Stability and Chaos Additional material We will be

Stability and Chaos

Additional material • We will be using two additional sources posted on TCU Online: • 1) Berry, M. V. , ‘Regular and Irregular Motion’ in Topics in Nonlinear Mechanics, ed. S. Jorna, Am. Inst. Ph. Conf. Proc No. 46, pp. 16 -120, 1978. (B-1) • 2) Berry, M. V. , ‘Semiclassical Mechanics of Regular and Irregular Motion’ in Les Houches Lecture Series Session XXXVI, eds. G. Iooss, R. H. G. Helleman and R. Stora, North Holland, Amsterdam, 1983. (B-2) http: //michaelberryphysics. wordpress. com/ Sir Michael Victor Berry (born 1941)

(B-2) 2. 1 Regular and irregular classical motion • Let us consider dynamics of a classical system with N degrees of freedom described by a Lagrangian (Hamiltonian) – i. e. system without dissipation • It turns out that motion of such a system can be of two types: regular and irregular • Regular motion (very small fraction of all systems in the universe): trajectories with neighboring initial conditions separate linearly • Irregular motion (overwhelming majority of all systems in the universe): trajectories with neighboring initial conditions separate exponentially, resulting in a sensitivity to initial conditions

Integrable systems (B-2) 2. 2 (B-1) 2 • The simplest type of a system with a regular motion is an integrable system, for which there exist N smooth independent functions that are constant along the trajectory of the system in the phase space • Such quantities are called constants of motion • For integrable systems, the number of constants of motion is equal to the number of degrees of freedom N

Integrable systems (B-2) 2. 2 (B-1) 2 • E. g. , for Lagranigans (Hamiltonians) without explicit time dependencies, one of the constants of motion is the total energy (Hamiltonian) • E. g. , for systems with central potentials, three constants of motion are the components of the total angular momentum • Along the trajectory: • These equations can be solved for momenta:

Integrable systems (B-2) 2. 2 (B-1) 2 • We can consider a canonical transformation to a new set of coordinates Qj and momenta Pj such that the Fj functions to be the new (conserved) momenta • Since the new momenta are constant, the new Hamiltonian will be independent of the generalized coordinates: • Therefore

Integrable systems (B-2) 2. 2 (B-1) 2 • The problem is solved if we can express the new generalized coordinates in terms of the old generalized coordinates • We choose a generating function of the form • For such generating function, the Legendre transformations yield • So, the problem is solved since the 2 N constants Cj and βj can be found

Integrable systems (B-2) 2. 2 (B-1) 2 • The described canonical transformations will work if the conserved quantities Fj satisfy conditions necessary to consider them as independent generalized canonical momenta: • 1) They should be independent (one cannot be derived from the other) • 2) Poisson brackets of all the pairs of the Fj functions vanish • The existence of N functions Fj implies that the system is limited to move on N-dimensional manifold in the 2 N-dimensional phase-space

Integrable systems (B-2) 2. 2 (B-1) 2 • To describe the motion-restricting N-dimensional manifold we introduce N vector fields in the 2 Ndimensional phase space • On each manifold, these vector fields will be smooth and independent (because Fj are smooth and independent)

Integrable systems • The normals defined as: (B-2) 2. 2 (B-1) 2 to the N-dimensional manifold can be • On each manifold, the vector fields will be tangential to the manifold (perpendicular to the normals):

Bounded integrable systems (B-2) 2. 2 (B-1) 2 11. 1 • Let us concentrate of systems with bound motion, i. e. motion with a finite accessible phase space • In this case, the motion-restricting manifold is compact • Topology theorem (without proof): A compact manifold “parallelizable” with N smooth independent fields must be an N-torus:

Bounded integrable systems (B-2) 2. 2 (B-1) 2 11. 1 • Such tori are called invariant tori, because the orbit starting on such a torus, remains on the torus forever • An N-dimensional torus has N independent irreducible circuits γj on it • We coordinatize the phase space using Qj and Pj, with {Pj} defining the invariant torus and {Qj} defining the coordinates on the torus • Standard set of such variables: actions and angles on tori

Example: 2 D harmonic oscillator • Let us consider a 2 D harmonic oscillator: • This Hamiltonian can be separated into two independent Hamiltonians, for each of which • Invariant tori: (B-1) 2 11. 1

