FRACTAL ANALYSIS OF HOPF BIFURCATION FOR A CLASS OF COMPLETELY INTEGRABLE NONLINEAR SCHRÖDINGER CAUCHY PROBLEMS

We study the complexity of solutions for a class of completely integrable, nonlinear integro-differential Schrödinger initial-boundary value problems on a bounded domain, depending on a real bifurcation parameter. The considered Schrödinger problem is a natural extension of the classical Hopf bifurcation model for planar systems into an infinite-dimensional phase space. Namely, the change in the sign of the bifurcation parameter has a consequence that an attracting (or repelling) invariant subset of the sphere in L(Ω) is born. We measure the complexity of trajectories near the origin by considering the Minkowski content and the box dimension of their finite-dimensional projections. Moreover we consider the compactness and rectifiability of trajectories, and box dimension of multiple spirals and spiral chirps. Finally, we are able to obtain the box dimension of trajectories of some nonintegrable Schrödinger evolution problems using their reformulation in terms of the corresponding (not explicitly solvable) dynamical systems in Rn. Corresponding author: Josipa Pina Milǐsić University of Zagreb, Faculty of Electrical Engineering and Computing Unska 3, 10000 Zagreb, Croatia pina.milisic@fer.hr

1. Introduction 1.1.Motivation and formulation of the problem.Dimension theory for dynamical systems has been rapidly developed due to its important applications in other natural, social and technical sciences.Since the early 1970's scientists have started to estimate and compute fractal dimensions of strange attractors for finite (Lorenz, Henon, Chua, Leonov, etc.), as well as for the infinite-dimensional dynamical systems (Ladyzhenskaya, Foiaş and Temam, Babin and Vishik, Ruelle, Lieb, etc.) Fractal dimensions in dynamics are discussed in a survey article [29].However, our approach to the question of fractal dimensions in dynamics is different in a way that we are interested in the complexity of trajectories of dynamical systems and study the dependence of the computed fractal dimension upon the bifurcation parameter.Knowing the information about fractal dimension of trajectories enables to measure the complexity of the studied dynamics which we mostly describe by calculating the corresponding box dimension and, in order to describe finer properties, its Minkowski content.Namely, the Hausdorff dimension of trajectories that we are interested in, is always trivial.
The second and third author have undertaken a systematic study of fractal properties of trajectories of vector fields near the weak focus in R 2 and analogously in R 3 .Furthermore, they considered a connection between the box dimension of the trajectory and the bifurcation of the related dynamical system.Here we mention that this interesting connection has been studied also in [10,16] but for one-dimensional discrete dynamical systems.
In this article, using methods developed in [20,26,27,28] we consider the trajectories of vector fields in infinite-dimensional case and we calculate their box dimensions.The original idea of this paper grounds on the connection between nonlocal Schrödinger evolution problems and the corresponding system of ODE's.More precisely, starting from completely integrable nonlinear integro-differential Schrödinger equation we arrive at an equivalent system of infinitely many nonlinear ODE's.Concerning the observed link between Schrödinger problems and vector fields it is interesting to point out that each planar system of ODE's with polynomial right-hand side can be interpreted as a nonlinear Schroedinger equation with an explicit corresponding nonlinear term.Moreover, as it is described in Section 6.3, this consideration is valid even for any dynamical system in R n .Therefore, it is worth noting that the 16th Hilbert problem about the search of an uniform upper bound for the number of limit cycles in polynomial vector fields can be considered in terms of a Schrödinger equation.
In Žubrinić and Županović [26,Theorem 9] it has been shown that the box dimension of spiral trajectories of the classical Hopf bifurcation system in the plane described by (4), viewed near the focus, is equal to d = 4/3 when the bifurcation parameter a 0 is equal 0. Furthermore, these trajectories are Minkowski nondegenerate, i.e. their d-dimensional Minkowski contents, are different from 0 and ∞.On the other hand, for a 0 = 0 all trajectories have trivial box dimension equal to 1, due to the fact that we have strong focus in this case.We shall see that it is possible to obtain analogous results in infinitedimensional case for problem (1).The main results are stated in Theorem 11 (Minkowski content), Theorem 15 (box dimension of multiple spirals), Theorem 17 (box dimension of solutions of NLS Cauchy problems), and Theorem 18 (box dimension of spiral chirps).In Theorem 11 we obtain some surprising relations between the sequence of Minkowski contents of projections of the solution of (1) at Hopf bifurcation, and the Sobolev and Lebesgue norms of the initial function u 0 .EJQTDE, 2010 No. 60, p. 2 We start with the Cauchy problem for the following nonlinear Schrödinger (NLS) initialboundary value problem: (1) Ω |u(t, x)| 2 dx + a 0 u(t, x) = 0 for x ∈ ∂Ω, t ∈ (t min , t max ) u(0, x) = u 0 (x) for x ∈ Ω.
where i is the imaginary unit, γ is a fixed nonzero complex number, a 0 a real bifurcation parameter, Ω a bounded domain in R N , N ≥ 1, and u 0 : Ω → C is a given initial function, u 0 ∈ L 2 (Ω, C).Here ∆u = N j=1 ∂ 2 u ∂x 2 j is the Laplace operator.For a fixed space variable x, the solutions u : (t min , t max ) → L 2 (Ω, C) of the NLS Cauchy problem (1) can be considered as trajectories in the Hilbert space L 2 (Ω, C) where L 2 (Ω, C) = L 2 (Ω) + iL 2 (Ω) is the complexification of the real space L 2 (Ω).We point out that t min and t max depend on γ, a 0 and u 0 , and 0 ∈ (t min , t max ).In this article, we are dealing with unbounded time intervals (t min , t max ) either of the form (t min , ∞), or (−∞, ∞), or (−∞, t max ).
