We study the dynamics of the spatial circular restricted three-body problem near class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167278916303001&_mathId=si12.gif&_user=111111111&_pii=S0167278916303001&_rdoc=1&_issn=01672789&md5=55babaf7fffcd66d6fdda62f8c0320fc" title="Click to view the MathML source">L1class="mathContainer hidden">class="mathCode">.
We describe a mechanism for Hamiltonian instability (Arnold diffusion).
We construct orbits that ‘diffuse’ across a 3-sphere near class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167278916303001&_mathId=si12.gif&_user=111111111&_pii=S0167278916303001&_rdoc=1&_issn=01672789&md5=55babaf7fffcd66d6fdda62f8c0320fc" title="Click to view the MathML source">L1class="mathContainer hidden">class="mathCode">.
These orbits change significantly the amplitude of motion relative to the ecliptic.
We provide analytical results and numerical evidence for diffusing orbits.