Example: 2 D harmonic oscillator (B-1) 2 11. 1 • If one frequency is a multiple of another, e. g. then the trajectory will close on itself and repeat the same pattern every period • If the frequencies are commensurate (n – rational), then the orbit still will be closed, but will trace out more than one path around p 1 and q 1 before closing • If the frequencies are incommensurate (n – irrational), then the orbit will never close, gradually covering the entire surface of the torus

Example: 2 D harmonic oscillator (B-1) 2 11. 1 • For the case of incommensurate frequencies, the orbit is called dense periodic • Dense periodic orbit does not pass through exactly the same point twice, but will eventually pass arbitrarily close to every point • So, the motion is confined to a toroidal surface – a 2 D manifold in a 4 D phase space

Nonintegrable systems (B-2) 2. 3 (B-1) 3 11. 2 • Is integrability the rule or the exception? • If all Hamiltonian systems were integrable, the constants of motion would always exist and our inability to determine them for all but the simplest problems would merely reflect our lack of analytical ingenuity • As a matter of fact, there are rigorous analytical and numerical methods showing that most Hamiltonian systems are not integrable, while integrable systems form a very small set (possibly of zero measure) • For nonintegrable systems, the trajectory in phase space fills a region of dimensionality greater than N

Perturbations (B-2) 2. 3 (B-1) 3 11. 2 • Many Hamiltonian systems can be modeled as a combination of a sum of an integrable system and a small nonintegrable perturbation • Does a small nonintegrable perturbation destroy the tori? • The answer is: it depends • In most cases, the tori persist under small perturbation albeit distorted • Some are destroyed and such tori form a finite set, which grows with the perturbation

Time-dependent perturbation theory (B-1) 3 11. 2 12. 2 • We start with an unperturbed integrable Hamiltonian • This system has N conserved quantities • We produce canonical transformations employing Hamilton’s principal function • The new Hamiltonian (Kamiltonian) is required to be identically zero, so for the new constant momenta and coordinates we have:

Time-dependent perturbation theory (B-1) 3 11. 2 12. 2 • The perturbed Hamiltonian is • We use the same functional dependence for S, only now the new canonical variables may not be constant, and the Kamiltonian does not vanish • Now the equations of motion are • These equations generally cannot be solved since the perturbation in nonintegrable

Time-dependent perturbation theory (B-1) 3 11. 2 12. 2 • Now we take advantage of the smallness of ΔH • In the first order approximation we can write • The notation with zero indicates that after differentiation we substitute αk and βk with their unperturbed (constant) values • These equations can be written in a symplectic form

Time-dependent perturbation theory (B-1) 3 11. 2 12. 2 • These equations can be integrated to yield γ 1(t) and by inverting the functional dependencies to obtain the time dependencies of qk and pk to the first order of perturbation • The second order perturbation equations are obtained by using γ 1(t) dependence in the right hand side: • And for the higher orders:

Poisson brackets formalism 12. 2 • Sometimes we need to know the time evolution of some functions of the new canonical set • For the perturbed system, the evolution is described by • We can invert the functional dependence and make K depend on ci • Then:

Poisson brackets formalism 12. 2 • If these equations cannot be solved exactly, we apply the perturbation theory approach, i. e. the righthand-sides are evaluated for the unperturbed motion, etc. • This is a generalization of the perturbation theory equations derived previously

Types of perturbed motion • As determined by the perturbation treatment, the parameters of the orbit may vary with time in two ways • There may be a small variation of the orbit around the unperturbed solution, which is not growing with time • There may be a perturbation of the orbit, which is slowly diverging form the unperturbed solution – secular change • The first type of perturbation does not change average parameters of the orbit; the secular perturbation slowly changes orbit parameters 12. 2

12. 2 Fake example 1: the harmonic oscillator potential as a perturbation • Unperturbed Hamiltonian: • The momentum is conserved • The Hamilton-Jacobi equation:

12. 2 Fake example 1: the harmonic oscillator potential as a perturbation

12. 2 Fake example 1: the harmonic oscillator potential as a perturbation • Let us assume that the small perturbation is • We pretend that we don’t know that the perturbation is integrable • The Kamiltonian is • The perturbed equations of motion:

12. 2 Fake example 1: the harmonic oscillator potential as a perturbation • First order perturbation approximation: • Assuming • Then: • Inverting for x and p:

12. 2 Fake example 1: the harmonic oscillator potential as a perturbation • Second order perturbation approximation: • Solutions: • Inverting for x and p:

12. 2 Fake example 1: the harmonic oscillator potential as a perturbation • If we continue to higher orders, it can be shown that • For • Big surprise!