Due to the integral term in (1), this NLS Cauchy problem is of nonlocal type.Similar nonlocal problems with the same integral term have been considered for the modified cubic wave equation, with homogeneous Dirichlet boundary condition, in Cazenave, Haraux and Weissler [3,4,5], in their study of completely integrable abstract wave equations.We consider the Schrödinger equation (1) for a 0 = 0 as an approximation of the equation u t = i∆u − γu|u| 2 following the approach of [5, p. 130].
The same integral term as in (1) can be seen in Christ [8, pp. 132 and 133].An analogous one appears in an equation of fourth order arising from the theory of aeroelasticity, x dx u xx + γu x + δu t + εu tt = 0 see Chicone [7, p. 310].Other integro-differential Schrödinger problems involving different integral operators on the right-hand side have been studied in numerous papers, see for example Chen and Guo [6].
In order to write down problem (1) as an infinite system of nonlinear ordinary differential equations we exploit a well known Fourier series expansion using the decomposition with respect to a Hilbert basis in L 2 (Ω).Let (ϕ j ) be the orthonormal basis of eigenfunctions of the operator −∆ with zero boundary data and with eigenvalues (λ j ) such that 0 < λ 1 < λ 2 ≤ λ 3 ≤ . . .(see e.g.Brezis [1, ersatz (12) on p. 209]).The basis (ϕ j ) is contained in the real Lebesgue space L 2 (Ω), and therefore it is also EJQTDE, 2010 No. 60, p. 3 an orthonormal base in L 2 (Ω, C).By writing the solution u(t, x) in the form: (2) the NLS Cauchy problem (1) formally reduces to a lattice Schrödinger equation on the Hilbert space of quadratically summable sequences of complex numbers, which we denote by ℓ 2 (C).More precisely, we obtain an infinite system of nonlinear ODE's: (3) żj = −iλ j z j − γz j ( z 2 + a 0 ), j = 1, 2, . . ., for each t ∈ (t min , t max ).Clearly, z j0 = u 0 , ϕ j = Ω u 0 ϕ j dx and • is a standard Euclid norm.For given u 0 ∈ L 2 (Ω, C), we interpret NLS equation (1) 1 as the lattice Schrödinger equation (3).To any v ∈ L 2 (Ω, C) one can assign z = (z j ) j ∈ ℓ 2 (C), where v = j z j ϕ j is the Fourier expansion of v, and this assignment is an isometric isomorphism.
Here, we note that the NLS Cauchy problem ( 1) is a natural extension of the Hopf bifurcation model for planar systems.The Hopf bifurcation is connected to 1-parameter families of vector fields where a limit cycle surrounding a singular point is born.It is well known that the Hopf bifurcation occurs at an equilibrium point x 0 of a planar system ẋ = f (x, a 0 ) depending on a parameter a 0 ∈ R when the matrix Df (x 0 , a 0 ) has a pair of pure imaginary eigenvalues, see [21, pg. 314.].In this sense we say that the point x 0 is the weak focus of the system.On the other side, if the eigenvalues are such that both their real and imaginary parts differ from zero, we are speaking about the strong focus.
Namely, when z = (z 1 , 0, 0, . . .), and assuming that λ 1 = 1, γ = 1, the system (3) reduces to the classical Hopf bifurcation system in the plane: where we denote x(t) = Re z(t), y(t) = Im z(t), real and imaginary part of the complex number z(t), respectively.System (4) written using the polar coordinates (r, θ) reads ṙ = r(r 2 + a 0 ), θ = −1.(5) and can be explicitly solved.Since θ = 0, the origin r = 0 is the only critical point.Since θ < 0 the flow is always clockwise.The phase portrait of (4) for a 0 < 0, a 0 = 0 and a 0 > 0 is shown by Figure 1.Viewing a 0 as the bifurcation parameter, we point out the following cases in which the stability of the origin changes, i.e. the bifurcation occurs: (i) For a 0 > 0 the origin is unstable strong focus and for all trajectories we have (t min , t max ) = (−∞, t 0 ) with t max = t 0 > 0. (ii) For a 0 = 0 the origin is unstable weak focus, and (t min , t max ) = (−∞, t 0 ), t 0 > 0. (iii) For a 0 < 0 the origin is a stable strong focus.The limit cycle is the circle of radius r = √ −a 0 , which is unstable.For trajectories inside the circle the solutions EJQTDE, 2010 No. 60, p. 4 are global, i.e. (t min , t max ) = (−∞, ∞), while for those outside the circle we have (t min , t max ) = (t 0 , ∞) with t 0 < 0.
Next, let us consider the Hopf bifurcation in a complex system (3), but in finitedimensional phase space.We start with two-dimensional complex system with the corresponding initial conditions z j (0) = z j0 , where a 0 is the bifurcation parameter.
To simplify, for the moment we assume that γ ∈ R and γ > 0. Near a 0 = 0 we have the qualitative change of the behavior of the system.Indeed, using the polar coordinates the system (6) can be written as ṙj = γr j (r 2 1 + r 2 2 + a 0 ), θj = −λ j , j = 1, 2, EJQTDE, 2010 No. 60, p. 5 where z j = r j exp(iθ j ).For a 0 < 0 an invariant set, 3-sphere in R 4 is born, where S 3 is the unit sphere in R 4 .In analogy with the case of R 2 , we call this phenomenon the Hopf bifurcation in R 4 .Clearly, if a 0 → 0 − , then the sphere shrinks to the origin.Moreover, if r 2 1 + r 2 2 < −a 0 , then ṙj < 0, and r(t) is decreasing as t → ∞.The α-limit set of the trajectory is the origin, while the ω-limit set is a subset of the sphere (7).More precisely, the corresponding limit set is the subset of the 2-torus, contained in the 3-sphere (7).The torus has the form r 1 S 1 × r 2 S 1 ⊂ √ −a 0 S 3 , where r 1,2 are positive scaling numbers depending on the initial point, r 2 1 +r 2 2 = −a 0 , and S 1 is the unit circle in the complex plane C. Clearly, if r 2 1 + r 2 2 > −a 0 , and r(t) monotonically increases, then the corresponding trajectories converge to the sphere and their ω-limit set is a subset of a 2-torus contained in the 3-sphere (7).
