User:Eugene M. Izhikevich/Proposed/Meander of spiral waves

• Dr. Claudia Wulff, University of Surrey, UK

Dr. Claudia Wulff accepted the invitation on 12 March 2007 (self-imposed deadline: 12 September 2007, now delayed to 12 March 2008).

Meander of spiral waves in reaction-diffusion system is a rotation of the spiral wave superimposed with a periodic motion which is caused by a Hopf instability or by external periodic forcing of a rigidly rotating spiral wave.

Spiral waves in reaction-diffusion systems

Spiral waves have been observed in various different biological, chemical, and physical systems. They occur, for instance, in the Belousov-Zhabotinsky reaction (see e.g. Steinbock et al 1993, Figure 1, and Winfree 1991) and in the catalysis on platinum surfaces, as observed by the group of the Nobel Laureate Ertl, see Nettesheim et al 1993.

Figure 1: Meandering spiral wave in the Belousov Zhabotinsky reaction. Adapted by permission from Maximilian publishers Ltd: Nature, Steinbock et al, copyright 1993. The tip trajectory is overlaid with a white curve.

These systems are modelled by reaction-diffusion equations on the plane $\tag{1} u_t = D\Delta u + f(u,\mu), \qquad x\in{\mathbb R}^2,\; u\in{\mathbb R}^N,$

where the diffusion matrix $D$ is diagonal with nonnegative entries.

We can rewrite (1) as an abstract differential equation $\tag{2} u_t = A u + B(u,\mu) = F(u,\mu)$

where $u = u(\cdot)$ lies in some infinite-dimensional phase space $Y\ ,$ e.g., the space of uniformly continuous functions.

Homogeneity implies that any solution behaves in the same fashion if we move it to a different location in the medium and rotate it about its center; its dynamical behavior does not depend upon its location in the medium. Rotating the pattern $u(t,x)$ by the angle $\varphi\in {\rm SO}(2)$ about $x=0 \ ,$ and subsequently translating it by the vector $a\in{\mathbb R}^2 \ ,$ which moves a point $x$ to $x+a \ ,$ results in a new solution of (1) given by

$[(\varphi,a)u](x,t) := \tilde{u}(x,t) = u(R_{-\phi}(x-a),t);$

Here $R_\phi$ is a rotation by $\phi$ in ${\mathbb R}^2\ .$

The set of all rotations and translations $(\varphi,a)\in {\rm SO}(2)\times\R^2$ constitutes the Euclidean symmetry group ${\rm SE}(2)$ of the plane. The combined effect of translating and rotating a solution first by $(\varphi,a)$ and then by $(\tilde{\varphi},\tilde{a})$ is expressed by the group multiplication

$\tag{3} (\phi_1,a_1)(\phi_2,a_2) = (\phi_1+\phi_2, a_1 + R_{\phi_1} a_2),~~ ~~\phi_i \in {\rm SO}(2),~a_i \in {\mathbb R}^2,~i=1,2.$

The simplest possible motion of spiral waves in the plane are rigid rotations. A rigidly-rotating spiral wave is periodic in time; in the laboratory frame, the spiral tip moves on a circle with uniform angular velocity while the spiral wave rotates about its tip with the same velocity. A rigidly-rotating spiral wave of a reaction-diffusion system (1), rotating around $x=0$ satisfies $u(t,x) = [(\omega_0^{\rm rot}t,0)u_0](x) = u_0( R_{\omega_0^{\rm rot} t} x).$ Consequently, it is an equilibrium in a coordinate frame which rotates with frequency $\omega_0^{\rm rot}\ .$ In these new coordinates, (1) is given by $\tag{4} u_t = D\Delta u + \omega_0^{\rm rot} \partial_\varphi u + f(u,\mu).$

Slightly more complicated are meandering or drifting spiral waves. The motion of a meandering wave is quasi-periodic in the laboratory and time-periodic in a co-rotating frame. Its tip traces out a flower pattern with inward or outward petals; see Figure 1 and Figure 2.

Figure 2: The patterns traced out by the tips of rigidly-rotating, meandering (outward petals), drifting, and meandering (inward petals) spiral waves (from left to right).

