one and all other eigenvalues have non-zero real parts Definition 52 Given F : R n R R n a saddle-node bifurcation occurs at the saddle-node ( x, λ) R n R if the following conditions are met SNB1 There exists a smooth curve g : ( δ, δ) R n R denoted by s (g 1 (s), g 2 (s)) such that g(0) = ( x, λ) and F (g 1 (s), g 2 (s)) = 0 SNB2 The curve defined by g has a quadratic tangency with R n { λ} at ( x, λ), that is g 2 (0) = λ, g 2(0) = 0, and g 2 (0) 0 SNB2 If s 0 then D x F (g 1 (s), g 2 (s)) is hyperbolic and if σ(s) is the eigenvalue of D x F (g 1 (s), g 2 (s)) that satisfies σ(0) = 0, then σ (0) 0 The following theorem which is proved in [9] as Theorem 812 provides sufficient generic conditions for a saddle-node bifurcation to occur in n-dimensions Theorem 53 (Saddle-node bifurcation theorem) Assume F : R n R R n is C 1, ( x, λ) R n R is a saddle-node, and the kernel of D x F ( x, λ) is spanned by the vector ṽ R n If D λ F ( x, λ) 0 and D 2 xf ( x, λ)(ṽ, ṽ) 0 20

21 and both are not in the range of D x F ( x, λ), then there is a saddle-node bifurcation at ( x, λ) Moreover, among all C one parameter families that have a saddle-node, those that undergo a saddle-node bifurcation form an open and dense subset The next lemma provides us with a way of checking if u 1, u 2 0, then they are not in the range of D x F ( x, λ) See Lemma 4 in [27] for the proof Lemma 54 Let D : R n R n be a linear transformation with dim(ker(d))=1 Then ker(d) T = w, for some w R n \ {0} Also, u is in the range of D if and only if u w = 0 The above Theorem 53 provides us with sufficient conditions to guarantee the occurrence of a saddle-node bifurcation, together with Lemma 54 provides a process for finding a saddle-node The following algorithm illustrates this process Algorithm 55 Let f : R n R R n be smooth Assume there exists ( x, λ) R n R and µ σ(d x f( x, λ)) such that f( x, λ) 0 and µ 0 Perform the following steps (i) Rigorously compute eigenpairs {( µ i, ṽ i ) i = 1,, n} such that µ i B ri ( µ i ) where (ii) Verify that µ 1 = 0 0 B r1 ( µ 1 ) and B ri ( µ i ) ir =, i = 2,, n (iii) Define u 1 := D λ f( x, λ) and u 2 := D 2 xf( x, λ)(ṽ, ṽ) and verify that where w R n \ {0} such that w T D x f( x, λ) = 0 w T u i 0, i = 1, 2, (18) If steps (i)-(iii) are carried out successfully, then a saddle-node bifurcation occurs at the saddle-node ( x, λ) B r ( x, λ) where r is obtained explicitly in the verification of step (ii) If the above algorithm is executed correctly one can verify that the hypotheses of Theorem 53 will be satisfied Next we discuss the rigorous numerical procedure used to carry out the above steps For step (i) we use the techniques found in section 31 in [27] We assume that this computation is successful For step (ii) we need to verify that µ 1 = 0, from the first step we know that there is at most one zero eigenvalue, hence it is enough to prove the existence of an eigenvector v that lies in the kernel of D x f( x, λ) Observe that if it exists, then v R n, so we can obtain isolation by requiring that v = 1 This leads to a modified version of the original f(x, λ) = 0 problem That is we need to prove existence of X = (x, λ, v) R 2n+1 satisfying f(x, λ) F (X) := v 2 1 = 0 (19) D x f(x, λ)v 21

22 To deal with the manifold parametrizations we consider the approximate map F N defined by F N (X) := f N (x, λ) v 2 1 D x f N (x, λ)v (20) where f N is the N th order approximating connecting orbit map The existence of such a zero can be addressed using the radii polynomial approach found in Section 3 in [27] A solution to (19) yields the existence of ( x, λ) B r ( x, λ) for some r > 0 This completes the necessary action for step (ii) Let D := D x f( x, λ), observe that step (iii) requires the existence of a vector w R n \ {0} such that w T D = 0 We next use the radii polynomial approach to compute rigorously a non-zero vector w such that ker(d T ) = w, therefore rank(d) = rank(d T ) = n 1 When it comes to solving D T w = 0 we can delete a row from D T and still retain the same solution space, to determine which row to delete we first assume that v 0 satisfies D v 0 Let k be the component of v with largest magnitude, that is v k = max { v i } 0 i=1,,n Let C 1,, C n denote the columns of D and R 1,, R n the corresponding rows of D T that is R i = Ci T for i = 1,, n Now from D v 0, we have C k 1 v v i C i = R k = Ck T 1 v v i R i k k i=1 i k We see that the k-th row R k of D T is a linear combination of the other rows, therefore we can delete it Similarly we can get rid of the k-th column C k of D Set M := ( T Dˆk), where Dˆk is defined to be the n (n 1) matrix defined by D without its k-th column C k A non-zero unit vector w such that kerd T = w is an isolated solution of ( w 2 ) 1 g(w) := = 0, (21) Mw which can be solved using the raddi polynomial approach outlined in [27] To complete algorithm 55 we need to check that condition 18 holds, this can be done using interval arithmetic Example 56 To demonstrate the above procedure for rigorously computing saddle-node bifurcations we consider the Henón map Using Newton s method we find an approximate zero to (20), that is X = ( x, λ, v) R 2(18)+1 and F N ( X, λ, v) 0 where λ = i=1 i k 22