In the similar way the Hopf bifurcation for the following system of k complex equations: can be understood.An invariant (2k − 1)-sphere is born in C k when a 0 < 0, defined by For the general theory of bifurcation problems in finite-dimensional dynamical systems see Guckenheimer and Holmes [13].
1.3.Hopf bifurcation in an infinite-dimensional phase space.Since we are interested in the phenomenon of the Hopf bifurcation in an infinite-dimensional complex system (3), we start with the description of the Hopf bifurcation for an ODE defined in a Banach space X (real or complex): Here F : D × R → X is a given map, where D is a dense subspace of X, and the initial value u 0 ∈ D is prescribed.For our purposes the mapping F is usually of the form as u → 0, and a 0 is the bifurcation parameter.Additionally we assume that for all a 0 we have F (0, a 0 ) = 0 so that u = 0 is an equilibrium point.For each initial point u 0 in a neighbourhood of 0 ∈ D let a trajectory of (9) be defined on an unbounded interval (t min , t max ) containing the origin.Definition 1.We say that a 0 = 0 is the point of Hopf bifurcation for the system (9) if the following conditions are fulfilled: EJQTDE, 2010 No. 60, p. 6 (i) For a 0 ≥ 0 small enough the system is unstable near the origin, and for u 0 from a neighbourhood of the origin we have (t min , t max ) = (−∞, t 0 ), t max = t 0 > 0. (ii) For a 0 < 0 with |a 0 | small enough, the origin is stable, and an unstable invariant set S(a 0 ) ⊂ X for ( 9) is born near the origin.There exists an open neighbourhood U(a 0 ) of the origin such that its boundary is S(a 0 ), and for any u 0 ∈ U(a 0 ) the corresponding solution is global, i.e. (t min , t max ) = (−∞, ∞), while for u 0 ∈ B \ U(a 0 ) we have that (t min , t max ) = (−∞, t 0 ) with t 0 > 0.
Definition 1 is analogous if the signs of a 0 are reversed, in which case the stability should be reversed as well as semi-infinite time intervals.Somewhat loosely we say that the Hopf bifurcation occurs if an invariant set is born near the origin when a 0 passes through the value of 0, and the origin changes from stable to unstable or vice versa.
Returning to the NLS Cauchy problem (1), the Hopf bifurcation consists in the birth of an invariant sphere in the space L 2 (Ω) when a 0 < 0.Here we introduce some notation.For a 0 < 0 we define R = √ −a 0 and let be the ball and the sphere in L 2 (Ω), respectively.In the same way, when a 0 < 0 for the corresponding system (3) in ℓ 2 (C), the Hopf bifurcation consists in the birth of an invariant sphere in the space ℓ 2 (C).Since ( 8) is also a special case of the NLS Cauchy problem (1), corresponding to the case when u 0 ∈ span {ϕ 1 , . . ., ϕ k }, we see that we can expect trajectories of (1) in the Hilbert space L 2 (Ω, C), which oscillate at infinitely many scales.We can achieve this by choosing u 0 so that u 0 , ϕ j = 0 for infinitely many j's.
Furthermore, we are interested in Hopf bifurcations from the point of view of fractal geometry.For that purpose, in the next subsection firstly we review some standard notation and definitions from fractal geometry and Sobolev spaces.
1.4.Notation and definitions.Let A be a bounded set in R k , and let d(x, A) be Euclidean distance from x to A. Then the Minkowski sausage A ε is A ε := {y ∈ R k d(y, A) < ε}, a term coined by B. Mandelbrot.By lower s-dimensional Minkowski content of A, s ≥ 0, we mean the following: where | • | is N-dimensional Lebesgue measure.Analogously for the upper s-dimensional Minkowski content of A. The lower box dimension of A is defined by dim B A = inf{s > 0 : M s * (A) = 0}, and analogously the upper box dimension dim B A. For various properties of fractal dimensions see Falconer [11].
EJQTDE, 2010 No. 60, p. 7 If A is such that dim B A = dim B A, then the common value is denoted by d := dim B A, and is called the box dimension of A. Furthermore, if both the upper and lower ddimensional Minkowski contents of A are different from 0 and ∞, we say that the set A is Minkowski nondegenerate.If in addition to this we have M d * (A) = M * d (A) =: M d (A) ∈ (0, ∞), then A is said to be Minkowski measurable.The notion of Minkowski content appears for example in the study of fractal drums and fractal strings, see He and Lapidus [14], Lapidus and Frankenhuysen [18].Furthermore, Minkowski content is essential for understanding some singular integrals, see [25].
If A is a subset of an infinite-dimensional vector space X, we say that dim B A = ∞ if there exists an increasing sequence of finite-dimensional subspaces We need this definition in Theorem 6(b).
We deal with the Sobolev spaces H 1 0 (Ω, C) and H 1 0 (Ω, C) ∩ H 2 (Ω, C) (in the sequel we omit C) equipped with the corresponding norms defined by ( 13) See Henry [1,15].All the results of this paper hold if in (1) we have −∆ instead of ∆.The j-th component of trajectory, viewed as a spiral in C, only changes the orientation for each j from negative to positive, see ( 18) below.

2.
Well-posedness and stability of solutions 2.1.Explicit solutions of the NLS problem.We consider the NLS initial-boundary value problem (1).For some special values of parameters γ and a 0 it is possible to find explicit solutions of problem (1) and then to consider their qualitative properties.In order to calculate directly the explicit solutions of problem (1), we need the following lemma.
