# A brief biography of Carmen Chicone

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## A brief biography of Carmen Chicone

by Yuri Latushkin, University of Missouri, Columbia, MO 65211
and Weishi Liu, University of Kansas, Lawrence, KS 66045
Carmen Chicone.

Carmen C. Chicone was born 4 March 1946 in Elmira, New York and grew up in Watkins Glen, New York, the son of Italian-American immigrants. He earned his undergraduate degree from the State University of New York at Albany in 1968 and earned his Ph.D. in mathematics at the University of Wisconsin in 1977. After one year as a visiting assistant professor at the University of Missouri-Columbia, he joined the faculty in 1978 and has been a Professor of Mathematics since 1989. Since 2005, he has held an Adjunct Professorship in Veterinary Pathobiology at the University of Missouri.

 Carmen in the 1980s. Carmen and his wife Jenny Chicone.

In summer 1985, Carmen visited the Instituto Nacional the Matemática Pura e Aplicada in Rio de Janeiro, Brazil, to collaborate with Jorge Sotomayor, and in the following summer he visited the Federal University of Rio de Janeiro, to collaborate with Izabel Camacho. The winter of 1989 he spent on sabbatical leave at Limburgs Universitair Centrum (now called Hasselt University) in Hasselt, Belgium, visiting Freddy Dumortier.

Carmen works in the areas of dynamical systems and applied mathematics. He has published over 70 research articles and two books, one of which is co-authored with Yuri Latushkin. During 19-21 May 2006 a

Dynamical Systems Workshop

was held at the University of Missouri-Columbia in honor of Carmen Chicone's 60th birthday (see also the report in DSWeb Magazine).

 Since the celebrated work of Anosov on diffeomorphisms, much research has been carried out relating to this subject. In a series of works, mainly in collaboration with Richard Swanson, Chicone examined relationships between Anosov diffeomorphisms, infinitesimal ergodicity and spectrum of the adjoint representation of diffeomorphism. For a diffeomorphism $$f$$ of a compact manifold $$M$$, a natural adjoint representation is the operator $$f_{*}: X \to Tf\cdot X\cdot f^{-1}$$ on the space $$\Gamma(TM)$$ of continuous sections of the tangent bundle of $$M$$. Jorge Sotomayor and Carmen Chicone.

If $$m$$ is a measure on $$M$$, one can define the space $$\Gamma^2(TM)$$ of square-integrable sections and the space of $$H^1(TM)$$ sections with one square-integrable weak derivative. The adjoint representation $$f_*$$ acts on both $$\Gamma^2(TM)$$ and $$H^1(TM)$$. A diffeomorphism $$f$$ is said to be infinitesimally ergodic with respect to an $$f$$-invariant measure $$m$$ if $$I-f_*: H^1(TM) \to H^1(TM)$$ has dense range. In their first joint work [3] Chicone and Swanson disproved a conjecture by Robbin that every Anosov diffeomorphism is infinitesimally ergodic. Continuing their work in [4], they established that the spectra of $$f_*$$ on $$\Gamma(TM)$$ and $$\Gamma^2(TM)$$ (with $$m$$ being $$f$$-invariant) are identical. As a consequence of a result of Mather that $$f$$ is Anosov if and only if $$f_*$$ acting on $$\Gamma(TM)$$ is hyperbolic [Nederl. Akad. Wetensch. Indag. Math. 30 (1968), 479-483], $$f$$ is Anosov if $$f_*$$ is hyperbolic on $$\Gamma^2(TM)$$. For a flow $$f^t$$ generated by the vector field $$X$$, they also established a very deep result -- the spectral mapping theorem: the spectrum $$f_*^1$$ on $$\Gamma(TM)$$ is the exponentiation of the spectrum of $$L_X$$ on $$\Gamma(TM\vert[X])$$ where $$L_X$$ is the Lie derivative with respect to $$X$$. These results have several applications: for example, the geodesic flow on the unit tangent bundle of a Riemannian manifold of negative curvature is Anosov ([5]) and, in the case of constant negative curvature, it is infinitesimally ergodic ([6]).