Drifting spiral waves arise if the petality of the flower pattern changes from inward to outward. At such a transition point, the radius of the circle traced out by the tip tends to infinity and the spiral-wave tip drifts along a line towards infinity while oscillating about the line, see Figure 3.

Figure 3: Phase diagram for the spiral wave dynamics for a reaction-diffusion system depending on the parameters a, b. Reprinted Fig. 1 by permission from Barkley (1994), copyright by the American Physical Society.]

A drifting spiral wave is time-periodic in a suitable moving frame. Meandering or drifting spiral waves of (1) occur via a transition from rigidly-rotating spiral waves which is described below.

Relative equilibria and relative periodic orbits

We say that an abstract differential equation (2) is $G$-equivariant if

$g F(u) = F(g u) \quad \forall g \in G.$

We conclude that the abstract differential equation (2) corresponding to the reaction diffusion system (1) is $G$-equivariant where $G = {\rm SE}(2)\ .$ Symmetry reduction of a $G$-equivariant differential equation (2) gives a system on the space of group orbits $Y/G \ .$

A rigidly-rotating spiral wave is an example of a relative equilibrium. A relative equilibrium of a $G$-equivariant differential equation (2) is an equilibrium in the space of group orbits, or, in other words, an invariant group orbit of (2). Since it becomes stationary in a corotating frame it is also called a rotating wave. Another example of a relative equilibrium of a reaction-diffusion system (1) on the plane is a travelling wave. It becomes an equilibrium in a comoving frame.

A solution $u(t)$ of a $G$-equivariant differential equation (2) is called relative periodic orbit if it is a periodic orbit in the space of group orbits. This means that there exists $T_0>0$ and $g_0 \in G$ such that $u(T_0) \ =\ g_0 u(0) \ ,$ see Figure 4.

Figure 4: A relative periodic orbit

We call $T_0>0$ the relative period of the relative periodic orbit and the corresponding group element $g_0$ the drift symmetry of the relative periodic orbit with respect to $u_0=u(0)\ .$ If $G = {\rm SE}(2)$ and $g_0$ is a translation we call the relative periodic orbit a modulated traveling wave (MTW); If $g_0 =\phi_0= \omega^{\rm rot}_0 T_0$ is a rotation we call the RPO a modulated rotating wave (MRW), see Figure 3. Note that a modulated traveling wave becomes periodic in a comoving frame, and a modulated rotating wave becomes periodic in a frame rotating with frequency $\omega^{\rm rot}_0\ .$

Hence a meandering spiral wave of a reaction-diffusion system (1) is a modulated rotating wave, and a drifting spiral wave a modulated traveling wave.

Centre manifold reduction

We saw above that a rigidly-rotating spiral wave $u_0$ is an equilibrium of (4). We linearize (4) about this pattern at $\mu=0$ and obtain the operator $L_0 = D \Delta + \omega_0^{\rm rot} \partial_\varphi + {\rm D}_u f(u_0(x),0).$

Hypothesis The spectrum of $L_0$ considered in the space $Y$ has $n+3$ eigenvalues on the imaginary axis, counted with multiplicity, and the rest of the spectrum is contained strictly in the left half-plane.

We emphasize that $\lambda=0$ and $\lambda=\pm{\rm i}\omega_0^{\rm rot}$ are always eigenvalues of $L_0$ on account of the Euclidean symmetry group. These eigenvalues correspond to the derivatives of $(\varphi,a)u_0$ with respect to $\varphi$ and $a$ at $(\varphi,a) = (0,0)\ .$ Throughout, we denote the generalized eigenspace associated with the remaining $n$ eigenvalues which are not related to the symmetry by $N\ .$

Theorem (Sandstede et al 1997) Under the above hypothesis any solution $u(t,x)$ of (1) which is close to the rotating wave $u_0(x)$ or a translated and rotated version of it for all times $t$ lies on a centre manifold $M_c$ diffeomorphic to ${\rm SE}(2) \times N\ .$

