I'd like to build a dynamical system where the fixed points depend on the initial condition. Meaning we have :

$\mathbf{v}'(t)=\mathbf{F}(\mathbf{v}(t))=\mathbf{F}[\{v_i(t)\}_{i \in N}]$

Where $\mathbf{v}$ is a vector.

The fixed points are defined as : $\mathbf{v}'(t)=0\Rightarrow \mathbf{v}=\mathbf{v}_f$.

Now we have $\mathbf{v}(t=0)=\mathbf{v^0}$.

I would like to obtain : $\mathbf{v}_f=\mathbf{v}_f(\{v_j^0\}_{j\in N_f})$.

Where $N_f \in N$.

So to sum up, I'd like my fixed points to depend on some initial condition for one of the variables. Is there a generical way to achieve that ? Or even a concrete example or a reference plz ?

I would be even interested in information as : what is the minimal dimension in order to achieve that phenomenon, what kind of tool can be used (I tried to use convolution but I didn't success).

Basically I want to introduce long-time memory in the dynamical system.

NOTE: When I'm sayong that it depends on initial values, I mean in a non trivial way. I mean that if there is memory, not only the the time $t=0$ is taken into account, but also all the other times.

I also think I understand what doesn't work : if you manage to write the dynamical system as $v_i'(t)=\sum_{i,j,k...}( \alpha_{i,j,k...} \Pi_h v_h^{n_h}) $ or other functions of that type, you fail since the fixed point depends only on $ \alpha_{i,j,k...}$. So you need something like an integral maybe, but for example convolution $l(t)=\int f(t')g(t-t') dt'$ may give a good behavior, if the function $g$ is not an exponential. Something that would describe a non reversible phenomena could be also good.


If your right hand side is a function that uses only the function values of $\mathbf{v}$ at time $t$ there is no way the initial conditions can influence the fixed points. The only thing the inital condition will do is to select which fixed point your solution will converge to (if it does at all). The easiest example would be $$ v'(t)=0 $$ where the whole real line is a fixed point and the solution will be $v(t)=v_0$.

If you want to "really" incorporate a memory your right hand side can not be a local function $F(v(t))$ but rather be a function $F[v](t)$. An easy example would be $$ v'(t)=2\cdot v(0)-v(t) $$ where the solution will be $$ v(t)=v_0(2-e^{-t}) $$

Edit: Concerning your note, prominent examples of nonlocal differential equations include integrals such as $$ v'(t)=\int_0^tv(s)\psi(t,s)ds $$


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