In collaboration with Paul Ehrlich, Chicone considered the following interesting question: Given a vector field $$X$$ on a manifold $$M$$, when can one find a metric $$g$$ (Riemannian or Lorentzian) so that the integral curves of $$X$$ are all geodesics of $$g$$? Their three main results are:

1. For a nonsingular vector field $$X$$ on a contractible subset $$S$$ of $${\mathbb{R}}^2$$, null geodesibility of $$X$$ is equivalent to $$X$$ being pre-Hamiltonian (i.e., Hamiltonian possibly after reparametrization) on $$S$$;
2. Timelike, spacelike or Riemannian pre-geodesibility of $$X$$ are all equivalent to $$X$$ being gradient-like (i.e., $$X$$ has a differentiable Lyapunov function) ([7]);
3. A vector field $$X$$ is null geodesible if and only if there exists a smooth one-form $$\omega$$ and a smooth line subbundle $$\bf {L}$$ of $$TM$$ such that $$\omega(X)=0$$, $$\omega(\bf {L})\neq 0$$ at any point, and $$i_Xd\omega=0$$ ([8]).
In a related result ([9]), Chicone and Ehrlich generalized results of Gromoll and Meyer [Ann. Math. 90 (1969), 75-90] and of Hawking and Penrose [Proc. Roy. Soc. London Ser. A 314 (1970), 529-548] and proved that if the integral of the Ricci curvature of the tangent vector field of a complete geodesic in a Riemannian manifold or a complete nonspacelike geodesic in a Lorentzian manifold is positive, then the geodesic contains a pair of conjugate points. It seems that further investigation of the relations between dynamical behavior and these geometric properties would be an interesting research project.

 In the 1980s, Chicone (collaborating with several researchers) made several important contributions to the second part of Hilbert's 16th problem. The problem is to express the number of limit cycles in terms of the degree of planar polynomial vector fields, and it stimulated a tremendous amount of research activity. Though simply stated, this problem turns out to be very difficult: even for quadratic polynomial vector fields the answer is still unknown. Carmen Chicone and Freddy Dumortier.

One of Chicone's results ([10] jointly with Douglas Shafer), combined with the results of Poincaré [J. Mathématiques 7 (1881), 375-422] and Andronov [Theory of bifurcations of dynamic systems on the plane. Halsted Press, New York, 1973], provides a partial solution of Dulac's problem on the finiteness of limit cycles for polynomial vector fields: If a quadratic system has infinitely many limit cycles, then the limit cycles have an accumulation point at infinity. Later, Dulac's problem was completely solved by Ecalle and Il'yashenko (see Finiteness Theorems for Limit Cycles. Transl. Math. Monogr., 94. Amer. Math. Soc., Providence, RI, 1991]).

An important problem related to the study of limit cycles is the understanding of the period functions of centers and the bifurcation of limit cycles from centers, especially the bifurcation of limit cycles from isochronous centers (i.e. the corresponding period function is constant). Inspired by the work of Bautin [Amer. Math. Soc. Transl. 100 (1954), 1-19], Chicone and Marc Jacobs considered families of plane analytic vector fields with $$N$$ parameters such that each member of the family has a center. At an isochronous center in such a family, they construct an ideal (in the polynomial ring in $$N$$ variables) from the Taylor coefficients of the period function and show that the number of generators of their ideal is the maximal number of critical points of the period function that can bifurcate from the isochronous center. These results are applied to quadratic systems with Bautin centers and to one-degree-of-freedom kinetic + potential' Hamiltonian systems with polynomial potentials. Carmen also obtained other results concerning the monotonicity ([11, 12]) and finiteness of critical values ([13]) of period functions of planar vector fields. In related work, Chicone and Jacobs showed that at most three limit cycles can bifurcate from the harmonic oscillator $$\dot x=-y$$ and $$\dot y=x$$ (which has an isochronous center) in a quadratic family ([14, 15]). This result is now in textbooks [L. Perko, Differential Equations and Dynamical Systems, 2nd Ed., New York, Springer-Verlag, 1996].

One of Chicone's conjectures (which appeared in Mathematical Reviews MR 94h:58072) has been the subject of several recent papers; it is still open: If a quadratic system has a center with a period function that is not monotonic, then, by an affine transformation and a constant rescaling of time, the system can be transformed to the Loud normal form $$\dot x =-y+B xy$$, $$\dot y= x+D x^2 +F y^2$$. Moreover, a system in Loud normal form has a center at the origin with a period function that has at most two critical points.