There is a map $\Psi:N \to Y$ such that the diffeomorphism $M_c \simeq {\rm SE}(2) \times N$ is given by $M_c = {\rm SE}(2)\Psi(N) \ ,$ and $\Psi(v) = u_0 + v + hot \ .$ So in particular, the rotating wave $u_0$ has coordinates $u_0 \simeq (g={\rm id}, v=0)\ .$

The above theorem holds more generally near relative equilibria of $G$-equivariant semilinear parabolic PDEs (2), see Sandstede et al (1997). In this case the centre manifold is diffeomorphic to $M_c \simeq G \times N \ .$

Centre bundle equations

Due to $G$-equivariance, the dynamics on the centre manifold $M_c \simeq G \times N$ takes the form $\tag{5} \dot{g} = g F_G(v),~~ \dot{v} = F_N(v),$

see (Fiedler et al 1996).

Let us consider the case that $G$ is the Euclidean symmetry ${\rm SE}(2)$ of the plane and that the relative equilibrium ${\rm SE}(2) u_0$ is a rotating wave with rotation frequency $\omega^{\rm rot}_0\ .$ Setting $F_G = (F_\phi,F_a)\ ,$ taking $\phi_1=\phi\ ,$ $\phi_2 = \omega^{\rm rot} t\ ,$ and $a_1 =a\ ,$ $a_2 = v t$ in (3), differentiating at $t= 0\ ,$ and setting $\omega^{\rm rot}=F_\phi\ ,$ $v=F_a\ ,$ we see that the first equation of (5), which models the drift dynamics near the rotating wave, takes the following form

$$\tag{6} \dot{\phi} = F_\phi(v),\quad \dot{a} = R_\phi F_a(v),$$

see (Fiedler et al 1996) and (Golubitsky et al 1997). Here $\phi$ models the angle of the spiral wave, $a$ the tip of the spiral and $v$ its shape.

Moreover $F_\phi(0) = \omega^{\rm rot}_0$ is the rotation frequency of the rotating wave. As in the general case, the rotating wave ${\rm SE}(2) u_0$ becomes an equilibrium of the slice equation

$$F_N(0) = 0\ .$$ These equations have first been formulated by Barkley (1993) and have then been derived by Fiedler et al (1996) and Golubitsky et al (1997).

Meandering transition

We now assume that both $F_N(\cdot, \mu)$ and $F_G(\cdot, \mu) = (F_\phi(\cdot, \mu),F_a(\cdot, \mu))$ depend on an external parameter $\mu \in \R\ .$ In a meandering transition the symmetry reduced system undergoes a Hopf bifurcation. This was first understood and numerically verified by Barkley (1994). Suppose that this bifurcation occurs at $v=0$ for $\mu=0\ ,$ let $\pm {\rm i} \omega^{\rm Hopf}_0$ be the Hopf eigenvalues of ${\rm D}_v F_N(0,0)$ and that ${\rm D} F_N(0,0)$ has no other eigenvalues in ${\rm i} \omega^{\rm Hopf}_0{\mathbb Z}\ .$

If the usual transversality condition for Hopf bifurcation is satisfied then there is a smooth path of points $v(s)=v(t=0,s)$ on periodic solutions $v(t,s)$ of the $\dot{v}$-equation with period $T(s) \approx T^{\rm Hopf}_0 = 2\pi/\omega_0^{\rm Hopf}$ and parameter $\mu(s)$ such that $v(0) = 0\ ,$ $T(0) = T^{\rm Hopf}_0\ ,$ $\mu(0) = 0\ .$

The periodic orbit through $v(s)$ of the $\dot{v}$-equation corresponds to a relative periodic orbit through $x(s) \simeq ({\rm id}, v(s))$ of the original ODE (6) with drift symmetry $\gamma(s) = (\phi(s), a(s))\ .$ Here $\phi(s)$ and $a(s)$ are obtained by integrating the $\dot\phi$ rsp. $\dot{a}$-equation of (6) from $0$ to $T(s)\ .$ Note that $(\phi, a)$ is a translation by $a$ if $\phi=0 ~{\rm mod}~2\pi\ .$ If $\phi \neq 0~{\rm mod}~2\pi$ then $(\phi, a)$ is a rotation around $x_c = (\phi, a)x_c = R_\phi x_c +a\ ,$ i.e. around $\tag{7} x_c= ({\rm id}-R_\phi)^{-1}a\ .$

We now distinguish two cases:

• If $\phi(s) \neq 0~{\rm mod}~2\pi$ then $x(s)$ lies on a meandering spiral wave (a modulated rotating wave), and this is the typical case;
• If $\phi(s) = 0~{\rm mod}~2\pi$ then $x( s)$ lies on a modulated travelling wave.