23 x = , v = Next we compute D X F N ( X), D X F N ( X) D x f N ( x, λ) D λ f N ( x, λ) 0 = ( v T D x Dx f N ( x, λ) ) D x D λ f N ( x, λ) v D x f N ( x, λ) from this we use interval arithmetic to compute A = D X F N ( X) 1, this can be done in practice using INTLAB in MATLAB which means A will have interval entries Define T : R 37 R 37 by T (X) := X AF N (X), Thus the Y bound can be determined by AF ( X) Y Next we compute the Z bound, for this we exploit the choice of A DT ( X + b)c = A ( DF N ( X + b) DF N ( X) ) c Defining h : [0, 1] R 37 by h(s) = D X F ( ) ( X + sb c, then h(1) h(0) = DFN ( X + b) DF N ( X) ) c, hence for k = 1,, 37 there exists s k [0, 1] such that ( DX FN k ( X + b) DFN( k X) ) c = h(1) h(0) = h k(s k ) = DXF 2 N( k X + s k b)(b, c) Where b, c B r (0) we can write b = br, c = cr for b, c B 1 (0) With this in mind we get ( DX F k N( X + b) D X F k N( X) ) c = D 2 XF k N ( X + s k b)( b, c)r 2 To assist in the computation of the Z-bound we set r = 10 1 an a-priori upper bound for the right endpoint of the existence interval of the radii polynomials, 23

24 this assumption will be justified if we show that r r Denote by b = [ r, r ] 37 an interval vector in R 37 with entries [ r, r ] Set X = x+b which has k th entry equal to [ Xk r, X k + r ], additionally we set δ = [ 1, 1] 37 Then for b, c B r (0) we get that A(DX F N ( X + b) D X F N ( X))c AD 2 X F N (X )(δ, δ) Using interval arithmetic, we compute Z R 37 such that AD 2 X F (X )(δ, δ) Z (2) With the above bounds we can compute the raddi polynomials as p k (r) = Z (2) k r2 r + Y k Using interval arithmetic we compute the following existence interval I = [ , ] We see that < r = 10 1, hence the existence interval is valid Applying Corollary 3 in [27] if we set r = there exists a unique isolated X = ( x, λ, ṽ) B r ( X) such that F ( X) = 0, with this we have f( x, λ) = 0 and dim(ker D x f( x, λ))=1 ( ) Let r be the smallest radius of existence for I, define B= B r ( x, λ) R 19 that is to say B = 16 k=1 [ x k r, x k + r] [ λ r, λ + r ] Let D := D x f( x, λ) and D:= D x f(b) a interval matrix computed with interval arithmetic Note that D D, we next use the radii polynomial for validating eigenvalue eigenvector pairs as outlined in Section 31 in [27] to show that σ(d) n j=1 B r j for some small balls B rj C Note that the matrix D is a matrix with interval entries hence the bounds Y, Z 0 and Z 1 all have to bound the possible errors from D as well, we use interval arithmetic to accomplish this We proved that the eigenvalues of D x f( x, λ) are enclosed in 10 j=1 B r j (see the table below), for i = 11,, 18 the eigenvalues are all equal to 1 which can be seen from the following Lemma Lemma 57 The matrix DF N in (9) has the eigenvalue λ = 1 with multiplicity n

25 r 1 r 2 r 3 r 4 r 5 r 6 r 7 r 8 r 9 r Proof DF N λi 2(n+2) = 1 λ Q1 Nu ( θ) θ λ 0 0 Q2 Nu ( θ) θ 0 28 x λ λ x n 1 1 λ P Ns 1 ( φ) φ P Ns 2 ( φ) φ λ 1 λ 0 0 Q2 Nu ( θ) θ λ λ 0 0 =( 1 λ) P x n 1 1 λ Ns ( φ) φ P Ns ( φ) φ λ 0 1 λ x λ λ 0 + Q1 Nu ( θ) θ P x n 1 1 Ns ( φ) φ P Ns 2 ( φ) φ λ 25