Nonlinear oscillation problems have played an important role in the development of dynamical system theory. In the 1990s, Chicone considered two types of problems:

1. continuation of periodic orbits in a family of periodic orbits of autonomous perturbed oscillations, and
2. for periodic perturbed oscillations, the perturbation of subharmonics from resonant periodic orbits (both isolated and non-isolated).
For problems of type (1), Chicone applied Lyapunov-Schmidt reduction to obtain bifurcation functions whose zeros correspond to periodic orbits that can be continued ([16, 17, 18, 19]). An important contribution of this work is the recasting of bifurcation problems in a form where their geometry is explicitly incorporated. For problems of type (2), Chicone constructed bifurcation functions whose zeros provide subharmonic orbits via a generalization of Melnikov's method ([20, 21]). He applied these techniques in many areas, for example, in the continuation of periodic orbits of resonantly coupled oscillators, the synchronization of inductively coupled Van der Pol oscillators, hydrodynamic instability of steady states of Euler's hydrodynamic partial differential equation, entrainment domains of periodically perturbed Van der Pol oscillatiors, and later, a Keplerian binary system perturbed by periodic gravitational radiation.

Hyperbolicity plays a central role in characterizing asymptotic behavior of dynamical systems. For finite-dimensional dynamical systems, this concept is well understood. In collaboration with Yuri Latushkin and Stephen Montgomery-Smith, Chicone investigated this problem for infinite-dimensional dynamical systems. For a $$C^0$$ semi-group $$\{T_t: t \ge 0\}$$ on a Hilbert space $$H$$, Chicone and Latushkin found a characterization of the hyperbolicity of $$T_t$$ in terms of the dissipativity of its generator ([22]). This result was obtained via a proper generalization of the concept of hyperbolicity to an evolution family $$\{U(t,s): t \ge s\}$$ using exponential dichotomy as, for example, the work of Daleckii and Krein [Stability of solutions of differential equations in Banach space, Transl. Math. Monogr., 43. Amer. Math. Soc., Providence, RI, 1974]. Extending their work, Chicone and Latushkin ([23]) examined hyperbolicity of linear skew-product flows of Hilbert bundles (bundles with a Hilbert space $$H$$ as fibers) over a compact metric space $$X$$. The hyperbolicity of such a linear skew-product flow is characterized by the hyperbolicity of a semi-group of weighted composition operators defined on $$L^2(X,H)$$. A quadratic Lyapunov function can be constructed using the equivalence of hyperbolicity and dissipativity.

Spectral theory for semi-groups generated by infinite-dimensional dynamical systems is more involved than the corresponding theory for finite-dimensional systems. One of most important problems is the validity of the spectral mapping property that relates the spectrum of the semi-group to the spectrum of its generator via exponentiation. In finite-dimensional systems the exponential of the spectrum of the generator is always the spectrum of the semi-group. Together with Latushkin and Montgomery-Smith ([24]), Chicone examined the spectrum of the kinematic dynamo operator $$L$$ for an ideally conducting fluid on a closed Riemannian manifold $$X$$. One of the difficulties associated with $$L$$ is working with a space of divergence-free vector fields. Inspired by the work of de la Llave [Geophys. Astrophys. Fluid Dynam. 73 (1993), 123-131], they obtained conditions under which the spectral mapping theorem holds for $$\dim X \ge 3$$. For $$\dim X \ge 2$$, the spectrum of $$e^{tL}$$ for $$t\neq 0$$ was shown to be exactly one annulus centered at the origin, which confirms a conjecture of de la Llave who obtained a similar result under a more restrictive assumption. Thus, the spectral mapping theorem implies that the spectrum of $$L$$ is a strip containing the purely imaginary axis. Consequently, the ideally conducting fluid does not have an exponential dichotomy. The boundary of the spectrum of $$L$$ was further characterized via Lyapunov exponents following previous work of Latushkin and Stëpin [Uspekhi Mat. Nauk 46(2) (1991), 85-143]. This work was generalized in [25] where the operator is generated by an arbitrary vector field $$u$$ (not necessarily divergence-free). Another aspect of the generalization is the consideration of the semi-group on divergence-free vector fields induced by a general cocycle. In this setting, they obtained the annular hull theorem: the exponentiation of the spectrum of the generator and its hull sandwich' the spectrum of the semi-group. As a corollary, the spectral radius of the generator is equal to the bound on the growth of the semi-group. This result is delicate; it is not true for general semi-groups and their generators.