Resonance drift

Note that $\phi(s) \approx \omega^{\rm rot}_0 T^{\rm Hopf}_0 = \frac{\omega^{\rm rot}_0}{\omega^{\rm Hopf}_0}2\pi.$ Hence modulated traveling waves bifurcate if

$\tag{8} \frac{\omega^{\rm rot}_0}{\omega^{\rm Hopf}_0} \in \Z,$

i.e., if there is a resonance between the rotation frequency $\omega^{\rm rot}_0$ and the Hopf frequency $\omega^{\rm Hopf}_0$ of the rotating wave, see Barkley (1994), Fiedler et al (1996), Golubitsky et al (1997), Wulff (1996). This phenomenon is called resonance drift. From (7) we see that the centre of rotation $x_c$ tends to infinity at a resonance.

To understand the meandering and drifting motion in more detail, we rewrite the $\dot\phi$-equation of (6) along the periodic orbit $v(t,s)$ (with $v(0,s) = v(s)$) at parameter $\mu(s)$ as

$\frac{{\rm d} }{ {\rm d} t} \phi(t,s) = \omega^{\rm rot}(s) + \tilde{F}_\phi(v(t,s),\mu(s)).$

where $\tilde{F}_\phi(v(t,s),\mu(s))$ has zero average, i.e.,

$\int_0^{T(s)}\tilde{F}_\phi(v(t,s),\mu(s)) {\rm d} t=0$

and $\phi(0,s) = 0\ ,$ $\phi(T(s),s) = \phi(s)\ .$ Integrating gives

$\phi(t,s) = \omega^{\rm rot}(s) t + \tilde\phi(t,s)$

where $\tilde\phi(t,s)$ is $T(s)$-periodic in $t\ .$ Inserting this into the $\dot{a}$-equation of (6) and integrating gives

$\tag{9} a(t,s) = a_0 + \int_0^t R_{\omega^{\rm rot}(s) t}R_{\tilde\phi(t,s)} F_a(v(t,s),\mu(s))\, {\rm d}t.$

Identifying ${\mathbb R}^2$ and ${\mathbb C}$ and expanding the term $R_{\tilde\phi(t,s)} F_a(v(t,s),\mu(s)) = \sum_{k=-\infty}^\infty B_k(s) {\rm e}^{{\rm i}k\omega(s)t }$ into a Fourier series and integrating, we get

$\tag{10} a(t,s) = a_0 + \sum_{k=-\infty}^\infty B_k(s) \frac{{\rm e}^{{\rm i}(\omega^{\rm rot}(s) +k\omega(s))t}-1} {{\rm i}(\omega^{\rm rot}(s) +k\omega(s))},$

where $\omega(s) = 2\pi/T(s)$ is the relative frequency of the RPO, and $\omega(0) =\omega^{\rm Hopf}_0\ .$ This gives in general a quasiperiodic tip motion with frequencies $\omega(s)$ and $\omega^{\rm rot}(s)\ .$

Therefore, the translation $a(t,s)$ is bounded, and in fact quasi-periodic in $t\ ,$ as long as $\omega^{\rm rot}(s)+k\omega(s)\neq0$ for all $k\in\Z\ .$ The resulting pattern is meandering. If, however, $\omega^{\rm rot}(s) + \tilde{k} \omega(s) = 0$ for some $\tilde{k}\in\Z\ ,$ then we have

$a(t,s) = a_0 + B_{\tilde{k}}(s) t + \sum_{k\neq\tilde{k}} B_k(s) \frac{{\rm e}^{{\rm i}(\omega^{\rm rot}(s)+k\omega(s))t}-1} {{\rm i}(\omega^{\rm rot}(s)+k\omega(s))}.$

The tip of the associated spiral wave moves in an oscillatory fashion along the direction $B_{\tilde{k}}(s)$towards infinity. Hence, the spiral wave is drifting.

