# Manifold flows and higher-order tangent bundles

Consider the flow on a manifold $$\mathcal{M}$$ defined by $$\dot{x} = f(x)$$ with $$x\in\mathcal{M}$$ and $$f : M\rightarrow TM$$. Associated to this flow I can define the variational dynamics $$\delta \dot{x} = \frac{\partial f(x)}{\partial x}\delta x$$ with $$\delta x \in T_xM$$. Then clearly $$\frac{\partial f(x)}{\partial x} : T_xM\rightarrow TTM$$ or more generally $$\frac{\partial f}{\partial x}\delta x : TM\rightarrow TTM$$. I have two primary questions about these two dynamics.

The "second order" variational dynamics given by $$\delta\delta\dot{x} = \frac{\partial \delta\dot{x}}{\partial \delta x}\delta\delta x = \frac{\partial f(x)}{\partial x}\delta\delta x$$ obeys the same dynamics as the first-order variation. By analogy to above, it is natural to think that $$\frac{\partial \delta\dot{x}}{\partial \delta x} : T_{(x, \delta x)}TM\rightarrow TTTM$$ and more generally $$\frac{\partial \delta\dot{x}}{\partial \delta x}\delta \delta x : TTM\rightarrow TTTM$$. However, from above, we have also said that $$\frac{\partial f(x)}{\partial x} : T_xM\rightarrow TTM$$, and that $$\frac{\partial \delta\dot{x}}{\partial \delta x} = \frac{\partial f}{\partial x}$$. How can I resolve this apparent paradox?

My second question builds on the first. I will call a system exponentially incrementally stable with rate $$\lambda$$ if $$\frac{d}{dt}\langle \delta x, M(x) \delta x\rangle \leq -2\lambda\langle \delta x, M(x)\delta x\rangle$$ where $$(x(t), \delta x(t))$$ is an arbitrary trajectory on $$TM$$ induced by the flow and $$M(x)$$ is a Riemannian metric. Because the second-order variational dynamics is identical to the first-order variational dynamics, I would like to say that the $$\delta\dot{x}$$ system is also exponentially incrementally stable if the $$\dot{x}$$ system is.

However, it is not clear how to think of a trajectory $$(\delta x(t), \delta\delta x(t))$$. Where does this live, and what would be the corresponding metric? One could think of a flow on $$TTM$$ but this would require $$(x, \delta x, \delta\delta x^h, \delta\delta x^v)$$ where $$\delta\delta x^h$$ and $$\delta\delta x^v$$ are the horizontal and vertical projections of an element of $$T_{(x, \delta x)}TM$$ respectively.

I have a less rigorous proof of this in the case where everything is $$\mathbb{R}^n$$. Here, the condition for incremental exponential stability reduces to negative semi-definiteness of $$\left(\frac{\partial f}{\partial x}\right)^TM + M\frac{\partial f}{\partial x} + \dot{M} + 2\lambda M$$. We also have $$x\in\mathbb{R}^n$$, $$\delta x\in\mathbb{R}^n$$, and $$\delta\delta x\in\mathbb{R}^n$$, so that one can, for example, look directly at $$\frac{d}{dt} \delta\delta x^T M(x) \delta\delta x$$. Implicit in this I'm sure is some isomorphism that I hope can be applied in the general case.