Chicone and Latushkin ([26]) also considered the linear differential operator $$D_\epsilon u=\epsilon\Delta u+{\rm curl}(v\times u)$$ on the unit tangent bundle of a compact surface of constant negative curvature $$k$$, where $$\Delta$$ is the Laplacian with respect to the Sasaki metric, $$v$$ the infinitesimal generator of the geodesic flow and $$\epsilon=1/R_{\rm m}>0$$ ($$R_{\rm m}$$ denotes the magnetic Reynolds number). This defines an idealization of the dynamo operator. Let $$s_\epsilon=\sup\{{\rm Re}\,\lambda\vert\lambda\in\sigma(D_\epsilon)\}$$ be the spectral radius of $$D_\epsilon$$. They showed that $$s_\epsilon>0$$ and $$\limsup_{\epsilon\to0}s_\epsilon>0$$ if $$R_{\rm m}\geq\sqrt{-k}$$, and hence, the associated kinematic dynamo given by $$\dot u=D_\epsilon u$$ and $${\rm div}\,u=0$$, is fast.

Another research direction of Chicone and Latushkin is their study of center manifolds of mild solutions of semi-linear nonautonomous differential equations ([27]). The novelty of this work lies in its generality and direct application of a Lyapunov-Perron type treatment. An interesting point is that they formulated the usual gap condition in spectral terms and showed that this condition is, in fact, a condition on the corresponding spaces of differentiable functions. This allowed them to obtain a direct proof of the existence of a smooth global center manifold.

The theory of normally hyperbolic invariant manifolds is important in the geometric study of global dynamical systems. An invariant manifold is normally hyperbolic if the generalized Lyapunov numbers are strictly less than 1. A fundamental fact is that normally hyperbolic invariant manifolds persist under $$C^1$$ perturbations. In practice, one often wishes to continue a normally hyperbolic invariant manifold with respect to some parameter beyond such a perturbation. What property of the underlying parametrized system allows the continuation of a normally hyperbolic invariant manifold? A natural guess is that the manifold can be continued as long as the generalized Lyapunov numbers are uniformly bounded away from 1. (In fact, this criterion is wrong, but it has been used by several authors in the early literature on this subject.) Collaborating with Weishi Liu [28], Chicone investigated the continuation of resonant periodic orbits of periodically perturbed systems. As a by-product, they found that the above condition on the generalized Lyapunov numbers is not sufficient for continuation of normally hyperbolic invariant manifolds.

An interest in two-body motion in modern physics (that is, no action at a distance) led Chicone to study approximations of functional differential equations by Newtonian equations, with post-Newtonian corrections [29]. Ryabov made an initiation in the study of delay equations with small delays and, among others, Driver further investigated the problem [SIAM Review 10 (1968), 329-341; J. Differential Equations 21 (1976), 148-166]. Chicone proved for a smooth delay equation with state variable in $${\mathbb{R}}^n$$ and small delay $$\tau$$ that the so-called Ryabov's special solutions form an $$n$$-dimensional smooth inertial manifold that varies smoothly with the delay. The reduced dynamical systems on the inertial manifold can be viewed as an approximation of the full delay equation. A widely employed different approach of approximating the delay equation is via a Taylor expansion of the vector field in $$\tau$$ at $$\tau=0$$. For every $$N$$th-order truncation of this expansion, the corresponding approximation, which is a system of ODEs, has an $$n$$-dimensional slow manifold. In [30, 31], Chicone established that the reduced vector fields on the $$n$$-dimensional inertial manifold and the $$n$$-dimensional slow manifold agree up to order $$N-1$$. This result provides a justification of the Taylor expansion approximation method.