Fake example 2: simple pendulum 12. 3 • Full Hamiltonian: • Expanding: • To obtain a harmonic oscillator approximation, we retain terms with i = 0, 1 • Let’s assume that the angles are small, but not small enough to use the harmonic oscillator approximation • Therefore, we have to retain a term with i = 2 as a small perturbation for the harmonic oscillator

Fake example 2: simple pendulum 12. 3 • Full (reduced) Hamiltonian: • We assume that we don’t know the solutions of the anharmonic oscillator and accept the anharmonicity as a perturbation • The harmonic oscillator has been solved previously employing the action-angle formalism

Fake example 2: simple pendulum • The first order time-dependence: • Averaging over a period Therefore, β is not a constant anymore, but it is changing with time in the following manner: • Averaging over a period • J (which is the measure of the amplitude) does not change with time 12. 3

(B-1) 3 10. 7 12. 4 Time-independent perturbation theory • We will consider conservative periodic separable systems with many degrees of freedom and a perturbation parameter ε • For the unperturbed system we introduce a set of action-angle variables {J 0 i}, {w 0 i} such that • The original generalized coordinates, which are multiple-periodic in {w 0 i} (with period unity) can be expanded in the Fourier series of the unperturbed angles

(B-1) 3 10. 7 12. 4 Time-independent perturbation theory • A compact form of the same Fourier expansion: • The perturbed Hamiltonian can be expanded in the powers of ε: • In the perturbed system, {J 0 i}, {w 0 i} remain a valid set of canonical variables, although they are no longer action-angle variables (since the full Hamiltonian depends on {w 0 i} now), and therefore {J 0 i} are not constants of motion

(B-1) 3 12. 4 Time-independent perturbation theory • If the invariant tori exist in the perturbed system, there must be a new set of action-angle variables {Ji}, {wi} such that • The sets {Ji}, {wi} and {J 0 i}, {w 0 i} are related by a canonical transformation generated by a function: • From the Legendre transformation it follows that • Thus the question of the continuing existence of tori reduces to the question of whether the latter equation can be solved (we have to find Y)

(B-1) 3 12. 4 Time-independent perturbation theory • We expand both the generating function and the new Hamiltonian in powers of ε: • On the other hand

(B-1) 3 12. 4 Time-independent perturbation theory • Using Taylor series expansion:

(B-1) 3 12. 4 Time-independent perturbation theory • Using Taylor series expansion:

(B-1) 3 12. 4 Time-independent perturbation theory • Therefore:

(B-1) 3 12. 4 Time-independent perturbation theory • Thus:

(B-1) 3 12. 4 Time-independent perturbation theory • Therefore, for the first order of ε: • If the system remains on the invariant tori, the coordinates and momenta should be periodic functions of {w 0 i} • Since H 1 is a function of q’s and p’s, it is a (given) periodic function of {w 0 i}

(B-1) 3 12. 4 Time-independent perturbation theory • Since Y is a function of q’s and p’s, it is a periodic function of {w 0 i}; so are all the expansion terms Yk: • The expansion terms Yk are defined up to an arbitrary constant, since Y is a generating function • Let’s choose Y 1 in the following form:

(B-1) 3 12. 4 Time-independent perturbation theory • Let us consider constant (time-independent) terms: • The remaining terms:

(B-1) 3 12. 4 Time-independent perturbation theory • In principle we can continue this algorithm for higher orders in ε • But we have to proceed with caution • What if the unperturbed orbit was closed? • Then we have a case of commensurate frequencies

(B-1) 3 12. 4 Time-independent perturbation theory • We can always find such set of ji that • The resonance! (Zero divisors) • The series diverges? ! • Actually, there are two concerns about convergence: of the series in powers of ε and the sum

(B-1) 3 12. 4 Time-independent perturbation theory • Moreover, even for an open orbit (noncommensurate frequencies), as we go higher and higher of the integer indices in the j-vector we can always find a combination of integers such that • And we have problems with convergence again! • Do all tori get destroyed? • Not so!