Lemma 2. Let u be a solution of the NLS Cauchy problem (1) where γ = γ 1 + iγ 2 .Furthermore, for u 0 = 0 we have Proof.In Section 1, we showed that the NLS Cauchy problem (1) can be written in the form on an infinite-dimensional ODE system EJQTDE, 2010 No. 60, p. 8 where z j = r j exp(iθ j ).The expressions (14) 2 and ( 14) 3 follow easily by multiplying the expression ( 16) by exp(−iθ j ).The first equation in (14) we get directly from the formal calculation.Namely, By dividing the first two expressions given in ( 14) we obtain ṙj r j = ρ ρ , hence r j = C j ρ, where C j is a constant depending on the initial value u 0 .Using the expression (2), we obtain that Next, we use the results given by lemma 2 in order to find the explicit solutions of the NLS Cauchy problem (1) for some special values of the parameters γ and a 0 .We use the notation ρ 0 = u 0 L 2 .For the sake of simplicity we take γ 2 = 0, i.e. γ ∈ R.
¿From lemma 2 directly follows that for γ 1 = 0 it holds u(t) L 2 = C(u 0 ), and moreover, r j (t) = C j (u 0 ) for each j ∈ N.
In this case V (ρ) = ρ 2 and the explicit solution of equation ( 14) 1 is given by ( 17) 15) and (14) 3 using ( 17), and noting θ j (0) = arg u 0 , ϕ j we obtain (18) r In this way, using the decomposition (2) we have the explicit solution of NLS problem (1) EJQTDE, 2010 No. 60, p. 9 where u 0 is a given initial function.Here we notice that the sign of the parameter γ 1 affects the maximal interval of the solution.More precisely, following the terminology introduced in Cazenave [2, Remark 3.1.6(ii)],we say that the solutions given by formula (19) are positively (negatively) global if γ 1 > 0 (γ 1 < 0).The solutions are global for γ 1 = 0 for any initial value u 0 , while for γ 1 = 0 the global solution exists only when u 0 = 0.
In this case one has V (ρ) = ρ 2 + a 0 and we obtain the Bernoulli equation ρ = −γ 1 ρV (ρ) with the solution ( 20) . We obtain the analogous series representation of solution u(t, x) of (1) as in (19), assuming again that γ 2 = 0: The ability to calculate the explicit solution of NLS boundary-value problem (1), given by formulas ( 19) and ( 21), guarantees its uniqueness.More precisely, it is obvious that the following result is true.
The explicit formulas (19) and (21) for the solution of the NLS problem (1) enable us to express the corresponding norms of the solution in the case when where ρ(t) is defined by ( 17) or ( 20) if a 0 = 0 or a 0 = 0 respectively.Similarly for Next, let us consider formula (17) again.It is obvious that for t → t + min where t min = −(2γ 1 ρ 2 0 ) −1 one has u(t) L 2 (Ω) → ∞, i.e. we have the blow up of the solution.Similar conclusion holds for a 0 = 0 and the expression (20).
EJQTDE, 2010 No. 60, p. 10 Finally, we note that directly from formulas ( 19) and ( 21) follows the invariance property of the solutions of the NLS problem.More precisely, if U 0 is the span of a given subset of {ϕ j : j ≥ 1} ⊂ L 2 (Ω), then the assumption u 0 ∈ U 0 implies that u(t) ∈ U 0 for all t > 0, where the closure is taken in L 2 (Ω).Moreover, the invariance property can be reformulated as follows: each subspace U 0 , where U 0 is spanned by a subset of ϕ j -s, is invariant for the nonlinear evolution operator T (t), T (t)u 0 = u(t), see (19), associated with the problem (1): T (t)U 0 ⊆ U 0 .In other words, a trajectory that starts in U 0 remains in this space forever.In particular, if u 0 ∈ span {ϕ 1 , . . ., ϕ k }, then Γ(u 0 ) ⊂ span {ϕ 1 , . . ., ϕ k }.This means that if u 0 , ϕ j = 0 for all but finitely many j's, then ( 1) is essentially a finite-dimensional problem, which can be viewed as (8).

2.2.
Well posedness and stability.The following proposition gives some stability results of the solution of NLS boundary-value problem (1) with respect to the value of the bifurcation parameter a 0 .Again, for the sake of simplicity we assume that γ 2 = 0 and γ 1 > 0. For γ 1 < 0 time intervals of the form (t min , ∞) should be changed to (−∞, t max ).The notation used in the following proposition (the ball and the sphere in L 2 (Ω)) we introduced in (11).Directly from the expressions ( 17) and ( 20) follow the power and the exponential rate of the convergence of u(t) L 2 to the origin in L 2 (Ω) (as the fixed point for problem (1)), respectively.More precisely, the following proposition holds.Proposition 4. (Hopf bifurcation for Schrödinger problem) Assume that γ = γ 1 > 1 and let u 0 ∈ L 2 (Ω) be the initial function for NLS initial-boundary value problem (1), where a 0 ∈ R is the bifurcation parameter.
(i) For a 0 > 0 then the origin is exponentially stable with respect to L 2 -topology for any u 0 = 0. (ii) For a 0 = 0 the origin is power stable in L 2 (Ω), with power α = 1/2.(iii) For a 0 < 0 let us denote R = √ −a 0 .Then we distinguish the following cases: (a) If u 0 ∈ B R (0) then the origin is exponentially unstable and the solutions are global with u(t) The continuous dependence on the initial condition, regularity and the continuous dependence on bifurcation parameter for NLS boundary-value problem (1) when u 0 ∈ L 2 (Ω) is given by the following proposition.Here, we point out that the same qualitative properties valid if The interval of the existence of u(t) we note by I = (t min , t max ).

Proposition 5. (a) (continuous dependence on initial condition)
Moreover, the convergence is uniform on each compact interval in I.