Chicone has devoted serious efforts to various applied aspects of dynamical systems in collaborating with researchers from other disciplines. The dynamics of continuous stirred tank reactors has many engineering and industrial applications. For example, continuous stirred tank reactors are widely used in labs to help design waste water treatment. With collaborators ([32, 33, 34, 35]), Chicone applied bifurcation and singularity theory to provide concrete results on maximal multiplicity of critical points for various models corresponding to several different physical situations. Early studies on critical points had been conducted by Dangelmayr and Stewart [SIAM J. Math. Anal. 15 (1984), 423-445; SIAM J. Appl. Math. 45 (1985), 895-918] and by Golubitsky and Keyfitz [SIAM J. Math. Anal. 11 (1980), 316-339]. Chicone and his collaborators further analyzed the dynamical behavior including the stability of critical points, existence of a circulatory attractor, and the route to chaos ([36]). With Latushkin and David Retzloff ([37]) he considered the dynamical behavior of certain tubular flow reactors. The mathematical models taken are coupled reaction-diffusion equations on a bounded interval. They established existence and persistence of multiple steady states with parameters. Most interestingly, the stability of steady states is completely determined: two special steady states corresponding to maximal and minimal energy are (locally) stable, and all others are unstable.

During the period 1996-2000, teamed with physicist Bahram Mashhoon and chemical engineer David Retzloff at MU, Chicone investigated global and long-term dynamical behavior of a Keplerian binary system perturbed by periodic gravitational radiation. Mathematically, the system is a periodically perturbed Hamiltonian system with all physical terms specified. They examined several important issues, including the existence of periodic orbits ([38, 39]), resonance capture followed by sustained resonance ([40, 41]), and chaotic behavior ([42, 43]). The mathematical analysis involves a large body of the theory of dynamical systems: KAM theory, the method of averaging, invariant manifolds, Melnikov method, etc. This work is a seminal example of applying mathematical theory in physical problems.

Most recently, in a series papers [44, 45, 46, 47, 48, 49, 50] co-authored with Mashhon, Chicone investigated the tidal and inertial effects of ultrarelativistic motion. Through their careful analysis they discovered, contrary to Newtonian expectations, that if the relative speed of free relativistic particles exceeds the critical value $$c/\sqrt{2}$$ where $$c$$ is the speed of the light, then the first-order gravitational tidal effects cause an acceleration (resp. deceleration) of the ultrarelativistic particles in the swarm moving in directions normal (resp. parallel) to the jet direction. Their results suggest a black hole tidal acceleration mechanism that could be relevant to the creation of highly energetic particles by micro-quasars in our galaxy as well as the origin of the highest energy cosmic rays reaching the Earth.

 Carmen is currently working with his graduate students on several research projects: modeling and optimization in cryobiology, minimal distortion transformations in elasticity, a two-body problem in acoustics, and Hertzian contact impact oscillators. In addition, he is writing a new book with working title An Invitation to Applied Mathematics. Chicone's current PhD students James Benson, Michael Heitzman, Kenneth Felts, and Oksana Bihun.

Finally, on a personal note, both Yuri and Weishi have had very pleasant collaborations with Carmen and they cherish their friendship with him.