(B-1) 3 (B-2) 2. 4 11. 2 KAM theorem • So, what happens to the perturbed invariant tori? • The answer to this question is given by the celebrated “KAM theorem” (Kolmogorov-Arnold. Moser theorem) Andrey Nikolaevich Kolmogorov Андрей Николаевич Колмогоров (1903 - 1987) Vladimir Igorevich Arnold Владимир Игоревич Арнольд (1937 - 2010) Jürgen Moser (1928 – 1999)

KAM theorem (B-1) 3 (B-2) 2. 4 11. 2 • KAM theorem (we mentioned this result earlier): “If the bounded motion of an integrable Hamiltonian H 0 with N degrees of freedom is disturbed by a small perturbation ΔH, that makes the total Hamiltonian, H = H 0 + ΔH, nonintegrable and if two conditions are satisfied: (a) the perturbation ΔH is small (b) the frequencies ωi of H 0 are incommensurate, then the motion in 2 N-dimensional phase space remains confined to an toroidal manifold of dimension N, except for a negligible set of initial conditions that result in a trajectory on a manifold with a dimension greater than N”

KAM theorem (B-1) 3 • Perturbation theory in the form discussed previously was known and employed for several centuries • However, it turned out that this theory was a very crude tool for studying the delicate problems arising from the small denominators • The central feature of KAM is the replacement of the series expansions of the conventional perturbation theory by a series of successive approximations to the suspected new tori • This approach has a vastly improved convergence leading to the proof of theorem

KAM theorem (B-1) 3 • Without going deep into technicalities of the proof, we will point out main features of this proof • Essentially, each new torus generated by the previous approximation is itself made the basis of the next approximation, rather than expressing all approximations in terms of the unperturbed torus • The central result is that the process of generating “perturbed” tori does converge for small but finite ε almost always • Another result is that the unperturbed tori in the neighborhood of those on which the orbits are closed (or partially closed) are almost all destroyed

KAM theorem (B-1) 3 • We will not discuss explicit forms of the KAM theorem expansion series because of the complex mathematics • Instead, we will illustrate the idea of improved convergence on a simple example from conventional calculus: finding the zero of a function f(x) • We start with a guess: x 0 (an unperturbed value) • Then we use a perturbation theory

KAM theorem • After standard series reversion: • This is analog of the standard perturbation theory • This is a very slowly converging method of finding a zero (B-1) 3

KAM theorem • In calculus to find a zero of a function, we usually use Newton’s method instead (B-1) 3

Example: calculating π (B-1) 3 • Let's compare the speed of conversion of both methods on the example of the following function: • The speed of convergence is best illustrating by the following table:

The anatomy of torus destruction (B-1) 3 (B-1) 4 • The tori that are destroyed correspond to commensurable frequencies • These destroyed tori give rise to zero denominators • How about tori with non-zero but small denominators? Are they all destroyed too? • No. KAM specifies the widths of the destroyed regions • To illustrate this, we will consider a simple example for N = 2

Simple example: N = 2 (B-1) 4 • For the tori with closed orbits: • For tori with closed orbits this rational • For tori with open orbits the frequency ratio is not rational (cannot be written as a ratio of two integers) • But it can be approximated arbitrarily closely by a rational number, e. g. • However there is a better way to approximate irrational tori by the rational ones

Simple example: N = 2 (B-1) 4 • Instead of a decimal ratio we represent the ratio as a continued fraction: • For π • Defining • The convergence is then

Simple example: N = 2 (B-1) 4 • For π: • KAM theorem proves convergence of such accelerated iteration-perturbation scheme for the torus generator Y for all initial tori whose frequency ratio is sufficiently irrational for the following relation to hold (for 2 D case): • The tori excluded: are mostly destroyed

Main result (revisited) (B-1) 4 • In a perturbed system, most orbits lie on tori in phase space • Those that do not, form a small but finite set pathologically disturbed in phase space near each unperturbed torus that supported closed or partially closed orbits • The motion in the narrow (0(s-μ)) gaps will be pushed out of the gaps onto a nearby preserved torus by a further random perturbation • The gaps resulting from low-order resonances are relatively wide and give rise to non-trivial observable and computable effects: deterministic chaos

