if a family court orders you to cantact i.e you must take the children to see their father at such a place at such a time.What happens is you do not follow this?When the father pays no child support.
The view taken by the court is about the child's right to know and see his father. When an order is made it must be kept and you can be penalised for disobeying it. There are circumstances which may alter this, for example the child refusing to go. This will also depend upon his age.
Paying no maintenance is a separate issue entirely. You can apply to the courty for mainteance and arears if the order is breached.