26 Now 1 λ 0 0 Q2 Nu ( θ) θ λ λ x n 1 1 λ P 1 Ns ( φ) φ P Ns 2 ( φ) φ λ =( 1 λ) 1 λ λ x n 1 1 λ P 2 Ns ( φ) φ The above determinants simplify as follows (i) (ii) 1 λ λ x n 1 1 λ 1 1 λ λ 0 0 P Ns 1 ( φ) + Q2 Nu ( θ) θ P 1 φ 0 28 x n 1 1 Ns ( φ) φ λ P Ns 2 ( φ) φ λ P 1 Ns ( φ) φ P 2 Ns ( φ) φ 1 1 λ λ x n 1 1 P 1 Ns ( φ) φ P 2 Ns ( φ) φ λ = λ =( 1 λ) 2n λ P 1 t 0 P 2 t λ 0 1 λ x λ λ x n 1 1 P 1 Ns ( φ) φ P 2 Ns ( φ) φ If we swap column 2k with column 2k 1 for k = 1,, n + 1 we obtain the matrix whose j th diagonal entry is 1 λ for j = 2k 1, k = 1,, n Moreover the entries a ij on the j th column are zero for i = 1,, j 1 additionally the entries a jk on the j th row are zero for k = j+1,, 2n+1 Therefore computing the determinant by cofactor expansion will yield the factor ( 1 λ) n Similarly for the (2n + 3) (2n + 3) determinant 0 1 λ x n x λ λ 0 P 1 Ns ( φ) φ P 2 Ns ( φ) φ λ, λ 26

27 There will be a factor of the form ( 1 λ) n+1 We now complete the last part of Algorithm 55, let w = , where w was numerically computed so that D x f( x, λ) T w 0 As discussed above by choosing D = D x f( x, λ) a nonzero vector w such that kerd T = w is an isolated solution to (21) Using the radii polynomial approach (cf Section 3 in [27]) we rigorously compute w B ( w) such that D x f( x, λ) w = 0 Lastly define u 1 = D λ f( x, λ) and u 2 = D 2 xf( x, λ)(ṽ, ṽ), we showed using interval arithmetic that u 1 w 0 and u 2 w 0 Next we validate the saddle node bifurcations for the Hénon map appearing in the diagram below 27

28 Figure 7: Saddle node bifurcations for the Hénon map The table below contains validated radii of existence for the saddle node bifurcation occurring at each parameter value a Note that at a = a r eig r zero r ker the saddle node bifurcation was validated in Example 56 Here r eig denotes the uniform radius of existence for the eigenpair validation, r zero denotes the uniform radius of existence for the zero of the map (19) which corresponds to the validation of µ 1 = 0 Lastly r ker denotes the radius of existence for the zero of the kernel map (21) Next we validate the branches of connecting orbits for each saddle node in figure 7 28

29 a r top r bottom [ , 14] [ , 133] [ , 125] [ , 1246] [ , 117] [ , 1175] Here r top, r bottom correspond to the radius of existence for the top branch and bottom branch of connecting orbits in figure 7 for the given range of parameter values a References [1] Zin Arai On hyperbolic plateaus of the Hénon map Experiment Math, 16(2): , 2007 [2] Zin Arai On loops in the hyperbolic locus of the complex Hénon map and their monodromies Phys D, 334: , 2016 [3] Zin Arai and Konstantin Mischaikow Rigorous computations of homoclinic tangencies SIAM J Appl Dyn Syst, 5(2): (electronic), 2006 [4] Gianni Arioli and Hans Koch Computer-assisted methods for the study of stationary solutions in dissipative systems, applied to the Kuramoto- Sivashinski equation Arch Ration Mech Anal, 197(3): , 2010 [5] W-J Beyn The numerical computation of connecting orbits in dynamical systems IMA J Numer Anal, 10(3): , 1990 [6] W-J Beyn and J-M Kleinkauf Numerical approximation of homoclinic chaos Numer Algorithms, 14(1-3):25 53, 1997 Dynamical numerical analysis (Atlanta, GA, 1995) [7] W-J Beyn and J-M Kleinkauf The numerical computation of homoclinic orbits for maps SIAM J Numer Anal, 34(3): , 1997 [8] Maciej J Capiński and J D Mireles James Validated computation of heteroclinic sets SIAM J Appl Dyn Syst, 16(1): , 2017 [9] Carmen Chicone Ordinary differential equations with applications, volume 34 Springer,New York, second edition, 2006 Texts in Applied Mathematics [10] Sarah Day, Jean-Philippe Lessard, and Konstantin Mischaikow Validated continuation for equilibria of PDEs SIAM J Numer Anal, 45(4): ,