(B-2) 2. 3 (B-1) 6 Back to dimensional considerations • Hamiltonian systems of N degrees of freedom with constant energy are restricted to move on a (2 N-1)dimensional manifold in the 2 N-dimensional phase space: energy hypersurface • For such systems that are non-integrable, there is an important class of motion – ergodic: orbits eventually pass through practically all points on the energy hypersurface • Case N = 1 is pathological (2 N – 1 = N) all orbits are both integrable and ergodic • Therefore true ergodic (and chaotic) motion can exist only for systems with N > 1

(B-2) 2. 3 (B-1) 6 Back to dimensional considerations • The first non-trivial dimension is N = 2 • The dimension of the energy hypersurface is 2 N - 1 =3 • This 3 D energy hypersurface is not an ordinary 3 D position space, because it is non-Euclidian, closed and may be multiple-connected (cf. 2 D surfaces in 3 D space may have a non-trivial topology)

Poincaré maps (B-2) 2. 6 (B-1) 6 11. 5 • One of the most efficient techniques to monitor the breakdown of integrabiltiy is Poincaré mapping • We study a 2 D slice (section) of the trajectory on 3 D hypersurface and calculate the locations of points where the orbits pass through the section • If the system is integrable (toroidal orbit) then the 2 D Poincaré map is a smooth closed curve Jules Henri Poincaré (1854 – 1912)

Example: Hénon-Heiles Hamiltonian 11. 5 • In 1960’s, Michel Hénon and Carl Heiles considered the Kepler astronomical problem equivalent to a 2 D harmonic oscillator with two cubic perturbations: • In polar coordinates: • The potential has a 3 -fold symmetry Carl Heiles (born 1939) Michel Hénon (born 1931)

Example: Hénon-Heiles Hamiltonian • Hénon and Heiles calculated the Poincaré maps in the plane for different values of E and λ = 1 Chaos! 11. 5

Example: Hénon-Heiles Hamiltonian • Hénon and Heiles calculated the Poincaré maps in the plane for different values of E 11. 5

Example: Hénon-Heiles Hamiltonian • How does KAM compare with the perturbation theory? (B-1) 6 11. 5

Open (non-conservative) systems • So far we considered Hamiltonian (closed) systems • Does deterministic chaos emerge in open (nonconservative) systems? • Yes • Before considering specific examples of such systems we will introduce several concepts helpful for quantitative description of deterministic chaos • Those are, among others: stationary orbits and points of the phase space, Lyapunov exponents, fractals, bifurcations

Stationary points • Let us recall a simple pendulum (closed system) • Its phase portrait • All orbits are stable

Stationary points • Let us recall a simple pendulum (closed system) • For large momenta, it will not oscillate anymore, but will start rotating instead • The special case of a trajectory separating oscillations and rotations – separatrix • Nodal points of the separatrix correspond to unstable equilibrium of a pendulum with its bob at the top

Stationary points • For a damped pendulum (open system) • Its phase portrait • Phase portrait converges to a stationary point

Stationary points • On the phase portrait, stationary points can be of several types: attractors, repellers, and saddle points • Attractors: solutions converge on them • Repellers: solutions escape from them • Saddle points: solutions converge on them along one direction and escape from them along the other

Examples of stationary points • For a damped oscillator, the phase diagram contains attractors and saddle points

Limit cycles • Of the phase diagram, in addition to stationary points there could be limit cycles • For instance, for the van der Pol oscillator: • Zero point is a repeller Balthasar van der Pol (1889 -1959) 11. 3

Lyapunov exponents 11. 4 • Let us consider two orbits in phase space • Initial separation between these orbits is s 0 • How does a separation between these two orbits s changes with time? • To measure this time change, we use Lyapunov exponents λ defined as: Aleksandr Mikhailovich Lyapunov Александр Михайлович Ляпунов (1857 – 1918)

11. 9 Hausdorff dimension • For evaluating trajectories of dynamic systems and describing natural objects it is useful to introduce a generalized treatment of dimension – Hausdorff dimension • For a line, a square, and a cube we reduce the scale of each object by a factor of r and add objects so that we fill the same space with N objects Felix Hausdorff (1868 – 1942)

