Love affairs and linear differential equations

Differential equations are a powerful tool for modeling how systems change over time, but they can be a little hard to get into. Love, on the other hand, is humanity’s perennial topic; some even claim it is all you need. In this blog post — inspired by Strogatz (1988, 2015) — I will introduce linear differential equations as a means to study the types of love affairs two people might find themselves in. Do opposites attract? What happens to a relationship if lovers are out of touch with their own feelings? We will answer these and other questions using two coupled linear differential equations. On our journey, we will use graphical as well as mathematical methods to classify the types of relationships this modeling framework can accommodate. In a follow-up blog post, we will also play around with non-linear terms and add a third wheel to the mix, which can lead to chaos — in the technical sense of the term, of course. Excited? Then let’s get started! Introducing Romeo A lovestruck Romeo sang the streets of serenade Laying everybody low with a love song that he made Finds a streetlight, steps out of the shade Says something like, "You and me, babe, how about it?" Romeo is quite the emotional type. Let $R(t)$ denote his feelings at time point $t$. Following common practice, we will usually write $R$ instead of $R(t)$, making the time dependence implicit. The process which describes how Romeo’s feelings change is rather simple: it depends only on Romeo’s current feelings. We write: [\frac{\mathrm{d}R}{\mathrm{d}t} = aR \enspace ,] which is a linear differential equation. Note that this implicitly encodes how Romeo’s feelings change over time, since when we know $R$ at time point $t$, we can compute the direction and speed with which $R$ will change — the derivative denotes velocity. Our goal, however, is to find an explicit, closed-form expression for Romeo’s feelings at time point $t$. In this particular case, we can do this analytically: [\begin{aligned} \frac{\mathrm{d}R}{\mathrm{d}t} &= aR \[.5em] \frac{1}{aR}\mathrm{d}R &= \mathrm{dt} \[.5em] \frac{1}{a}\int \frac{1}{R}\mathrm{d}R &= \int \mathrm{dt} \[.5em] \frac{1}{a} \left[\text{log} \, R + C \right] &= t \[.5em] \text{log} \, R &= a t - C \[.5em] R &= e^{at - C} \enspace . \end{aligned}] A differential equation describes how something changes; to kickstart the process, we need an initial condition $R_0$. This allows us to find the constant of integration $C$. In particular, assume that $R = R_0$ at $t = 0$, which leads to: [\begin{aligned} R_0 &= e^{-C} \[.5em] \text{log} \, R_0 &= -C \enspace , \end{aligned}] such that: [\begin{aligned} R &= e^{at + \text{log} \, R_0} \[.5em] R &= R_0 e^{at} \enspace . \end{aligned}] The left two panels of the figure below visualize how Romeo’s feelings change over time for $a > 0$ with initial condition $R_0 = 1$ (top) or $R_0 = -1$ (bottom). The right two panels show how his feelings change for $a < 0$ with $R_0 = 100$ (top) or $R_0 = -100$ (bottom). We conclude: Romeo is a simple guy, albeit with an emotion regulation problem. When the object of his affection is such that $a > 0$, his feelings will either grow exponentially towards mad love if he starts out with a positive first impression ($R_0 > 0$), or grow exponentially towards mad hatred if he starts out with a negative first impression ($R_0 < 0$). On the other hand, if $a < 0$, then regardless of his initial feelings, they will decay exponentially towards indifference. For $R_0 = 0$, Romeo’s feelings never change. For any other initial condition, we have uhindered, exponential growth when $a > 0$; it never stops. For any other initial condition and $a < 0$, we crash down to zero very rapidly. Thus $R = 0$ is a fixed point in both cases, which is stable for $a < 0$ but becomes unstable if $a > 0$. We can visualize this in phase space on a line. The phase space is filled with all possible trajectories because each point can serve as the initial condition. In the next section, a wonderful new episode in Romeo’s life begins: he meets Juliet. Introducing Juliet Juliet says, "Hey, it's Romeo, you nearly gave me a heart attack" He's underneath the window, she's singing, "Hey, la, my boyfriend's back You shouldn't come around here singing up at people like that Anyway, what you gonna do about it?" Life becomes more complicated for Romeo now that Juliet is in his life. It is their first real relationship, and they have much to learn. We start simple. Let $J$ denote Juliet’s feelings for Romeo, and let $R$ denote Romeo’s feelings for Juliet. We can extend our single linear differential equation from above to a system of two linear differential equations: [\begin{aligned} \frac{\mathrm{d}R}{\mathrm{d}t} &= aR\[.5em] \frac{\mathrm{d}J}{\mathrm{d}t} &= dJ \enspace . \end{aligned}] Using the results from above, the solutions to the two differential equations are: [\begin{aligned} R(t) &= R_0 e^{at} \[.5em] J(t) &= J_0 e^{dt} \enspace , \end{aligned}] where $R(t)$ and $J(t)$ give the trajectories of love for Romeo and Juliet, respectively, and $J_0$ is Juliet’s initial feeling towards Romeo at $t = 0$. In contrast to the one-dimensional phase diagram from above, we now have a two-dimensional picture which is known as a vector field.