30 [11] E J Doedel, B W Kooi, G A K van Voorn, and Yu A Kuznetsov Continuation of connecting orbits in 3D-ODEs I Point-to-cycle connections Internat J Bifur Chaos Appl Sci Engrg, 18(7): , 2008 [12] E J Doedel, B W Kooi, G A K Van Voorn, and Yu A Kuznetsov Continuation of connecting orbits in 3D-ODEs II Cycle-to-cycle connections Internat J Bifur Chaos Appl Sci Engrg, 19(1): , 2009 [13] Eusebius J Doedel and Mark J Friedman Numerical computation of heteroclinic orbits J Comput Appl Math, 26(1-2): , 1989 Continuation techniques and bifurcation problems [14] J-P Eckmann, H Koch, and P Wittwer A computer-assisted proof of universality for area-preserving maps Mem Amer Math Soc, 47(289):vi+122, 1984 [15] X Cabré, E Fontich, and R de la Llave The parameterization method for invariant manifolds I manifolds associated to non-resonant subspaces Indiana Univ Math J, 52: , 2003 [16] X Cabré, E Fontich, and R de la Llave The parameterization method for invariant manifolds II regularity with respect to parameters Indiana Univ Math J, 52: , 2003 [17] X Cabré, E Fontich, and R de la Llave The parameterization method for invariant manifolds III overview and applications J Differential Equations, 218: , 2005 [18] Jorge Gonzalez and J D Mireles James High-order parameterization of stable/unstable manifolds for long periodic orbits of maps (To appear in SIAM Journal on Applied Dynaical Systems), pages 1 46, 2017 [19] À Haro and R de la Llave A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: numerical algorithms Discrete Contin Dyn Syst Ser B, 6(6): , 2006 [20] A Haro and R de la Llave A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: rigorous results J Differential Equations, 228(2): , 2006 [21] A Haro and R de la Llave A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: explorations and mechanisms for the breakdown of hyperbolicity SIAM J Appl Dyn Syst, 6(1): , 2007 [22] Àlex Haro, Marta Canadell, Jordi-Lluí s Figueras, Alejandro Luque, and Josep-Maria Mondelo The parameterization method for invariant manifolds, volume 195 of Applied Mathematical Sciences Springer, [Cham], 2016 From rigorous results to effective computations 30

31 [23] Hans Koch, Alain Schenkel, and Peter Wittwer Computer-assisted proofs in analysis and programming in logic: a case study SIAM Rev, 38(4): , 1996 [24] Oscar E Lanford, III Computer-assisted proofs in analysis Phys A, 124(1-3): , 1984 Mathematical physics, VII (Boulder, Colo, 1983) [25] Oscar E Lanford, III Computer-assisted proofs in analysis In Proceedings of the International Congress of Mathematicians, Vol 1, 2 (Berkeley, Calif, 1986), pages Amer Math Soc, Providence, RI, 1987 [26] Jean-Philippe Lessard Delay differential equations and continuation (Preliminary lecture notes for the AMS short course on rigorous numerics in dynamics), page [27] Jean-Philippe Lessard Rigorous verification of saddle-node bifurcations in ODEs Indag Math (NS), 27(4): , 2016 [28] Jean-Philippe Lessard, Jason D Mireles James, and Christian Reinhardt Computer assisted proof of transverse saddle-to-saddle connecting orbits for first order vector fields J Dynam Differential Equations, 26(2): , 2014 [29] Rudolf J Lohner Computation of guaranteed enclosures for the solutions of ordinary initial and boundary value problems In Computational ordinary differential equations (London, 1989), volume 39 of Inst Math Appl Conf Ser New Ser, pages Oxford Univ Press, New York, 1992 [30] J D Mireles James Polynomial approximation of one parameter families of (un)stable manifolds with rigorous computer assisted error bounds Indag Math (NS), 26(1): , 2015 [31] J D Mireles James and Konstantin Mischaikow Rigorous a posteriori computation of (un)stable manifolds and connecting orbits for analytic maps SIAM J Appl Dyn Syst, 12(2): , 2013 [32] JD Mireles James and Konstantin Mischaikow Rigorous a-posteriori computation of (un)stable manifolds and connecting orbits for analytic maps SIAM J Numer Anal, 12(2): , 2013 [33] Evelyn Sander and Thomas Wanner Validated saddle-node bifurcations and applications to lattice dynamical systems SIAM J Appl Dyn Syst, 15(3): , 2016 [34] Warwick Tucker Validated numerics for pedestrians In European Congress of Mathematics, pages Eur Math Soc, Zürich, 2005 [35] Warwick Tucker Validated numerics Princeton University Press, Princeton, NJ, 2011 A short introduction to rigorous computations 31