EJQTDE, 2010 No. 60, p. 11 (c) (continuous dependence on bifurcation parameter) Let u 0 ∈ L 2 (Ω) be fixed, and let u(t) and u a 0 (t) be defined by ( 19) and ( 21) respectively.Then for any t ∈ I, where I is the interval of existence of u(t), Moreover, the convergence is uniform on compact intervals J contained in I.
Proof.(a) If v 0 = 0 then the claim follows easily from proposition 4. Let v 0 = 0 and u 0 → v 0 .Denoting û0 = u 0 / u 0 L 2 and using |a + b| 2 ≤ 2|a| 2 + 2|b| 2 for a, b ∈ C, we obtain: , where we note , and from this (22) Since ρ(t) is uniformly continuous, one has that ρ(t) → ρ(s) for t ≥ s and it suffices to show that, for given ε the expression can be made less than ε if |t − s| is small enough.For any m > 1, the sum is less than or equal to: EJQTDE, 2010 No. 60, p. 12 3. Compactness and non-rectifiability of trajectories 3.1.Compactness.Since we consider trajectories in infinite-dimensional spaces, it is not at all clear if they are compact sets, or even rectifiable.Using the notation given in section 1.2, the solution of NLS problem (1) can be written in the following way where u 0 is a given initial function.By Γ(u 0 ) = {u(t) : t ≥ t 0 } we note the trajectories of solutions.The notation (23) enables us to reduce the problem of compactness of the trajectories to the problem of the compactness of the bounded sets in ℓ 2 (C).In this sense, we review the following characterization of relatively compact sets in ℓ 2 (C) (Weidmann [24,p.

135]):
A subset Y of ℓ 2 (C) is relatively compact in ℓ 2 (C) if and only if Y is bounded and for every ε > 0 there exists j 0 ∈ N such that for all sequences (f j ) in Y the following condition is fulfilled: The following theorem establishes not only compactness of individual trajectories Γ(u 0 ) of solutions of the NLS Cauchy problem (1) in L 2 (Ω), but also for some bundles of trajectories.If a trajectory generated by u 0 is positively global, defined for t ∈ (t min , ∞), then by Γ(u 0 ) we denote its part corresponding to t ≥ 0. We do analogously for negatively global trajectories.
For a given nonempty base set of initial functions A ⊂ L 2 (Ω) we can define the corresponding bundle of trajectories by The following theorem provides some sufficient conditions on A that ensure compactness of Γ(A).To simplify, we assume that a 0 = 0 and γ 1 > 0 in (1).Theorem 6.(a) For any u 0 ∈ L 2 (Ω) the corresponding trajectory Γ(u 0 ) of ( 1) starting with t 0 = 0 is relatively compact in L 2 (Ω).
(b) Let us first prove that A(u 0 ) = Γ(A(u 0 )).The inclusion ⊆ is clear.To prove the converse inclusion, let v(t) ∈ Γ(A(u 0 )), where v(t) is a trajectory generated by v 0 ∈ A(u 0 ).Then where we used the monotonicity of ρ(t) for γ 1 > 0, so that for t ≥ 0 we have ρ(t) ≤ ρ(0) = u 0 L 2 .Hence v(t) ∈ A(u 0 ).The compactness of A(u 0 ) is proved using a slight change in the proof of (a).Denoting by A ′ (u 0 ) the set in ℓ 2 (C) corresponding to A(u 0 ) in L 2 (Ω), then where To prove (25), it suffices to note that if v = j z j ϕ j , then z = (z j ) j ∈ A(u 0 ) if and only if z j ∈ B r j (0) for all j.
Remark 7. Similar results like those presented in Theorem 6 valid if . More precisely, in then the trajectory Γ(u 0 ) is relatively compact in H 1 0 (Ω) and H 1 0 (Ω) ∩ H 2 (Ω), respectively.Indeed, the subspace It is easy to see that the analogous characterization of compact sets as for ℓ 2 in (a) holds also in ℓ ′ 2 with respect to the new norm.Similarly for the subspace ℓ ′′ 2 corresponding to 3.2.Rectifiability.Assuming that t max (u 0 ) = ∞ and u(t) L 2 → 0 as t → ∞, we define the length of trajectory Γ(u 0 ) over the interval [0, t) by where , ( 2) and (3).We consider rectifiability and nonrecitifiability of Γ(u 0 ) only for t ∈ [0, ∞) or t ∈ (−∞, 0].Moreover, we are interested in the asymptotic behaviour of l(u 0 , t) as t → ∞.For that purpose, we introduce here some notation.We say that: The following theorem states the dependence of the (non)-rectifiability of trajectories Γ(u 0 ) near the origin of the NLS initial-boundary value problem (1) due to the value of the bifurcation parameter a 0 .
Remark 10.Let A be a bounded subset of H 1 0 (Ω) ∩ H 2 (Ω) such that 0 / ∈ A, with closure taken in this space.Let us consider the bundle of trajectories Γ(A), and let us define lower and upper length of bundle Γ(A) in time interval [0, t) by: Assuming that a 0 = 0, γ 1 > 0, u 0 = 0, it can be shown by reconsidering the proof of Theorem 8 that l(A, t) ≃ t 1/2 , l(A, t) ≃ t 1/2 as t → ∞.EJQTDE, 2010 No. 60, p. 16

Minkowski sequence associated to a trajectory
In this section we consider NLS boundary-initial value problem (1) for a 0 = 0. Let Γ(u 0 ) be a trajectory generated by u 0 = 0, corresponding to t ≥ 0 if the solution u(t) is positively global, and to t ≤ 0 if it is negatively global.Denote by Γ j (u 0 ) the orthogonal projection of Γ(u 0 ) ⊂ L 2 (Ω) onto the ϕ j -component, where ϕ j are defined in section 1.2.Here, we recall that the solution of (1) can be written in form u(t, x) = ∞ j=1 z j (t)ϕ j (x).Now, Γ j (u 0 ) can be viewed as a curve in the complex plane defined by z j (t) = u(t), ϕ j = r j (t) exp(iθ j (t)).Figure 2 shows an example of Γ j (u 0 ), j = 1, 2, 3, 4, where z j (t) = (t + 1) −1/2 c j e −iλ j t with c j = u 0 , ϕ j .For some chosen eigenvalues λ j and Fourier coefficients c j we plotted z j (t) on the time interval [0, ∞).The trajectory t → z(t) of the considered system in ℓ 2 (C) is equal to a sequence t → (z j (t)) j∈N .In this way Figure 2   The following result describes some properties of the sequence of d-dimensional Minkowski contents of curves Γ j (u 0 ) for d EJQTDE, 2010 No. 60, p. 17 Hence, since dim B Γ j (u 0 ) = 4/3 for all j, see Theorem 11(a) below, it has sense to consider M d (Γ j (u 0 )) for d = 4/3 only.