Blog — Andrei A. Klishin, Ph.D.

On the Arnold Diffusion Mechanism in Medium Earth Orbit - Journal of Nonlinear Science

Space debris mitigation guidelines represent the most effective method to preserve the circumterrestrial environment. Among them, end-of-life disposal solutions play a key role. In this regard, effective strategies should be conceived not only on the basis of novel technologies, but also following an advanced theoretical understanding. A growing effort is devoted to exploit natural perturbations to lead the satellites toward an atmospheric reentry, reducing the disposal cost, also if departing from high-altitude regions. In the case of the Medium Earth Orbit region, home of the navigation satellites (like GPS and Galileo), the main driver is the gravitational perturbation due to the Moon, that can increase the eccentricity in the long term. In this way, the pericenter altitude can get into the atmospheric drag domain and the satellite can eventually reenter. In this work, we show how an Arnold diffusion mechanism can trigger the eccentricity growth. Focusing on the case of Galileo, we consider a hierarchy of Hamiltonian models, assuming that the main perturbations on the motion of the spacecraft are the oblateness of the Earth and the gravitational attraction of the Moon. First, the Moon is assumed to lay on the ecliptic plane and periodic orbits and associated stable and unstable invariant manifolds are computed for various energy levels, in the neighborhood of a given resonance. Along each invariant manifold, the eccentricity increases naturally, achieving its maximum at the first intersection between them. This growth is, however, not sufficient to achieve reentry. By moving to a more realistic model, where the inclination of the Moon is taken into account, the problem becomes non-autonomous and the satellite is able to move along different energy levels. Under the ansatz of transversality of the stable and unstable manifolds in the autonomous case, checked numerically, Poincaré–Melnikov techniques are applied to show how the Arnold diffusion can be attained, by constructing a sequence of homoclinic orbits that connect invariant tori at different energy levels on the normally hyperbolic invariant manifold.

On Nested Central Configurations of the 3n Body Problem - Journal of Nonlinear Science

In this work, we consider the existence of (3, n)–crowns in the classical Newtonian 3n–body problem, which are central configurations formed by three groups of n bodies with the same mass within each group, located at the vertices of three concentric regular polygons. We consider the case with dihedral symmetry, called nested (3, n)–crowns, where the vertices of the polygons are aligned. We characterize the set of admissible radii for the polygons for which nested (3, n)–crowns exist. We conclude with numerical evidences that suggest uniqueness for each set of three masses.

Oscillatory Motions, Parabolic Orbits and Collision Orbits in the Planar Circular Restricted Three-Body Problem - Communications in Mathematical Physics

In this paper we consider the planar circular restricted three body problem (PCRTBP), which models the motion of a massless body under the attraction of other two bodies, the primaries, which describe circular orbits around their common center of mass. In a suitable system of coordinates, this is a two degrees of freedom Hamiltonian system. The orbits of this system are either defined for all (future or past) time or eventually go to collision with one of the primaries. For orbits defined for all time, Chazy provided a classification of all possible asymptotic behaviors, usually called final motions. By considering a sufficiently small mass ratio between the primaries, we analyze the interplay between collision orbits and various final motions and construct several types of dynamics. In particular, we show that orbits corresponding to any combination of past and future final motions can be created to pass arbitrarily close to the massive primary. Additionally, we construct arbitrarily large ejection-collision orbits (orbits which experience collision in both past and future times) and periodic orbits that are arbitrarily large and get arbitrarily close to the massive primary. Furthermore, we also establish oscillatory motions in both position and velocity, meaning that as time tends to infinity, the superior limit of the position or velocity is infinity while the inferior limit remains a real number.