Strangely Attractive Relationships

All happy relationships are alike, but every unhappy relationship is unhappy in its own fractally self-similar way

Supposedly it takes an average of seven attempts to leave an abusive relationship.

This is a “fact” that I heard several times during my training as a therapist, never with any attribution. I don’t know what studies have been done, or if those studies were well designed. I can say, though, that the statistic sounds true, based on my clinical and personal experience.

If you look it up, you’ll find plenty of articles explaining why it’s hard to leave, ranging from fear of retribution, lack of resources, uncertainty about the unknown, and lots more reasons. I’m not going to rehash those here.

Instead, let’s zoom out. Why do people stay in abusive relationships? Because change is hard. Changing the relationship from abusive to healthy is hard. But changing the relationship from close to distant is just as hard. So people stay stuck.

Strange Attractors

There’s this idea in math of a strange attractor that’s a perfect analogy for an abusive relationship. I’m a little worried that there’s not very much overlap between people who want to read about a niche branch of math and a niche branch of the theory of relationships, but I’m going to try it anyway.

A dynamical system in math is a function that describes how a point moves around in an n-dimensional space over time.¹ Why did they call this a dynamical system and not a dynamic system? I don’t know. Poincaré was working on the earliest theorems in this area around 1890, so maybe he was humming the first few stanzas of I Am the Very Model of a Modern Major General (1879) to himself as he worked, trying to get dynamical to rhyme with quadratical.

These are some linear vector fields showing some points bopping around in space according to various functions. You don’t need to understand what you’re looking at to get the rest of this article, but you do get to feel smart because you’re reading an article with linear vector fields right now. You’re welcome. The image comes from the wikipedia article about dynamical systems and is shared under the Creative Commons Attribution 2.5 Generic license by user XaosBits.

Okay, so in the dynamical system, you’ve got this point bopping around in space, and there are rules (the function) governing which way it bops. Maybe the point represents how a relationship is going right now. How much time is the couple spending together, how much are they yelling at each other, how much time are they spending watching TV and snuggling, how truthful they are being with each other, and a hundred other variables. If they yell at each other a lot on Tuesday, they spend less time watching TV and snuggling on Wednesday. That’s part of the function that says how that point bops around.

Some dynamical systems have attractors. An attractor is sort of like a path that the point keeps carving out in space over and over again as time goes on.² Some systems have a predictable loop or cycle they keep making. Some systems find a stable spot and just stick there forever once they find it (that’s called a fixed point).³ Anytime the point finds its way into a repeating cycle of any kind, that’s an attractor.⁴

In this system, there’s a cycle attractor, meaning points in this system really like to go in a specific sort of oval shape. If you start too far to the left, like the outer curve does, you end up squishing inward. And if you start too far to the right, like the inner curve does, you end up expanding outward. Everybody ends up in the middle as they go along. This image comes from the wikipedia article about limit cycles and it’s in the public domain.

The cycle in an abusive relationship goes like this:

• building tension

• abusive incident

• reconciliation

• calm / honeymoon period

Again, there are a lot of articles out there about this and I’m not going to rehash them here. For our purposes, the important thing is that each phase of the cycle evokes the next. If lashing out is your go-to move when tensions are high, then a phase of building tension is going to lead inexorably to an abusive incident. The other phases of the cycle are just as predictable; each one is the next step in a dance that couples repeat over and over.

Finally, a strange attractor is an attractor with a really complicated fractal structure. I’m not sure abusive relationships really qualify in a mathematical sense - it would depend how we operationalized and measured the variables describing the relationship, I guess - but I liked the phrase because it captures the confusion we all experience when we look at an abusive relationship and say, “WTF? Why do they just keep doing that?”

This is a rad-looking strange attractor. It was made by user Wiscdragon and shared under the Creative Commons Attribution 4.0 International license.

First-Order vs. Second-Order Change

The thing about dynamical systems with attractors is, it doesn’t really matter what’s going on right now in the system. As time goes on and the point bops around in space, it always finds its way back to the attractor, that same old loop. You can try yelling less and snuggling more, or spending more or less time together, or any other sort of surface-level change to your behavior, but if the underlying function controlling your dynamical system hasn’t changed, your point is going to end up bopping around in the same old pattern eventually. As long as tension leads to abuse, you’re eventually going to find yourselves tense and then there’s going to be an abusive incident.