The sequence (M d (Γ j (u 0 ))) j will be called the Minkowski sequence associated to the trajectory Γ(u 0 ).More generally, for any trajectory Γ(u 0 ) in L 2 (Ω) the corresponding Minkowski sequence is (M d j (Γ j (u 0 ))) j , where d j = dim B Γ j (u 0 ), provided the box dimension exists for each j.The following result deals with properties of the Minkowski sequence associated to the trajectory of the NLS initial-boundary value problem (1) at the point of the bifurcation.As the cruical step in the proof of Theorem 11 we use the result of Žubrinić and Županović [26,Theorem 6].The simpler equivalent formulation can be found in [17,Theorem 3].Because of the completeness, we briefly recall that result here.
In particular, , and equality is achieved if and only if u 0 ∈ span {ϕ j }, u 0 = 0. Furthermore, we have the following asymptotic behaviour (32) max EJQTDE, 2010 No. 60, p. 18 (b) For any u 0 ∈ H 1 0 (Ω), u 0 = 0, we have the following identity In particular, and equality is achieved if and only if , u 0 = 0, besides (33) we have the following identity:
Proof.(a) We consider the case γ 1 > 0 (for γ 1 < 0 the proof is analogous).After eliminating variable t from the system (18) one obtains where , and θ 0 is a constant.The expression (38) enables to represent the spiral Γ j (u 0 ) ⊂ C in polar coordinates (r j , θ j ) in a form r j = f (θ j ), where f (θ j ) = m(θ 0 −θ j ) −1/2 .Now, the expression (30) follows directly from the result of Žubrinić and Županović [26,Theorem 6], which we briefly recalled at the beginning of this Section.We use the mentioned result in our situation by taking α = 1/2.Direct calculation gives lim and the expression (30) follows directly from (28).Furthermore, inequality (31) follows from (30) since The last claim follows from (31) and the Weyl asymptotic result (29) for the eigenvalues of −∆ subject to zero boundary data.
The last claim follows from inequality u 0 H 1 0 ∩H 2 ≥ λ 1 u 0 L 2 , which is an immediate consequence of (13).
Remark 13.For a 0 = 0 all curves Γ j (u 0 ) are rectifiable near the origin, so that dim B Γ j (u 0 ) = 1.Moreover, M 1 (Γ j (u 0 )) is equal to the length of the curve up to a multiplicative constant independent of j and u 0 , see Federer [12, 3.2.39.Theorem], and its length depends on the choice of t 0 = t 0j > t min (u 0 ).If d = dim B Γ j (u 0 ) > 1 (like in Theorem 11), then M d (Γ j (u 0 )) does not depend on the choice t 0j due to excision property of d-dimensional Minkowski content, see [25,Lemma 5.6(b)].
Remark 14.To see how the Minkowski sequence depends on the domain, let us denote by Ω ε = εΩ the domain obtained from Ω by scaling using the factor ε > 0. To any u 0 ∈ L 2 (Ω) we assign u 0ε ∈ L 2 (Ω ε ) with u 0ε (x) = u 0 (x/ε), and similarly for ϕ jε .It is easy to see that λ jε = ε −2 λ j , and from this using (30) we obtain:

Box dimension of the trajectory
For calculation of the box dimension of trajectories of NLS initial-boundary value problem (1) we use the well known fact that the box dimension is invariant with the respect to the bilipshitz mappings.As the references for this important results we refer to Falconer The main result of this section is given by theorem 17 and says that at the bifurcation value a 0 = 0 the box dimension of trajectories of (1) viewed near the origin in L 2 (Ω) (or near the origin of ℓ 2 (C) for ( 3)) has a jump.More precisely, for u 0 = 0 the mapping a 0 → dim B Γ a 0 (u 0 ) is discontinuous at a 0 = 0, where Γ a 0 (u 0 ) is the trajectory of (1) corresponding to u 0 and a 0 .
5.1.Multiple spirals.For a given trajectory Γ(u 0 ) in L 2 (Ω) we define its box dimension via finite-dimensional approximations: where Π k is the orthogonal projection of the Lebesgue space onto span{ϕ 1 , . . ., ϕ k }, or equivalently, from ℓ 2 (C) onto C k , corresponding to the first k components of ℓ 2 (C).The above limit exists due to the monotonicity property of box dimension, see Falconer [11, p. 37].By the multiple spiral (or n-spiral) Γ 2N we mean a curve in where t 0 > 0. The following result will be fundamental for the computation of box dimension of trajectories of problem (1).Its consequence is that box dimension of multiply oscillating trajectories in C N is always less then 2. The claim extends the formula of box dimension of planar spirals due to Tricot [23, p. 121] to oscillating curves in C N .Since the result seems to be interesting for itself, we state it in a slightly more general form.