Inward and outward petals of meandering spirals

Near a $k:1$-resonance the dominating term is the $k$-th term of (10). From (9) and the fact that $\|v(t,s)\| = O(s)$ we see that the $0$th term of (10) is $O(1)$ if $F_a(0,0)\neq 0$ and that the other terms of (10) are $O(s)\ .$ So $a(t,s)$ performs an epicyclic motion consisting of a large circle rotating with frequency $\omega^{\rm rot}(s)+k\omega(s)$ and an $O(1)$ rotation with frequency $\omega^{\rm rot}(s)\ .$ Assume that the rotation frequency of the rotating wave does not vanish $\omega^{\rm rot}_0 \neq 0\ .$ Then for $s\approx 0$ also $\omega^{\rm rot}(s)\neq 0\ .$ If the rotating wave rotates counterclockwise (rsp. clockwise) and the large circle is traversed by the spiral tip counterclockwise (rsp. clockwise), so that $\omega^{\rm rot}(s)$ and $\omega^{\rm rot}(s)+k\omega(s)$ have the same sign then the meandering motion has inward petals; if $\omega^{\rm rot}(s)$ and $\omega^{\rm rot}(s)+k\omega(s)$ have different sign the meandering pattern has outward petals. Consequently, there is a change of petality when a $k:1$ resonance is passed transversely.

Other mechanisms of meandering

• A transition from rigidly rotating to meandering and drifting spiral waves is also caused by periodic external forcing of the reaction-diffusion system when the reaction term $f(u,\mu,\omega^{\rm ext} t)$ in (1) is independent of $t$ for $\mu=0$ and $2\pi$-periodic in $\tau=\omega^{\rm ext} t$ for $\mu\neq 0\ .$ The same conditions for resonance drift hold, with $\omega^{\rm Hopf}_0$ replaced by $\omega^{\rm ext}\ ,$ see e.g. Wulff (1996, 2000) and references therein.

• Perturbations of the system (1) which break the translational symmetry, but preserve rotational symmetry around $x=0\ ,$ induce a transition of the family of spirals rotating rigidly around a point different from the origin, to, typically, finitely many families of meandering spirals, see LeBlanc and Wulff (2000). Corresponding experiments on spiral waves of the BZ reaction forced by light pulses have been performed by Grill et al (1996).

• The meandering transition from rotating waves to modulated rotating waves and modulated traveling waves has also been studied in systems with spherical symmetry $G={\rm SO}(3)$ where analogous results apply, see Wulff (2000), Chan (2006) and references therein.

• The meandering transition and conditions for the bifurcation have been studied for $m$-armed spiral waves as well. In this case resonance drift occurs under more restrictive conditions, see e.g. Fiedler et al (1996), Wulff (2000).

Meandering of Archimedean spirals

The above hypothesis is only satisfied for spiral waves $u_0(x)$ which decay at infinity, $u_0(x) \to 0$ as $x \to \infty \ ,$ see {Sandstede et al 1997). For Archimedean type spiral waves, i.e., spiral waves of the form $u(r,\phi) \simeq u(r-\kappa\phi)$ for $r \to \infty\ ,$ the above theorem does not apply, and other methods, have to be employed, see (Sandstede and Scheel 2001, 2006) for details.

Hyper-Meandering

• Neimark-Sacker bifurcation (secondary Hopf bifurcation) from a meandering spiral wave or external forcing of meandering spirals leads to relative invariant tori, i.e., invariant tori of the symmetry reduced dynamics. This is a form of generalized meandering. Generalized drifting spiral waves, i.e. solutions which are quasiperiodic in a comoving frame, occur if the resonance condition $\omega_{\rm rot} = k \omega_1 + m \omega_2, k,m \in {\mathbb Z},$ is satisfied. Typically the motion is bounded, see (Lamb et al 2006).

• LeBlanc and Wulff (2000) showed that under perturbations of the system which break the translational symmetry typically finitely many families of meandering spirals which do not rotate around zero persist as relative invariant tori.