## Bibliography

 1 C. Chicone, Quadratic gradients on the plane are generically Morse-Smale.'' J. Differential Equations 33 (1979), 159-166. 2 C. Chicone and D. S. Shafer, Quadratic Morse-Smale vector fields which are not structurally stable.'' Proc. Amer. Math. Soc. 85 (1982), 125-134. 3 R. C. Swanson and C. Chicone, Anosov does not imply infinitesimally ergodic.'' Proc. Amer. Math. Soc. 75 (1979), 169-170. 4 C. Chicone and R. C. Swanson, The spectrum of the adjoint representation and the hyperbolicity of dynamical systems.'' J. Differential Equations 36 (1980), 28-39. 5 C. Chicone and R. C. Swanson, Infinitesimal hyperbolicity implies hyperbolicity.'' Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979), pp. 50-64, Lecture Notes in Math., 819, Springer, Berlin-New York, 1980. 6 C. Chicone, Tangent bundle connections and the geodesic flow.'' Rocky Mountain J. Math. 11 (1981), 305-317. 7 C. Chicone and P. Ehrlich, Gradient-like and integrable vector fields on $$R\sp 2$$.'' Manuscripta Math. 49 (1984), 141-164. 8 C. Chicone and P. Ehrlich, Lorentzian geodesibility.'' Differential topology-geometry and related fields, and their applications to the physical sciences and engineering, 75-99, Teubner-Texte Math., 76, Teubner, Leipzig, 1985. 9 C. Chicone and P. Ehrlich, Line integration of Ricci curvature and conjugate points in Lorentzian and Riemannian manifolds.'' Manuscripta Math. 31 (1980), 297-316. 10 C. Chicone and D. S. Shafer, Separatrix and limit cycles of quadratic systems and Dulac's theorem.'' Trans. Amer. Math. Soc. 278 (1983), 585-612. 11 C. Chicone, The monotonicity of the period function for planar Hamiltonian vector fields.'' J. Differential Equations 69 (1987), 310-321. 12 C. Chicone and F. Dumortier, A quadratic system with a nonmonotonic period function.'' Proc. Amer. Math. Soc. 102 (1988), 706-710. 13 C. Chicone and F. Dumortier, Finiteness for critical periods of planar analytic vector fields.'' Nonlinear Anal. 20 (1993), 315-335. 14 C. Chicone and M. Jacobs, Bifurcation of critical periods for plane vector fields.'' Trans. Amer. Math. Soc. 312 (1989), 433-486. 15 C. Chicone and M. Jacobs, Bifurcation of limit cycles from quadratic isochrones.'' J. Differential Equations 91 (1991), 268-326. 16 C. Chicone, Lyapunov-Schmidt reduction and Melnikov integrals for bifurcation of periodic solutions in coupled oscillators.'' J. Differential Equations 112 (1994), 407-447. 17 C. Chicone, A geometric approach to regular perturbation theory with an application to hydrodynamics.'' Trans. Amer. Math. Soc. 347 (1995), 4559-4598. 18 C. Chicone, Periodic solutions of a system of coupled oscillators near resonance.'' SIAM J. Math. Anal. 26 (1995), 1257-1283. 19 C. Chicone, Periodic orbits of coupled oscillators near resonance.'' Planar nonlinear dynamical systems (Delft, 1995). Differential Equations Dynam. Systems 5 (1997), 203-227. 20 C. Chicone, Bifurcations of nonlinear oscillations and frequency entrainment near resonance.'' SIAM J. Math. Anal. 23 (1992), 1577-1608. 21 M.B.H. Rhouma and C. Chicone, On the continuation of periodic orbits.'' Methods Appl. Anal. 7 (2000), 85-104. 22 C. Chicone and Y. Latushkin, Hyperbolicity and dissipativity.'' Evolution equations (Baton Rouge, LA, 1992), 95-106, Lecture Notes in Pure and Appl. Math., 168, Dekker, New York, 1995. 23 C. Chicone and Y. Latushkin, Quadratic Lyapunov functions for linear skew-product flows and weighted composition operators.'' Differential Integral Equations 8 (1995), 289-307. 24 C. Chicone, Y. Latushkin, and S. Montgomery-Smith, The spectrum of the kinematic dynamo operator for an ideally conducting fluid.'' Comm. Math. Phys. 173 (1995), 379-400. 25 C. Chicone, Y. Latushkin, and S. Montgomery-Smith, The annular hull theorems for the kinematic dynamo operator for an ideally conducting fluid.'' Indiana Univ. Math. J. 45 (1996), 361-379. 