Hausdorff dimension • Hausdorff dimension is defined as: 11. 9

11. 9 Cantor set • We begin with a straight line • We remove the middle third of the line • Iteratively removing the middle third the remaining segments, we generate a Cantor set • What is the Hausdorff dimension of it? Georg Ferdinand Ludwig Philipp Cantor (1845 – 1918)

11. 9 Cantor set • We begin with a straight line • We remove the middle third of the line • Iteratively removing the middle third the remaining segments, we generate a Cantor set • What is the Hausdorff dimension of it? • Fractal (non-integer) Housdorff dimension Georg Ferdinand Ludwig Philipp Cantor (1845 – 1918)

11. 9 Sierpinski carpet • We reproduce iterations similar to Cantor set in 2 D to generate a Sierpinski carpet • What is the Hausdorff dimension of it? Wacław Franciszek Sierpiński (1882 – 1969)

Menger sponge • We reproduce iterations similar to Cantor set in 3 D to generate a Menger sponge Karl Menger (1902 - 1985)

Other related examples • Cantor dust

Other related examples • Sierpinski triangle

Other related examples • Sierpinski pyramids

Self-similarity • Important feature of all these fractal objects – they are self-similar

Self-similarity • In nature, there are plenty of examples of self-similar structures with essentially fractal dimensions

Self-similarity • Self-similarity has been with the humankind for centuries

Fractals and chaos • There is a direct relationship between the fractal geometry and deterministic chaos • To demonstrate this connection we will first consider a discrete form of evolution equations • One of the simplest discrete evolution equations generating chaos is the logistic equation: • The variable is restricted to the following domain: • a - is called a control parameter 11. 8

Logistic equation • Successive iterations of the logistic equation are expected to bring x closer and closer to a limiting value x∞, so that further iterations produce no additional change in x • Stationary points of this equation lie at the intersection of a straight line (l. h. s. ) and a parabola (r. h. s. ) 11. 8

Logistic equation • As the value of the control parameter grows, the maximum of the parabola rises and the value of x∞ grows 11. 8

Logistic equation • As the value of the control parameter grows, the maximum of the parabola rises and the value of x∞ grows 11. 8

Logistic equation • How stable is the stationary point x∞? • For convergence we need: • What happens if a > 3? 11. 8

Logistic equation • The stationary point loses its stability and a stable solution is a cycle with two alternating stable points • Transition from one stable points to two stable points is called a bifurcation 11. 8

Logistic equation 11. 8

Logistic equation 11. 8 • With the increase of the control parameter the cycle becomes unstable and generates a double cycle (two more bifurcations)

Logistic equation 11. 8 • With the increase of the control parameter the cycle becomes unstable and generates a double cycle (two more bifurcations)

Logistic equation • With further growth of the control parameter, the process of multiplication of bifurcations continues until we reach a chaotic type of evolution 11. 8

Logistic equation • With further growth of the control parameter, the process of multiplication of bifurcations continues until we reach a chaotic type of evolution 11. 8

11. 8 Logistic equation • Such type of transition to chaos is called the Feigenbaum scenario Mitchell Jay Feigenbaum (born 1944)

Logistic equation • Despite its chaotic nature, the Feigenbaum set has a rich structure, which is fractal 11. 8

Logistic equation 11. 8

Logistic equation • For the Feigenbaum set we introduce a discrete equivalent of the Lyapunov exponent 11. 8

Feigenbaum scenario for continuous systems 11. 7 • We now consider a non-conservastive system of a driven and damped simple pendulum • It is equivalent to a system of three first-order equations (recall: chaos exists in systems with N > 1)

Feigenbaum scenario for continuous systems • Remarkably, with growing g this system exhibits a transition to chaos extremely similar to that for the logistic equation 11. 7

Feigenbaum scenario for continuous systems • Remarkably, with growing g this system exhibits a transition to chaos extremely similar to that for the logistic equation 11. 7

Feigenbaum scenario for continuous systems • Remarkably, with growing g this system exhibits a transition to chaos extremely similar to that for the logistic equation 11. 7

Feigenbaum scenario for continuous systems • Remarkably, with growing g this system exhibits a transition to chaos extremely similar to that for the logistic equation 11. 7

Feigenbaum scenario for continuous systems • Remarkably, with growing g this system exhibits a transition to chaos extremely similar to that for the logistic equation 11. 7

Questions?