Theorem 15.Let α k > 0 and λ k = 0 be given numbers, k = 1, . . ., N. Then the corresponding multiple spiral Γ 2N has box dimension Proof.(a) We assume without loss of generality that α 1 is minimal among all α i .Let Γ 2 be the curve in C defined by t → t −α 1 e iλ 1 t .Let us introduce the mapping F : Γ 2 → Γ 2N in the following way.First, we view the curves Γ 2 and Γ 2N as subsets of R 2 and R 2N respectively, and define (writing where Here C 1 and C 2 are real constants.It follows: and ∂g k ∂y 1 .This proves that the mapping F is Lipschitzian.It is easy to see that the inverse Minkowski nondegeneracy of Γ 2N for α 1 ∈ (0, 1) follows from [27, Theorem 1], since Γ 2 is Minkowski nondegenerate, and moreover Minkowski measurable, see [26,Corollary 2], and Γ 2N is lipeomorphic (i.e.bilipshitz equivalent) to Γ Remark 16.The condition on α i to be positive is essential.Indeed, if N = 2 and α 1 = α 2 = 0, then we obtain the curve on the 2-torus, and assuming that λ 1 /λ 2 is rational we obtain that the curve is periodic, hence its box dimension is 1.Therefore, formula (42) does not hold in this case.A more general formulation of the first part of the theorem is as follows: if α j ≥ 0 for all j and the set J = {j : and analogously for the second part.Now, using the result given by theorem 15 we are able to show that at the bifurcation value a 0 = 0 the box dimension of trajectories of (1) viewed near the origin in L 2 (Ω) (or near the origin of ℓ 2 (C) for ( 3)) has a jump.Theorem 17. Assume that u 0 = 0 and γ ∈ R \ {0}.Let Γ(u 0 ) be the trajectory of (1) viewed in a bounded neighbourhood of the origin.If EJQTDE, 2010 No. 60, p. 22 Proof.We consider the case of γ = γ 1 + iγ 2 with γ 2 = 0 and γ 1 > 0 (for γ 1 < 0 the proof is analogous).The case when γ 1 = 0 and γ 2 = 0, see (18), can be treated with slight modifications.
The projection of solution u(t) defined by (19), viewed in ℓ 2 (C), onto its first n components, defines the following curve in C N : where c j = u 0 ,ϕ j u 0 L 2 , t ≥ t 0 with t 0 large enough.We assume for simplicity that c j = 0 for all j.It is natural to define the curve The claim follows from Theorem 15.The case when c j = 0 for at least one j, is treated with minor modifications of the above proof.
If a 0 = 0, then all components z j (t) are rectifiable, hence any finite-dimensional projection π k (Γ 0 ) is rectifiable, where π k : ℓ 2 (C) → C k it the natural projection onto the first k components of ℓ 2 (C).This implies that dim B π k (Γ 0 ) = 1 for each k, and the claim follows.
As we saw before, assuming that a 0 = 0 and γ 1 = 0, the natural projection of any trajectory Γ(u 0 ) into j-th component, that is, into the eigenspace spanned by ϕ j , can be viewed as a curve in the Gauss plane either of box dimension 4/3 if u 0 , ϕ j = 0 (in this case the projection is a spiral), or zero if u 0 , ϕ j = 0 (in this case the projection is simply a point).

Applications
In this section we introduce the notion of spiral chirp.The aim is to study the box dimension of spiral chirps and associated trajectories of NLS problems with different nonlinearities.To achieve this goal, we shall use the results on box dimensions of trajectories obtained in previous sections.Finally, in Subsection 6.3 we show that a class of planar polynomial dynamical systems can be interpreted in terms of a NLS problem of nonlocal type.
6.1.Box dimension of spiral chirps.Results on box dimension of trajectories of NLS initial-boundary value problem (1) obtained in the previous section can be used in order to calculate the box dimension of some other curves, for example spiral chirps.More precisely, in the preceding subsection we have studied box dimension of projections of Γ 0 into subspaces of the form span {ϕ 1 , . . ., ϕ N }, that is, on R 2N .Now we would like to EJQTDE, 2010 No. 60, p. 23 consider projections onto R 2N −1 .The basic model in R 3 is the curve that we call spiral chirp: (45) Γ = (t −α cos t, t −α sin t, t −α cos t) ∈ R 3 : t ≥ t 0 , with α ∈ (0, 1).A more general model could be (α, β)-spiral chirp: for fixed positive α, β, t 0 , and λ j = 0, j = 1, 2. Of course, box dimension of Γ 3 will remain the same if we replace the last component with t −β sin λ 2 t.
Theorem 18.Let α and β be fixed positive numbers such that either Furthermore, assume that λ 1 , λ 2 are nonzero real numbers, and t 0 > 0. Then for the spiral chirp Γ 3 defined by expression (46) we have: The curve Γ 3 is rectifiable if and only if min{α, β} > 1.
At the end of the proof, let us consider the rectifiability in the case when min{α, β} > 1.The spiral Γ 2 is rectifiable, as well as the graph of the chirp z 1 (t) = t β cos t −1 for t near zero, corresponding to the third component of Γ 3 .In this case the curve Γ 3 is rectifiable as well.Indeed, since z 1 (t) has local extrema for t k such that t This implies that dim B Γ 3 = 1.The first inequality in (48) can be justified by isometrically transforming Γ 3 onto the chirp defined in rectangular coordinates, defined on the interval of length of the spiral Γ 2 .
6.2.Hopf bifurcation for other types of nonlinearities.Up to now we studied the initial-boundary value problem for Schrödinger equation where Ω ⊂ R N and γ = γ 1 + iγ 2 ∈ C.More precisely, we considered the nonlinearity V (ρ) = ρ 2 + a 0 , where a 0 was a real bifurcation parameter.In this section we consider some polynomial types of nonlinearity given by the expression (50) where a i , i = 0, . . ., l − 1 are prescribed real parameters.The NLS equation (49) corresponds to the standard generic generalized Hopf bifurcation, see Takens [22], also called the standard Hopf-Takens bifurcation.The following result extends Theorem 17.