The only way out of the attractor is to change the function itself.

In family systems theory, changing what you’re doing at the moment while keeping the same function intact is called first-order change, and changing the function itself is called second-order change.

First-order change: “Through sheer force of will I will refrain from hitting my partner today.”

Second-order change: “I will learn to notice my discomfort when my partner and I disagree and accept that disagreement is expected; rather than getting angry and lashing out about it I will figure out what my underlying fear or sadness is about and I will discuss that calmly with my partner.”

You can see why one of those is harder to bring about than the other, but also why second-order change sticks and first-order doesn’t.

Leaving for Good is Second-Order Change

Looking in from the outside, it seems so obvious. If your partner is abusing you, you should just leave. It doesn’t matter if it’s going to be expensive or scary. All the best possible futures happen outside that abusive relationship, so it’s better to go. Right?

For the abuser, it takes a tremendous amount of work to achieve second-order change, to really get it at a fundamental level, to develop all the layers of awareness and skill it takes to do something other than lashing out. Turning an abusive relationship into a healthy one is a hell of a lot of work.

For the one being abused, it actually takes just as much work to achieve second-order change, to really get why leaving is better at a fundamental level, to develop all the insight and strength it takes to chart a different path. Extricating yourself from all the ways you are intertwined with your abuser, all the ways you depend on them and they depend on you, is just as hard.

When you’re the one who maybe should leave, it’s hard to decide to begin your own process of change. It doesn’t look appealing, it doesn’t look right. It seems like there must be a better choice, something within the relationship, some way to nudge or shift the cycle to trim the worst parts off of it and keep the rest. It seems cold and cruel and abrupt and unjust to leave.

First-order change: “I will spend the night at a friend’s house.”

Second-order change: “I now truly believe that I am better off apart from my abuser, that I have or can reconstruct everything I need to be okay in this world, and that my freedom is worth whatever pain they experienced because I left.”

That’s hard.

A Note for Supporters

When you’re on the outside looking in, whether you’re a friend, a family member, a therapist, or some other kind of supporter, it’s maddening watching people you care go around in circles in their strangely attractive relationship. You can see how broken and bad it is. You want to stick your hands in and start rearranging the parts - separate the people, divide up the assets, and give them each a friendly nudge on their new paths. It’s so clear what should happen. There’s sometimes a disrespect for or unawareness of the psychological obstacles to change on either participant’s part.

So for you, it takes patience. It takes acceptance that you are not in control of this situation and that you would not want to violate your loved one’s autonomy and decide for them, against their will - and that even if you could or would do that, you would simply be imposing first-order change, because there’s no such thing as imposing second-order change.

However impossible you think it would be to get the abuser to change, know that the task of shifting the thoughts of the other person is just as hard.

Rest. Take care of yourself. This is going to take awhile.

Palate-Cleansing Fractal

I am not going to write about dynamical systems without leaving you with a gorgeous public domain fractal picture of unknown origin. Let’s party like math nerds in 1993.

1. Oh, hey, I’ve already said wrong things! What I’ve said here describes only a subset of dynamical systems. They don’t have to be over n-dimensional space. 10 points to Ravenclaw if you said “Well, ackshually,” when you read this. Otherwise don’t worry about it.

2. This is also false! This also describes a subset of all the things an attractor can be. An attractor can be any set, and there don’t have to be cycles or repeats in which points get visited. However, my description here covers a lot of what people mean/visualize when they talk about attractors.

3. Oh jeez this is getting embarrassing. “Stable” has a mathematical meaning as well as a layperson meaning, and the mathematical meaning is more precise. A fixed point need not be stable in the mathematical sense. All I meant here is that once you get to the fixed point, you stay there.

4. I think this one is true but can’t be turned inside out; stable cycles are attractors but not all attractors are cycles. (All of these footnotes are Alex Altair’s fault. Thanks, Alex.)