26 C. Chicone and Y. Latushkin, The geodesic flow generates a fast dynamo: an elementary proof.'' Proc. Amer. Math. Soc. 125 (1997), 3391-3396. 27 C. Chicone and Y. Latushkin, Center manifolds for infinite-dimensional nonautonomous differential equations.'' J. Differential Equations 141 (1997), 356-399. 28 C. Chicone and W. Liu, On the continuation of an invariant torus in a family with rapid oscillations.'' SIAM J. Math. Anal. 31 (1999-2000), 386-415. 29 C. Chicone, What are the equations of motion of classical physics?'' Can. Appl. Math. Q. 10 (2002), 15-32. 30 C. Chicone, Inertial and slow manifolds for delay equations with small delays.'' J. Differential Equations 190 (2003), 364-406. 31 C. Chicone, Inertial flows, slow flows, and combinatorial identities for delay equations.'' J. Dynam. Differential Equations 16 (2004), 805-831. 32 C. Chicone and D. G. Retzloff, Dynamics of the CR equations modeling a constant flow stirred tank reactor.'' Nonlinear Anal. 6 (1982), 983-1000. 33 D. G. Retzloff, P. C.-H. Chan, R. Mohamed, D. Offin, and C. Chicone, Maximal multiplicity of the critical points of the CR equation modeling a constant flow stirred tank reactor.'' J. Math. Anal. Appl. 124 (1987), 327-338. 34 D. G. Retzloff, P. C.-H. Chan, C. Chicone, and I. Papick, Maximal multiplicity for sequential bifurcations of a first-order reaction occurring in continuous stirred tank reactors coupled in series.'' SIAM J. Appl. Math. 52 (1992), 1136-1147. 35 D. G. Retzloff, C. Chicone, and G.-H. Hsu, Multiple solutions of a nonlinear boundary value problem with application to chemical reactor dynamics.'' J. Math. Anal. Appl. 185 (1994), 501-519. 36 D. G. Retzloff, P. C.-H. Chan, C. Chicone, D. Offin, and R. Mohamed, Chaotic behavior in the dynamical system of a continuous stirred tank reactor.'' Phys. D 25 (1987), 131-154. 37 C. Chicone, Y. Latushkin, and D.G. Retzloff, Chemical reactor dynamics: stability of steady states.'' Math. Methods Appl. Sci. 19 (1996), 381-400. 38 C. Chicone, B. Mashhoon, and D.G. Retzloff, Gravitational ionization: periodic orbits of binary systems perturbed by gravitational radiation.'' Ann. Inst. H. Poincaré Phys. Théor. 64 (1996), 87-125. 39 C. Chicone, B. Mashhoon, and D.G. Retzloff, On the ionization of a Keplerian binary system by periodic gravitational radiation.'' J. Math. Phys. 37 (1996), 3997-4016. 40 C. Chicone, B. Mashhoon, and D.G. Retzloff, Evolutionary dynamics while trapped in resonance: a Keplerian binary system perturbed by gravitational radiation.'' Classical Quantum Gravity 14 (1997), 1831-1850. 41 C. Chicone, B. Mashhoon, and D.G. Retzloff, Sustained resonance: a binary system perturbed by gravitational radiation.'' J. Phys. A 33 (2000), 513-530. 42 C. Chicone, B. Mashhoon, and D.G. Retzloff, Chaos in the Kepler system.'' Classical Quantum Gravity 16 (1999), 507-527. 43 C. Chicone, B. Mashhoon, D.G. Retzloff, Chaos in the Hill system.'' Helv. Phys. Acta 72 (1999), 123-157. 44 C. Chicone and B. Mashhoon, A gravitational mechanism for the acceleration of ultrarelativistic particles. '' Ann. Phys. (8) 14 (2005), 751-763. 45 C. Chicone and B. Mashhoon, Tidal dynamics of relativistic flows near black holes.'' Ann. Phys. (8) 14 (2005), 290-308. 46 C. Chicone and B. Mashhoon, Ultrarelativistic motion: inertial and tidal effects in Fermi coordinates.'' Classical Quantum Gravity 22 (2005), 195-205. 47 C. Chicone and B. Mashhoon, Significance of $$c/\sqrt 2$$ in relativistic physics.'' Classical Quantum Gravity 21 (2004), L139-L144. 48 C. Chicone and B. Mashhoon, The generalized Jacobi equation.'' Classical Quantum Gravity 19 (2002), 4231-4248. 49 C. Chicone and B. Mashhoon, Acceleration-induced nonlocality: uniqueness of the kernel.'' Phys. Lett. A 298 (2002), 229-235. 50 C. Chicone and B. Mashhoon, Acceleration-induced nonlocality: kinetic memory versus dynamic memory.'' Ann. Phys. (8) 11 (2002), 309-332.
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