• As shown by Fiedler and Turaev (1998), a Takens-Bogdanov bifurcation of the base dynamics (i.e. of the $\dot{v}$-equation of (6)) induces a Brownian motion like dynamics of the spiral tip (i.e. of the $\dot{a}$-equation of (6)).

• Chaotic movements of the symmetry reduced dynamics caused by break up of invariant tori also induce Brownian-like motion of the spiral tip, see Ashwin et al (2001).

References

• P. Ashwin, I. Melbourne and M. Nicol. "Hypermeander of spirals; local bifurcations and statistical properties". Physica D, 156:364-382, 2001.

• D. Barkley. "Euclidean symmetry and the dynamics of rotating spiral waves", Phys. Rev. Lett., 72:164-167, 1994.

• M. Braune and H. Engel. "Compound rotation of spiral waves in a light-sensitive Belousov-Zhabotinsky medium". Chem. Phys. Lett., 204(3,4):257-264, 1993.

• D. Chan. "Hopf bifurcations from relative equilibria in spherical geometry". J. Differential Equations, 226:118-134, 2006.

• B. Fiedler and D. Turaev. "Normal forms, resonances, and meandering tip motions near relative equilibria of Euclidean group actions." Arch. Rat. Mech. Anal., 145:129-159, 1998.

• B. Fiedler, B. Sandstede, A. Scheel, C. Wulff. "Bifurcation from relative equilibria of non-compact group actions: Skew products, meanders and drifts". Doc. Math. J. DMV, 1:479-505, 1996.

• M. Golubitsky, V. LeBlanc, and I. Melbourne. "Meandering of the spiral tip - an alternative approach". J. Nonl. Sci., 7: 557-586, 1997.

• S. Grill, V. S. Zykov, and S. C. Müller. "Spiral Wave Dynamics under Pulsatory Modulation of Excitability." J. Phys. Chem.: 100: 19082--19088, 1996.

• J. Lamb, I. Melbourne, C. Wulff. "Hopf bifurcation from relative periodic solutions: Secondary bifurcations from meandering spirals". J. Difference Equations and Applications, 12(11):1127-1145, 2006.

• V. LeBlanc, C. Wulff. "Translational symmetry breaking for spiral waves". J. Nonlinear Sci. 10:569-601, 2000.

• S. Nettesheim, A. von Oertzen, H.H. Rotermund, and G. Ertl. "Reaction diffusion patterns in the catalytic CO-oxidation on Pt(110) -- front propagation and spiral waves", J. Chem. Phys., 98:9977-9985, 1993.

• B. Sandstede, A. Scheel, and C. Wulff, "Dynamics of spiral waves on unbounded domains using center-manifold reductions". J. Diff. Eq., 141:122-149, 1997.

• B Sandstede and A Scheel. "Superspiral structures of meandering and drifting spiral waves". Physical Review Letters, 86:171-174, 2001.

• B. Sandstede and A. Scheel. "Curvature effects on spiral spectra: Generation of point eigenvalues near branch points". Physical Review E, 73:016217-016224, 2006.

• O. Steinbock, S.C. Müller and V.S. Zykov. "Control of spiral wave dynamics in active media by periodic modulation of excitability". Nature, 366:322-324, 1993.

• A.T. Winfree. "Varieties of spiral wave behaviour: an experimentalist's approach to the theory of excitable media". Chaos, 1:303-334, 1991.

• C.Wulff. "Theory of meandering and drifting spiral waves in reaction-diffusion systems". Dissertation, Berlin, 1996.

• C. Wulff. "Transition from relative equilibria to relative periodic orbits". Doc. Math. J. DMV, 5: 227-274, 2000.

Further Reading

• M. Golubitsky and I. Stewart. "The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space". Birkhäuser, 2002.

• P. Chossat and R. Lauterbach. "Methods in Equivariant Bifurcations and Dynamical Systems". World Scientific, 2000.

See also

Dynamical Systems, Equilibrium, Periodic Orbit, Normal Form, Equivariant dynamical Systems, Equivariant Bifurcation Theory.