Up to now we studied trajectories with spiral components z j (t) converging to zero with equal rate α > 0, that is, |z j (t)| ≃ t −α as t → ∞, for all j, see for example expressions (45) or (46).Next, we are interested in dynamical systems yielding solutions in which spiral components converge to zero with different rates.
A. As the simplest model we consider a polynomial system in in C 2 , defined by where m j are fixed positive integers and a 0 is the real bifurcation parameter.It is easy to see that for a 0 < 0 an attracting invariant set S(a 0 ) is born, defined by |z 1 | 2m Assume that a 0 = 0 and let z(0) = (z 01 , z 02 ) ∈ C 2 be a given initial value, z(0) = 0.It can be shown that the unique solution z(t) = (z j (t)) of system (52) is given by Here z j (t) is a spiral converging to zero with rate α j = 1/(2m j ).Assume that both z 01 and z 02 are different from zero.According to Theorem 15 the box dimension of the corresponding trajectory Γ 52) is a special case of the following Schrödinger equation: with initial and boundary values as in (1), and with scalar products in L (Ω).System (52) is obtained from (54) by considering solutions of the form u(t, x) = z 1 (t)ϕ 1 (x) + z 2 (t)ϕ 2 (x), where ϕ 1 and ϕ 2 are defined in the subsection 1.2.B. Various extensions of ( 52) and (54) are possible, like for example a cyclic polynomial system in R 2k , which we view as a bifurcation problem in C k : For a 0 < 0 an invariant set S(a 0 ) is born in C k defined by , which is a subset the surface defined by j |z j | 2m j = − k 2 a 0 , diffeomorphic to the (2k − 1)-sphere in C k .
Denote by Γ(u 0 ) the trajectory in L 2 (Ω) corresponding to t ≥ 0. Using the definition of dim B Γ(u 0 ) in (40) and Theorem 15 we obtain the following result.
Corollary 22.Let a 0 = 0. Let u(t) be the solution of (54) with F (u) defined by (57), and a 0 = 0, satisfying initial condition u(0) = u 0 ∈ L 2 (Ω).Then dim B Γ(u 0 ) = 2 1 + 0.5 sup{m j : u 0 , ϕ j = 0} −1 .(58) In particular, if m j → ∞ as j → ∞, and u 0 , ϕ j = 0 for infinitely many j's, then dim B Γ(u 0 ) = 2. EJQTDE, 2010 No. 60, p. 28 6.3.Box dimension of trajectories of some nonintegrable Schrödinger problems.Many dynamical systems important for applications, like for example Liénard systems, weakly damped systems etc., are not explicitly solvable.However, in spite of their nonintegrability it is possible to get the information about the box dimension of corresponding trajectories, see [30] for some examples.Here, using the connection between the nonlinear system of ODE's and the Schrödinger equation pointed out already in Section 1.2, we would like to study the box dimension of trajectories of some nonintegrable Schrödinger evolution problems.For this purpose let us consider the following polynomial planar system: ẋ = f (x, y) = λ 1 y + p(x, y) ẏ = g(x, y) = −λ 1 x + q(x, y), (59) where we define p(x, y) = f (x, y) − λ 1 y and q(x, y) = g(x, y) + λ 1 x.Here λ 1 is the first eigenvalue of the −∆| H 1 0 ∩H 2 corresponding to an open and bounded domain Ω ⊂ R N .An initial point in the phase space is prescribed by x(0) = x 10 and y(0) = y 10 .Naturally, we say that the system (59) is equivalent to (54) if there is a bijection between the corresponding solution sets.In this sense it is easy to see that the problem (59) is equivalent to Schrödinger equation (54) with F : Cϕ 1 → Cϕ 1 defined by (60) [F (u)](ξ) = p( Re u, ϕ 1 , Im u, ϕ 1 ) + iq( Re u, ϕ 1 , Im u, ϕ 1 ) ϕ 1 (ξ), ξ ∈ Ω, and with initial function u(0) = z 01 ϕ 1 , where z 01 = x 10 + iy 10 ∈ C. Note that the nonlinear term F (u) is of nonlocal type, i.e.F it is not defined in pointwise manner, but via scalar products, that is, by means of two integrals.The solution of (54) in this case has the form u(t, x) = z(t)ϕ 1 (x), where z(t) = x(t) + iy(t), and the components satisfy (59).Note that (54) is of nonlocal type, and in general not explicitly solvable.Moreover, it is interesting that the study of any system of ODE's can be considered as the study of a Schrödinger problem.
Proposition 23.Any Cauchy problem for the system ẋ = f (x), where x ∈ R n and f : R n → R n is a Lipschitz function, can be naturally interpreted as a Schrödinger problem of the form (54), with F (u) written explicitly, see (62) below.
Proof.We can assume without loss of generality that n is even, n = 2k (if n = 2k − 1, we can add a new trivial equation ẋ2k = 0, x 2k (0) = 0).Now the system can be written in the form (61) ẋj = f j (x, y) ẏj = g j (x, y) j = 1, . . ., k where x, y ∈ R k .Defining z j = x j + iy j and p j (x, y) = f j (x, y) − λ j y j , q j (x, y) = g j (x, y) + λ j x j , j = 1, . . ., k, where λ j are the first k eigenvalues of −∆ (counting the multiplicity) with zero boundary data on a given bounded open domain Ω in R N , then we have żj = −iλ j z j + (p j + iq j ).Multiplying with eigenfunctions ϕ j = ϕ j (ξ) normalized in L 2 (Ω), and EJQTDE, 2010 No. 60, p. 29
should indicate the projection of the considered trajectory into C 4 .−0.3 + i ⋅ 0